Copyright ©2002 RSA Security Inc. License to copy this document is granted provided that it is identified
as “RSA Security Inc. Public-Key Cryptography Standards (PKCS)” in all material mentioning or
referencing this document.
003-100000-210-001-000
PKCS #1 v2.1: RSA Cryptography Standard
RSA Laboratories
June 14, 2002
Table of Contents
TABLE OF CONTENTS ..............................................................................................................................1
1 INTRODUCTION................................................................................................................................2
2 NOTATION..........................................................................................................................................3
3 KEY TYPES .........................................................................................................................................6
3.1 RSA PUBLIC KEY................................................................................................................................6
3.2 RSA PRIVATE KEY..............................................................................................................................6
4 DATA CONVERSION PRIMITIVES ...............................................................................................8
4.1 I2OSP................................................................................................................................................8
4.2 OS2IP................................................................................................................................................9
5 CRYPTOGRAPHIC PRIMITIVES...................................................................................................9
5.1 ENCRYPTION AND DECRYPTION PRIMITIVES .....................................................................................10
5.1.1 RSAEP ...................................................................................................................................10
5.1.2 RSADP...................................................................................................................................10
5.2 SIGNATURE AND VERIFICATION PRIMITIVES......................................................................................12
5.2.1 RSASP1..................................................................................................................................12
5.2.2 RSAVP1 .................................................................................................................................13
6 OVERVIEW OF SCHEMES ............................................................................................................14
7 ENCRYPTION SCHEMES ..............................................................................................................14
7.1 RSAES-OAEP.................................................................................................................................15
7.1.1 Encryption operation.............................................................................................................17
7.1.2 Decryption operation.............................................................................................................20
7.2 RSAES-PKCS1-V1_5......................................................................................................................22
7.2.1 Encryption operation.............................................................................................................23
7.2.2 Decryption operation.............................................................................................................24
8 SIGNATURE SCHEMES WITH APPENDIX................................................................................25
8.1 RSASSA-PSS..................................................................................................................................26
8.1.1 Signature generation operation.............................................................................................27
8.1.2 Signature verification operation............................................................................................28
8.2 RSASSA-PKCS1-V1_5 ...................................................................................................................29
8.2.1 Signature generation operation.............................................................................................30
8.2.2 Signature verification operation............................................................................................31
9 ENCODING METHODS FOR SIGNATURES WITH APPENDIX.............................................32
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9.1 EMSA-PSS......................................................................................................................................33
9.1.1 Encoding operation ...............................................................................................................35
9.1.2 Verification operation............................................................................................................36
9.2 EMSA-PKCS1-V1_5.......................................................................................................................37
A. ASN.1 SYNTAX .................................................................................................................................39
A.1 RSA KEY REPRESENTATION .............................................................................................................39
A.1.1 RSA public key syntax............................................................................................................39
A.1.2 RSA private key syntax...........................................................................................................39
A.2 SCHEME IDENTIFICATION..................................................................................................................41
A.2.1 RSAES-OAEP ........................................................................................................................41
A.2.2 RSAES-PKCS1-v1_5..............................................................................................................43
A.2.3 RSASSA-PSS ..........................................................................................................................43
A.2.4 RSASSA-PKCS1-v1_5............................................................................................................44
B. SUPPORTING TECHNIQUES ........................................................................................................46
B.1 HASH FUNCTIONS .............................................................................................................................46
B.2 MASK GENERATION FUNCTIONS .......................................................................................................47
B.2.1 MGF1.....................................................................................................................................48
C. ASN.1 MODULE................................................................................................................................49
D. INTELLECTUAL PROPERTY CONSIDERATIONS ..................................................................55
E. REVISION HISTORY.......................................................................................................................56
F. REFERENCES...................................................................................................................................57
G. ABOUT PKCS....................................................................................................................................61
1 Introduction
This document provides recommendations for the implementation of public-key
cryptography based on the RSA algorithm [42], covering the following aspects:
• Cryptographic primitives
• Encryption schemes
• Signature schemes with appendix
• ASN.1 syntax for representing keys and for identifying the schemes
The recommendations are intended for general application within computer and
communications systems, and as such include a fair amount of flexibility. It is expected
that application standards based on these specifications may include additional
constraints. The recommendations are intended to be compatible with the standard IEEE-
1363-2000 [26] and draft standards currently being developed by the ANSI X9F1 [1] and
IEEE P1363 [27] working groups.
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This document supersedes PKCS #1 version 2.0 [44] but includes compatible techniques.
The organization of this document is as follows:
• Section 1 is an introduction.
• Section 2 defines some notation used in this document.
• Section 3 defines the RSA public and private key types.
• Sections 4 and 5 define several primitives, or basic mathematical operations. Data
conversion primitives are in Section 4, and cryptographic primitives (encryptiondecryption,
signature-verification) are in Section 5.
• Sections 6, 7, and 8 deal with the encryption and signature schemes in this document.
Section 6 gives an overview. Along with the methods found in PKCS #1 v1.5, Section
7 defines an OAEP-based [3] encryption scheme and Section 8 defines a PSS-based
[4][5] signature scheme with appendix.
• Section 9 defines the encoding methods for the signature schemes in Section 8.
• Appendix A defines the ASN.1 syntax for the keys defined in Section 3 and the
schemes in Sections 7 and 8.
• Appendix B defines the hash functions and the mask generation function used in this
document, including ASN.1 syntax for the techniques.
• Appendix C gives an ASN.1 module.
• Appendices D, E, F and G cover intellectual property issues, outline the revision
history of PKCS #1, give references to other publications and standards, and provide
general information about the Public-Key Cryptography Standards.
2 Notation
c ciphertext representative, an integer between 0 and n – 1
C ciphertext, an octet string
d RSA private exponent
di additional factor ri’s CRT exponent, a positive integer such that
e · di ≡ 1 (mod (ri – 1)), i = 3, …, u
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dP p’s CRT exponent, a positive integer such that
e · dP ≡ 1 (mod (p – 1))
dQ q’s CRT exponent, a positive integer such that
e · dQ ≡ 1 (mod (q – 1))
e RSA public exponent
EM encoded message, an octet string
emBits (intended) length in bits of an encoded message EM
emLen (intended) length in octets of an encoded message EM
GCD (. , .) greatest common divisor of two nonnegative integers
Hash hash function
hLen output length in octets of hash function Hash
k length in octets of the RSA modulus n
K RSA private key
L optional RSAES-OAEP label, an octet string
LCM (. , … , .) least common multiple of a list of nonnegative integers
m message representative, an integer between 0 and n – 1
M message, an octet string
mask MGF output, an octet string
maskLen (intended) length of the octet string mask
MGF mask generation function
mgfSeed seed from which mask is generated, an octet string
mLen length in octets of a message M
n RSA modulus, n = r1 · r2 ⋅ … · ru , u ≥ 2
(n, e) RSA public key
p, q first two prime factors of the RSA modulus n
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qInv CRT coefficient, a positive integer less than p such that
q · qInv ≡ 1 (mod p)
ri prime factors of the RSA modulus n, including r1 = p, r2 = q, and
additional factors if any
s signature representative, an integer between 0 and n – 1
S signature, an octet string
sLen length in octets of the EMSA-PSS salt
ti additional prime factor ri’s CRT coefficient, a positive integer less
than ri such that
r1 · r2 · … · ri–1 · ti ≡ 1 (mod ri) , i = 3, …, u
u number of prime factors of the RSA modulus, u ≥ 2
x a nonnegative integer
X an octet string corresponding to x
xLen (intended) length of the octet string X
0x indicator of hexadecimal representation of an octet or an octet
string; “0x48” denotes the octet with hexadecimal value 48;
“(0x)48 09 0e” denotes the string of three consecutive octets with
hexadecimal value 48, 09, and 0e, respectively
λ(n) LCM (r1 – 1, r2 – 1, … , ru – 1)
⊕ bit-wise exclusive-or of two octet strings
 .
 ceiling function; x is the smallest integer larger than or equal to
the real number x
|| concatenation operator
≡ congruence symbol; a ≡ b (mod n) means that the integer n divides
the integer a – b
Note. The CRT can be applied in a non-recursive as well as a recursive way. In this document a recursive
approach following Garner’s algorithm [22] is used. See also Note 1 in Section 3.2.
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3 Key types
Two key types are employed in the primitives and schemes defined in this document: RSA
public key and RSA private key. Together, an RSA public key and an RSA private key
form an RSA key pair.
This specification supports so-called “multi-prime” RSA where the modulus may have
more than two prime factors. The benefit of multi-prime RSA is lower computational cost
for the decryption and signature primitives, provided that the CRT (Chinese Remainder
Theorem) is used. Better performance can be achieved on single processor platforms, but
to a greater extent on multiprocessor platforms, where the modular exponentiations
involved can be done in parallel.
For a discussion on how multi-prime affects the security of the RSA cryptosystem, the
reader is referred to [49].
3.1 RSA public key
For the purposes of this document, an RSA public key consists of two components:
n the RSA modulus, a positive integer
e the RSA public exponent, a positive integer
In a valid RSA public key, the RSA modulus n is a product of u distinct odd primes ri, i =
1, 2, …, u, where u ≥ 2, and the RSA public exponent e is an integer between 3 and n – 1
satisfying GCD (e, λ(n)) = 1, where λ(n) = LCM (r1 – 1, …, ru – 1). By convention, the
first two primes r1 and r2 may also be denoted p and q respectively.
A recommended syntax for interchanging RSA public keys between implementations is
given in Appendix A.1.1; an implementation’s internal representation may differ.
3.2 RSA private key
For the purposes of this document, an RSA private key may have either of two
representations.
1. The first representation consists of the pair (n, d), where the components have the
following meanings:
n the RSA modulus, a positive integer
d the RSA private exponent, a positive integer
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2. The second representation consists of a quintuple (p, q, dP, dQ, qInv) and a (possibly
empty) sequence of triplets (ri, di, ti), i = 3, …, u, one for each prime not in the quintuple,
where the components have the following meanings:
p the first factor, a positive integer
q the second factor, a positive integer
dP the first factor’s CRT exponent, a positive integer
dQ the second factor’s CRT exponent, a positive integer
qInv the (first) CRT coefficient, a positive integer
ri the ith
factor, a positive integer
di the ith
factor’s CRT exponent, a positive integer
ti the ith
factor’s CRT coefficient, a positive integer
In a valid RSA private key with the first representation, the RSA modulus n is the same as
in the corresponding RSA public key and is the product of u distinct odd primes ri, i = 1,
2, …, u, where u ≥ 2. The RSA private exponent d is a positive integer less than n
satisfying
e · d ≡ 1 (mod λ(n)) ,
where e is the corresponding RSA public exponent and λ(n) is defined as in Section 3.1.
In a valid RSA private key with the second representation, the two factors p and q are the
first two prime factors of the RSA modulus n (i.e., r1 and r2), the CRT exponents dP and
dQ are positive integers less than p and q respectively satisfying
e · dP ≡ 1 (mod (p – 1))
e · dQ ≡ 1 (mod (q – 1)) ,
and the CRT coefficient qInv is a positive integer less than p satisfying
q · qInv ≡ 1 (mod p) .
If u > 2, the representation will include one or more triplets (ri, di, ti), i = 3, …, u. The
factors ri are the additional prime factors of the RSA modulus n. Each CRT exponent di (i
= 3, …, u) satisfies
e · di ≡ 1 (mod (ri – 1)) .
Each CRT coefficient ti (i = 3, …, u) is a positive integer less than ri satisfying
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Ri · ti ≡ 1 (mod ri) ,
where Ri = r1 · r2 · … · ri–1.
A recommended syntax for interchanging RSA private keys between implementations,
which includes components from both representations, is given in Appendix A.1.2; an
implementation’s internal representation may differ.
Notes.
1. The definition of the CRT coefficients here and the formulas that use them in the primitives in Section
5 generally follow Garner’s algorithm [22] (see also Algorithm 14.71 in [37]). However, for
compatibility with the representations of RSA private keys in PKCS #1 v2.0 and previous versions, the
roles of p and q are reversed compared to the rest of the primes. Thus, the first CRT coefficient, qInv, is
defined as the inverse of q mod p, rather than as the inverse of R1 mod r2, i.e., of p mod q.
2. Quisquater and Couvreur [40] observed the benefit of applying the Chinese Remainder Theorem to
RSA operations.
4 Data conversion primitives
Two data conversion primitives are employed in the schemes defined in this document:
• I2OSP – Integer-to-Octet-String primitive
• OS2IP – Octet-String-to-Integer primitive
For the purposes of this document, and consistent with ASN.1 syntax, an octet string is an
ordered sequence of octets (eight-bit bytes). The sequence is indexed from first
(conventionally, leftmost) to last (rightmost). For purposes of conversion to and from
integers, the first octet is considered the most significant in the following conversion
primitives.
4.1 I2OSP
I2OSP converts a nonnegative integer to an octet string of a specified length.
I2OSP (x, xLen)
Input: x nonnegative integer to be converted
xLen intended length of the resulting octet string
Output: X corresponding octet string of length xLen
Error: “integer too large”
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Steps:
1. If x ≥ 256xLen
, output “integer too large” and stop.
2. Write the integer x in its unique xLen-digit representation in base 256:
x = xxLen–1 256xLen–1
+ xxLen–2 256xLen–2
+ … + x1 256 + x0 ,
where 0 ≤ xi < 256 (note that one or more leading digits will be zero if x is less
than 256xLen–1
).
3. Let the octet Xi have the integer value xxLen–i for 1 ≤ i ≤ xLen. Output the octet
string
X = X1 X2 … XxLen.
4.2 OS2IP
OS2IP converts an octet string to a nonnegative integer.
OS2IP (X)
Input: X octet string to be converted
Output: x corresponding nonnegative integer
Steps:
1. Let X1 X2 … XxLen be the octets of X from first to last, and let xxLen–i be the integer
value of the octet Xi for 1 ≤ i ≤ xLen.
2. Let x = xxLen–1 256xLen–1
+ xxLen–2 256xLen–2
+ … + x1 256 + x0.
3. Output x.
5 Cryptographic primitives
Cryptographic primitives are basic mathematical operations on which cryptographic
schemes can be built. They are intended for implementation in hardware or as software
modules, and are not intended to provide security apart from a scheme.
Four types of primitive are specified in this document, organized in pairs: encryption and
decryption; and signature and verification.
The specifications of the primitives assume that certain conditions are met by the inputs,
in particular that RSA public and private keys are valid.
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5.1 Encryption and decryption primitives
An encryption primitive produces a ciphertext representative from a message
representative under the control of a public key, and a decryption primitive recovers the
message representative from the ciphertext representative under the control of the
corresponding private key.
One pair of encryption and decryption primitives is employed in the encryption schemes
defined in this document and is specified here: RSAEP/RSADP. RSAEP and RSADP
involve the same mathematical operation, with different keys as input.
The primitives defined here are the same as IFEP-RSA/IFDP-RSA in IEEE Std 1363-
2000 [26] (except that support for multi-prime RSA has been added) and are compatible
with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation.
5.1.1 RSAEP
RSAEP ((n, e), m)
Input: (n, e) RSA public key
m message representative, an integer between 0 and n – 1
Output: c ciphertext representative, an integer between 0 and n – 1
Error: “message representative out of range”
Assumption: RSA public key (n, e) is valid
Steps:
1. If the message representative m is not between 0 and n – 1, output “message
representative out of range” and stop.
2. Let c = me
mod n.
3. Output c.
5.1.2 RSADP
RSADP (K, c)
Input: K RSA private key, where K has one of the following forms:
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 11
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 a pair (n, d)
 a quintuple (p, q, dP, dQ, qInv) and a possibly empty sequence
of triplets (ri, di, ti), i = 3, …, u
c ciphertext representative, an integer between 0 and n – 1
Output: m message representative, an integer between 0 and n – 1
Error: “ciphertext representative out of range”
Assumption: RSA private key K is valid
Steps:
1. If the ciphertext representative c is not between 0 and n – 1, output “ciphertext
representative out of range” and stop.
2. The message representative m is computed as follows.
a. If the first form (n, d) of K is used, let m = cd
mod n.
b. If the second form (p, q, dP, dQ, qInv) and (ri, di, ti) of K is used, proceed
as follows:
i. Let m1 = cdP
mod p and m2 = cdQ
mod q.
ii. If u > 2, let mi = cdi mod ri, i = 3, …, u.
iii. Let h = (m1 – m2) · qInv mod p.
iv. Let m = m2 + q · h.
v. If u > 2, let R = r1 and for i = 3 to u do
1. Let R = R · ri–1.
2. Let h = (mi – m) · ti (mod ri).
3. Let m = m + R · h.
3. Output m.
Note. Step 2.a can be rewritten as a single loop, provided that one reverses the order of p and q. For
consistency with PKCS #1 v2.0, however, the first two primes p and q are treated separately from the
additional primes.
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5.2 Signature and verification primitives
A signature primitive produces a signature representative from a message representative
under the control of a private key, and a verification primitive recovers the message
representative from the signature representative under the control of the corresponding
public key. One pair of signature and verification primitives is employed in the signature
schemes defined in this document and is specified here: RSASP1/RSAVP1.
The primitives defined here are the same as IFSP-RSA1/IFVP-RSA1 in IEEE 1363-2000
[26] (except that support for multi-prime RSA has been added) and are compatible with
PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation, as in the encryption
and decryption primitives of Section 5.1. RSASP1 and RSAVP1 are the same as RSADP
and RSAEP except for the names of their input and output arguments; they are
distinguished as they are intended for different purposes.
5.2.1 RSASP1
RSASP1 (K, m)
Input: K RSA private key, where K has one of the following forms:
 a pair (n, d)
 a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
sequence of triplets (ri, di, ti), i = 3, …, u
m message representative, an integer between 0 and n – 1
Output: s signature representative, an integer between 0 and n – 1
Error: “message representative out of range”
Assumption: RSA private key K is valid
Steps:
1. If the message representative m is not between 0 and n – 1, output “message
representative out of range” and stop.
2. The signature representative s is computed as follows.
a. If the first form (n, d) of K is used, let s = md
mod n.
b. If the second form (p, q, dP, dQ, qInv) and (ri, di, ti) of K is used, proceed
as follows:
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i. Let s1 = mdP
mod p and s2 = mdQ
mod q.
ii. If u > 2, let si = mdi mod ri, i = 3, …, u.
iii. Let h = (s1 – s2) · qInv mod p.
iv. Let s = s2 + q · h.
v. If u > 2, let R = r1 and for i = 3 to u do
1. Let R = R · ri–1.
2. Let h = (si – s) · ti (mod ri).
3. Let s = s + R · h.
3. Output s.
Note. Step 2.a can be rewritten as a single loop, provided that one reverses the order of p and q. For
consistency with PKCS #1 v2.0, however, the first two primes p and q are treated separately from the
additional primes.
5.2.2 RSAVP1
RSAVP1 ((n, e), s)
Input: (n, e) RSA public key
s signature representative, an integer between 0 and n – 1
Output: m message representative, an integer between 0 and n – 1
Error: “signature representative out of range”
Assumption: RSA public key (n, e) is valid
Steps:
1. If the signature representative s is not between 0 and n – 1, output “signature
representative out of range” and stop.
2. Let m = se
mod n.
3. Output m.
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6 Overview of schemes
A scheme combines cryptographic primitives and other techniques to achieve a particular
security goal. Two types of scheme are specified in this document: encryption schemes
and signature schemes with appendix.
The schemes specified in this document are limited in scope in that their operations
consist only of steps to process data with an RSA public or private key, and do not
include steps for obtaining or validating the key. Thus, in addition to the scheme
operations, an application will typically include key management operations by which
parties may select RSA public and private keys for a scheme operation. The specific
additional operations and other details are outside the scope of this document.
As was the case for the cryptographic primitives (Section 5), the specifications of scheme
operations assume that certain conditions are met by the inputs, in particular that RSA
public and private keys are valid. The behavior of an implementation is thus unspecified
when a key is invalid. The impact of such unspecified behavior depends on the
application. Possible means of addressing key validation include explicit key validation
by the application; key validation within the public-key infrastructure; and assignment of
liability for operations performed with an invalid key to the party who generated the key.
A generally good cryptographic practice is to employ a given RSA key pair in only one
scheme. This avoids the risk that vulnerability in one scheme may compromise the
security of the other, and may be essential to maintain provable security. While RSAESPKCS1-v1_5
(Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally
been employed together without any known bad interactions (indeed, this is the model
introduced by PKCS #1 v1.5), such a combined use of an RSA key pair is not
recommended for new applications.
To illustrate the risks related to the employment of an RSA key pair in more than one
scheme, suppose an RSA key pair is employed in both RSAES-OAEP (Section 7.1) and
RSAES-PKCS1-v1_5. Although RSAES-OAEP by itself would resist attack, an opponent
might be able to exploit a weakness in the implementation of RSAES-PKCS1-v1_5 to
recover messages encrypted with either scheme. As another example, suppose an RSA
key pair is employed in both RSASSA-PSS (Section 8.1) and RSASSA-PKCS1-v1_5.
Then the security proof for RSASSA-PSS would no longer be sufficient since the proof
does not account for the possibility that signatures might be generated with a second
scheme. Similar considerations may apply if an RSA key pair is employed in one of the
schemes defined here and in a variant defined elsewhere.
7 Encryption schemes
For the purposes of this document, an encryption scheme consists of an encryption
operation and a decryption operation, where the encryption operation produces a
ciphertext from a message with a recipient’s RSA public key, and the decryption
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operation recovers the message from the ciphertext with the recipient’s corresponding
RSA private key.
An encryption scheme can be employed in a variety of applications. A typical application
is a key establishment protocol, where the message contains key material to be delivered
confidentially from one party to another. For instance, PKCS #7 [45] employs such a
protocol to deliver a content-encryption key from a sender to a recipient; the encryption
schemes defined here would be suitable key-encryption algorithms in that context.
Two encryption schemes are specified in this document: RSAES-OAEP and RSAESPKCS1-v1_5.
RSAES-OAEP is recommended for new applications; RSAES-PKCS1v1_5
is included only for compatibility with existing applications, and is not
recommended for new applications.
The encryption schemes given here follow a general model similar to that employed in
IEEE Std 1363-2000 [26], combining encryption and decryption primitives with an
encoding method for encryption. The encryption operations apply a message encoding
operation to a message to produce an encoded message, which is then converted to an
integer message representative. An encryption primitive is applied to the message
representative to produce the ciphertext. Reversing this, the decryption operations apply a
decryption primitive to the ciphertext to recover a message representative, which is then
converted to an octet string encoded message. A message decoding operation is applied to
the encoded message to recover the message and verify the correctness of the decryption.
To avoid implementation weaknesses related to the way errors are handled within the
decoding operation (see [6] and [36]), the encoding and decoding operations for RSAESOAEP
and RSAES-PKCS1-v1_5 are embedded in the specifications of the respective
encryption schemes rather than defined in separate specifications. Both encryption
schemes are compatible with the corresponding schemes in PKCS #1 v2.0.
7.1 RSAES-OAEP
RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1 and 5.1.2)
with the EME-OAEP encoding method (step 1.b in Section 7.1.1 and step 3 in Section
7.1.2). EME-OAEP is based on Bellare and Rogaway’s Optimal Asymmetric Encryption
scheme [3]. (OAEP stands for “Optimal Asymmetric Encryption Padding.”). It is
compatible with the IFES scheme defined in IEEE Std 1363-2000 [26], where the
encryption and decryption primitives are IFEP-RSA and IFDP-RSA and the message
encoding method is EME-OAEP. RSAES-OAEP can operate on messages of length up to
k – 2hLen – 2 octets, where hLen is the length of the output from the underlying hash
function and k is the length in octets of the recipient’s RSA modulus.
Assuming that computing eth
roots modulo n is infeasible and the mask generation
function in RSAES-OAEP has appropriate properties, RSAES-OAEP is semantically
secure against adaptive chosen-ciphertext attacks. This assurance is provable in the sense
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that the difficulty of breaking RSAES-OAEP can be directly related to the difficulty of
inverting the RSA function, provided that the mask generation function is viewed as a
black box or random oracle; see [21] and the note below for further discussion.
Both the encryption and the decryption operations of RSAES-OAEP take the value of a
label L as input. In this version of PKCS #1, L is the empty string; other uses
of the label are outside the scope of this document. See Appendix A.2.1 for the relevant
ASN.1 syntax.
RSAES-OAEP is parameterized by the choice of hash function and mask generation
function. This choice should be fixed for a given RSA key. Suggested hash and mask
generation functions are given in Appendix B.
Note. Recent results have helpfully clarified the security properties of the OAEP encoding method [3]
(roughly the procedure described in step 1.b in Section 7.1.1). The background is as follows. In 1994,
Bellare and Rogaway [3] introduced a security concept that they denoted plaintext awareness (PA94). They
proved that if a deterministic public-key encryption primitive (e.g., RSAEP) is hard to invert without the
private key, then the corresponding OAEP-based encryption scheme is plaintext-aware (in the random
oracle model), meaning roughly that an adversary cannot produce a valid ciphertext without actually
“knowing” the underlying plaintext. Plaintext awareness of an encryption scheme is closely related to the
resistance of the scheme against chosen-ciphertext attacks. In such attacks, an adversary is given the
opportunity to send queries to an oracle simulating the decryption primitive. Using the results of these
queries, the adversary attempts to decrypt a challenge ciphertext.
However, there are two flavors of chosen-ciphertext attacks, and PA94 implies security against only one of
them. The difference relies on what the adversary is allowed to do after she is given the challenge
ciphertext. The indifferent attack scenario (denoted CCA1) does not admit any queries to the decryption
oracle after the adversary is given the challenge ciphertext, whereas the adaptive scenario (denoted CCA2)
does (except that the decryption oracle refuses to decrypt the challenge ciphertext once it is published). In
1998, Bellare and Rogaway, together with Desai and Pointcheval [2], came up with a new, stronger notion
of plaintext awareness (PA98) that does imply security against CCA2.
To summarize, there have been two potential sources for misconception: that PA94 and PA98 are equivalent
concepts; or that CCA1 and CCA2 are equivalent concepts. Either assumption leads to the conclusion that
the Bellare-Rogaway paper implies security of OAEP against CCA2, which it does not.
1
OAEP has never
been proven secure against CCA2; in fact, Victor Shoup [48] has demonstrated that such a proof does not
exist in the general case. Put briefly, Shoup showed that an adversary in the CCA2 scenario who knows how
to partially invert the encryption primitive but does not know how to invert it completely may well be able
to break the scheme. For example, one may imagine an attacker who is able to break RSAES-OAEP if she
knows how to recover all but the first 20 bytes of a random integer encrypted with RSAEP. Such an attacker
does not need to be able to fully invert RSAEP, because she does not use the first 20 octets in her attack.
Still, RSAES-OAEP is secure against CCA2, which was proved by Fujisaki, Okamoto, Pointcheval, and
Stern [21] shortly after the announcement of Shoup’s result. Using clever lattice reduction techniques, they
managed to show how to invert RSAEP completely given a sufficiently large part of the pre-image. This
1
It might be fair to mention that PKCS #1 v2.0 cites [3] and claims that “a chosen ciphertext attack is
ineffective against a plaintext-aware encryption scheme such as RSAES-OAEP” without specifying the kind
of plaintext awareness or chosen ciphertext attack considered.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 17
Copyright © 2002 RSA Security Inc.
observation, combined with a proof that OAEP is secure against CCA2 if the underlying encryption
primitive is hard to partially invert, fills the gap between what Bellare and Rogaway proved about RSAESOAEP
and what some may have believed that they proved. Somewhat paradoxically, we are hence saved by
an ostensible weakness in RSAEP (i.e., the whole inverse can be deduced from parts of it).
Unfortunately however, the security reduction is not efficient for concrete parameters. While the proof
successfully relates an adversary A against the CCA2 security of RSAES-OAEP to an algorithm I inverting
RSA, the probability of success for I is only approximately ε
2
/ 2
18
, where ε is the probability of success for
A.
1
In addition, the running time for I is approximately t
2
, where t is the running time of the adversary. The
consequence is that we cannot exclude the possibility that attacking RSAES-OAEP is considerably easier
than inverting RSA for concrete parameters. Still, the existence of a security proof provides some assurance
that the RSAES-OAEP construction is sounder than ad hoc constructions such as RSAES-PKCS1-v1_5.
Hybrid encryption schemes based on the RSA-KEM key encapsulation paradigm offer tight proofs of
security directly applicable to concrete parameters; see [30] for discussion. Future versions of PKCS #1 may
specify schemes based on this paradigm.
7.1.1 Encryption operation
RSAES-OAEP-ENCRYPT ((n, e), M, L)
Options: Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
Input: (n, e) recipient’s RSA public key (k denotes the length in octets of the
RSA modulus n)
M message to be encrypted, an octet string of length mLen, where
mLen ≤ k – 2hLen – 2
L optional label to be associated with the message; the default value
for L, if L is not provided, is the empty string
Output: C ciphertext, an octet string of length k
Errors: “message too long”; “label too long”
Assumption: RSA public key (n, e) is valid
Steps:
1
In [21] the probability of success for the inverter was ε2
/ 4. The additional factor 1 / 216
is due to the eight
fixed zero bits at the beginning of the encoded message EM, which are not present in the variant of OAEP
considered in [21] (I must apply A twice to invert RSA, and each application corresponds to a factor 1 / 28
).
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 18
Copyright © 2002 RSA Security Inc.
1. Length checking:
a. If the length of L is greater than the input limitation for the hash function
(261
– 1 octets for SHA-1), output “label too long” and stop.
b. If mLen > k – 2hLen – 2, output “message too long” and stop.
2. EME-OAEP encoding (see Figure 1 below):
a. If the label L is not provided, let L be the empty string. Let lHash = Hash
(L), an octet string of length hLen (see the note below).
b. Generate an octet string PS consisting of k – mLen – 2hLen – 2 zero octets.
The length of PS may be zero.
c. Concatenate lHash, PS, a single octet with hexadecimal value 0x01, and
the message M to form a data block DB of length k – hLen – 1 octets as
DB = lHash || PS || 0x01 || M .
d. Generate a random octet string seed of length hLen.
e. Let dbMask = MGF (seed, k – hLen – 1)
f. Let maskedDB = DB ⊕ dbMask.
g. Let seedMask = MGF (maskedDB, hLen).
h. Let maskedSeed = seed ⊕ seedMask.
i. Concatenate a single octet with hexadecimal value 0x00, maskedSeed, and
maskedDB to form an encoded message EM of length k octets as
EM = 0x00 || maskedSeed || maskedDB.
3. RSA encryption:
a. Convert the encoded message EM to an integer message representative m
(see Section 4.2):
m = OS2IP (EM) .
b. Apply the RSAEP encryption primitive (Section 5.1.1) to the RSA public
key (n, e) and the message representative m to produce an integer
ciphertext representative c:
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 19
Copyright © 2002 RSA Security Inc.
c = RSAEP ((n, e), m) .
c. Convert the ciphertext representative c to a ciphertext C of length k octets
(see Section 4.1):
C = I2OSP (c, k) .
4. Output the ciphertext C.
PSlHash M
seed
DB =
EM =
a
’
MGF
’
MGF
00
00 maskedSeed maskedDB
Figure 1: EME-OAEP encoding operation. lHash is the hash of the optional label L.
Decoding operation follows reverse steps to recover M and verify lHash and PS.
Note. If L is the empty string, the corresponding hash value lHash has the following hexadecimal
representation for different choices of Hash:
SHA-1: (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
SHA-256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c a495991b 7852b855
SHA-384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743 4c0cc7bf 63f6e1da
274edebf e76f65fb d51ad2f1 4898b95b
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 20
Copyright © 2002 RSA Security Inc.
SHA-512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc 83f4a921 d36ce9ce
47d0d13c 5d85f2b0 ff8318d2 877eec2f 63b931bd 47417a81 a538327a f927da3e
7.1.2 Decryption operation
RSAES-OAEP-DECRYPT (K, C, L)
Options: Hash hash function (hLen denotes the length in octets of the hash function
output)
MGF mask generation function
Input: K recipient’s RSA private key (k denotes the length in octets of the
RSA modulus n)
C ciphertext to be decrypted, an octet string of length k, where k •
2hLen + 2
L optional label whose association with the message is to be verified;
the default value for L, if L is not provided, is the empty string
Output: M message, an octet string of length mLen, where mLen ≤ k – 2hLen – 2
Error: “decryption error”
Steps:
1. Length checking:
a. If the length of L is greater than the input limitation for the hash function
(261
– 1 octets for SHA-1), output “decryption error” and stop.
b. If the length of the ciphertext C is not k octets, output “decryption error”
and stop.
c. If k < 2hLen + 2, output “decryption error” and stop.
2. RSA decryption:
a. Convert the ciphertext C to an integer ciphertext representative c (see
Section 4.2):
c = OS2IP (C) .
b. Apply the RSADP decryption primitive (Section 5.1.2) to the RSA private
key K and the ciphertext representative c to produce an integer message
representative m:
m = RSADP (K, c) .
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 21
Copyright © 2002 RSA Security Inc.
If RSADP outputs “ciphertext representative out of range” (meaning that c
≥ n), output “decryption error” and stop.
c. Convert the message representative m to an encoded message EM of
length k octets (see Section 4.1):
EM = I2OSP (m, k) .
3. EME-OAEP decoding:
a. If the label L is not provided, let L be the empty string. Let lHash = Hash
(L), an octet string of length hLen (see the note in Section 7.1.1).
b. Separate the encoded message EM into a single octet Y, an octet string
maskedSeed of length hLen, and an octet string maskedDB of length k –
hLen – 1 as
EM = Y || maskedSeed || maskedDB .
c. Let seedMask = MGF (maskedDB, hLen).
d. Let seed = maskedSeed ⊕ seedMask.
e. Let dbMask = MGF (seed, k – hLen – 1).
f. Let DB = maskedDB ⊕ dbMask.
g. Separate DB into an octet string lHash’ of length hLen, a (possibly empty)
padding string PS consisting of octets with hexadecimal value 0x00, and a
message M as
DB = lHash’ || PS || 0x01 || M .
If there is no octet with hexadecimal value 0x01 to separate PS from M, if
lHash does not equal lHash’, or if Y is nonzero, output “decryption error”
and stop. (See the note below.)
4. Output the message M.
Note. Care must be taken to ensure that an opponent cannot distinguish the different error conditions in Step
3.f, whether by error message or timing, or, more generally, learn partial information about the encoded
message EM. Otherwise an opponent may be able to obtain useful information about the decryption of the
ciphertext C, leading to a chosen-ciphertext attack such as the one observed by Manger [36].
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 22
Copyright © 2002 RSA Security Inc.
7.2 RSAES-PKCS1-v1_5
RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections 5.1.1 and
5.1.2) with the EME-PKCS1-v1_5 encoding method (step 1 in Section 7.2.1 and step 3 in
Section 7.2.2). It is mathematically equivalent to the encryption scheme in PKCS #1 v1.5.
RSAES-PKCS1-v1_5 can operate on messages of length up to k – 11 octets (k is the octet
length of the RSA modulus), although care should be taken to avoid certain attacks on
low-exponent RSA due to Coppersmith, Franklin, Patarin, and Reiter when long
messages are encrypted (see the third bullet in the notes below and [10]; [14] contains an
improved attack). As a general rule, the use of this scheme for encrypting an arbitrary
message, as opposed to a randomly generated key, is not recommended.
It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts without knowing the
corresponding plaintexts, with a reasonable probability of success. This ability can be
exploited in a chosen-ciphertext attack as shown in [6]. Therefore, if RSAES-PKCS1v1_5
is to be used, certain easily implemented countermeasures should be taken to thwart
the attack found in [6]. Typical examples include the addition of structure to the data to
be encoded, rigorous checking of PKCS #1 v1.5 conformance (and other redundancy) in
decrypted messages, and the consolidation of error messages in a client-server protocol
based on PKCS #1 v1.5. These can all be effective countermeasures and do not involve
changes to a PKCS #1 v1.5-based protocol. See [7] for a further discussion of these and
other countermeasures. It has recently been shown that the security of the SSL/TLS
handshake protocol [17], which uses RSAES-PKCS1-v1_5 and certain countermeasures,
can be related to a variant of the RSA problem; see [32] for discussion.
Note. The following passages describe some security recommendations pertaining to the use of RSAESPKCS1-v1_5.
Recommendations from version 1.5 of this document are included as well as new
recommendations motivated by cryptanalytic advances made in the intervening years.
• It is recommended that the pseudorandom octets in step 2 in Section 7.2.1 be generated independently
for each encryption process, especially if the same data is input to more than one encryption process.
Håstad's results [24] are one motivation for this recommendation.
• The padding string PS in step 2 in Section 7.2.1 is at least eight octets long, which is a security
condition for public-key operations that makes it difficult for an attacker to recover data by trying all
possible encryption blocks.
• The pseudorandom octets can also help thwart an attack due to Coppersmith et al. [10] (see [14] for an
improvement of the attack) when the size of the message to be encrypted is kept small. The attack
works on low-exponent RSA when similar messages are encrypted with the same RSA public key.
More specifically, in one flavor of the attack, when two inputs to RSAEP agree on a large fraction of
bits (8/9) and low-exponent RSA (e = 3) is used to encrypt both of them, it may be possible to recover
both inputs with the attack. Another flavor of the attack is successful in decrypting a single ciphertext
when a large fraction (2/3) of the input to RSAEP is already known. For typical applications, the
message to be encrypted is short (e.g., a 128-bit symmetric key) so not enough information will be
known or common between two messages to enable the attack. However, if a long message is
encrypted, or if part of a message is known, then the attack may be a concern. In any case, the RSAESOAEP
scheme overcomes the attack.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 23
Copyright © 2002 RSA Security Inc.
7.2.1 Encryption operation
RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)
Input: (n, e) recipient’s RSA public key (k denotes the length in octets of the
modulus n)
M message to be encrypted, an octet string of length mLen, where
mLen ≤ k – 11
Output: C ciphertext, an octet string of length k
Error: “message too long”
Steps:
1. Length checking: If mLen > k – 11, output “message too long” and stop.
2. EME-PKCS1-v1_5 encoding:
a. Generate an octet string PS of length k – mLen – 3 consisting of pseudorandomly
generated nonzero octets. The length of PS will be at least eight
octets.
b. Concatenate PS, the message M, and other padding to form an encoded
message EM of length k octets as
EM = 0x00 || 0x02 || PS || 0x00 || M .
3. RSA encryption:
a. Convert the encoded message EM to an integer message representative m
(see Section 4.2):
m = OS2IP (EM) .
b. Apply the RSAEP encryption primitive (Section 5.1.1) to the RSA public
key (n, e) and the message representative m to produce an integer
ciphertext representative c:
c = RSAEP ((n, e), m) .
c. Convert the ciphertext representative c to a ciphertext C of length k octets
(see Section 4.1):
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 24
Copyright © 2002 RSA Security Inc.
C = I2OSP (c, k) .
4. Output the ciphertext C.
7.2.2 Decryption operation
RSAES-PKCS1-V1_5-DECRYPT (K, C)
Input: K recipient’s RSA private key
C ciphertext to be decrypted, an octet string of length k, where k is
the length in octets of the RSA modulus n
Output: M message, an octet string of length at most k – 11
Error: “decryption error”
Steps:
1. Length checking: If the length of the ciphertext C is not k octets (or if k < 11),
output “decryption error” and stop.
2. RSA decryption:
a. Convert the ciphertext C to an integer ciphertext representative c (see
Section 4.2):
c = OS2IP (C) .
b. Apply the RSADP decryption primitive (Section 5.1.2) to the RSA private
key (n, d) and the ciphertext representative c to produce an integer
message representative m:
m = RSADP ((n, d), c) .
If RSADP outputs “ciphertext representative out of range” (meaning that c
≥ n), output “decryption error” and stop.
c. Convert the message representative m to an encoded message EM of
length k octets (see Section 4.1):
EM = I2OSP (m, k) .
3. EME-PKCS1-v1_5 decoding: Separate the encoded message EM into an octet
string PS consisting of nonzero octets and a message M as
EM = 0x00 || 0x02 || PS || 0x00 || M .
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 25
Copyright © 2002 RSA Security Inc.
If the first octet of EM does not have hexadecimal value 0x00, if the second octet
of EM does not have hexadecimal value 0x02, if there is no octet with
hexadecimal value 0x00 to separate PS from M, or if the length of PS is less than
8 octets, output “decryption error” and stop. (See the note below.)
4. Output M.
Note. Care shall be taken to ensure that an opponent cannot distinguish the different error conditions in Step
3, whether by error message or timing. Otherwise an opponent may be able to obtain useful information
about the decryption of the ciphertext C, leading to a strengthened version of Bleichenbacher’s attack [6];
compare to Manger’s attack [36].
8 Signature schemes with appendix
For the purposes of this document, a signature scheme with appendix consists of a
signature generation operation and a signature verification operation, where the
signature generation operation produces a signature from a message with a signer’s RSA
private key, and the signature verification operation verifies the signature on the message
with the signer’s corresponding RSA public key. To verify a signature constructed with
this type of scheme it is necessary to have the message itself. In this way, signature
schemes with appendix are distinguished from signature schemes with message recovery,
which are not supported in this document.
A signature scheme with appendix can be employed in a variety of applications. For
instance, the signature schemes with appendix defined here would be suitable signature
algorithms for X.509 certificates [28]. Related signature schemes could be employed in
PKCS #7 [45], although for technical reasons the current version of PKCS #7 separates a
hash function from a signature scheme, which is different than what is done here; see the
note in Appendix A.2.3 for more discussion.
Two signature schemes with appendix are specified in this document: RSASSA-PSS and
RSASSA-PKCS1-v1_5. Although no attacks are known against RSASSA-PKCS1-v1_5,
in the interest of increased robustness, RSASSA-PSS is recommended for eventual
adoption in new applications. RSASSA-PKCS1-v1_5 is included for compatibility with
existing applications, and while still appropriate for new applications, a gradual transition
to RSASSA-PSS is encouraged.
The signature schemes with appendix given here follow a general model similar to that
employed in IEEE Std 1363-2000 [26], combining signature and verification primitives
with an encoding method for signatures. The signature generation operations apply a
message encoding operation to a message to produce an encoded message, which is then
converted to an integer message representative. A signature primitive is applied to the
message representative to produce the signature. Reversing this, the signature verification
operations apply a signature verification primitive to the signature to recover a message
representative, which is then converted to an octet string encoded message. A verification
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 26
Copyright © 2002 RSA Security Inc.
operation is applied to the message and the encoded message to determine whether they
are consistent.
If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5), the verification
operation may apply the message encoding operation to the message and compare the
resulting encoded message to the previously derived encoded message. If there is a match,
the signature is considered valid. If the method is randomized (e.g., EMSA-PSS), the
verification operation is typically more complicated. For example, the verification
operation in EMSA-PSS extracts the random salt and a hash output from the encoded
message and checks whether the hash output, the salt, and the message are consistent; the
hash output is a deterministic function in terms of the message and the salt.
For both signature schemes with appendix defined in this document, the signature
generation and signature verification operations are readily implemented as “single-pass”
operations if the signature is placed after the message. See PKCS #7 [45] for an example
format in the case of RSASSA-PKCS1-v1_5.
8.1 RSASSA-PSS
RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the EMSA-PSS
encoding method. It is compatible with the IFSSA scheme as amended in the IEEE
P1363a draft [27], where the signature and verification primitives are IFSP-RSA1 and
IFVP-RSA1 as defined in IEEE Std 1363-2000 [26] and the message encoding method is
EMSA4. EMSA4 is slightly more general than EMSA-PSS as it acts on bit strings rather
than on octet strings. EMSA-PSS is equivalent to EMSA4 restricted to the case that the
operands as well as the hash and salt values are octet strings.
The length of messages on which RSASSA-PSS can operate is either unrestricted or
constrained by a very large number, depending on the hash function underlying the
EMSA-PSS encoding method.
Assuming that computing eth
roots modulo n is infeasible and the hash and mask
generation functions in EMSA-PSS have appropriate properties, RSASSA-PSS provides
secure signatures. This assurance is provable in the sense that the difficulty of forging
signatures can be directly related to the difficulty of inverting the RSA function, provided
that the hash and mask generation functions are viewed as black boxes or random oracles.
The bounds in the security proof are essentially “tight”, meaning that the success
probability and running time for the best forger against RSASSA-PSS are very close to
the corresponding parameters for the best RSA inversion algorithm; see [4][13][31] for
further discussion.
In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash function identifier is
not embedded in the EMSA-PSS encoded message, so in theory it is possible for an
adversary to substitute a different (and potentially weaker) hash function than the one
selected by the signer. Therefore, it is recommended that the EMSA-PSS mask generation
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 27
Copyright © 2002 RSA Security Inc.
function be based on the same hash function. In this manner the entire encoded message
will be dependent on the hash function and it will be difficult for an opponent to
substitute a different hash function than the one intended by the signer. This matching of
hash functions is only for the purpose of preventing hash function substitution, and is not
necessary if hash function substitution is addressed by other means (e.g., the verifier
accepts only a designated hash function). See [34] for further discussion of these points.
The provable security of RSASSA-PSS does not rely on the hash function in the mask
generation function being the same as the hash function applied to the message.
RSASSA-PSS is different from other RSA-based signature schemes in that it is
probabilistic rather than deterministic, incorporating a randomly generated salt value. The
salt value enhances the security of the scheme by affording a “tighter” security proof than
deterministic alternatives such as Full Domain Hashing (FDH); see [4] for discussion.
However, the randomness is not critical to security. In situations where random
generation is not possible, a fixed value or a sequence number could be employed instead,
with the resulting provable security similar to that of FDH [12].
8.1.1 Signature generation operation
RSASSA-PSS-SIGN (K, M)
Input: K signer’s RSA private key
M message to be signed, an octet string
Output: S signature, an octet string of length k, where k is the length in octets
of the RSA modulus n
Errors: “message too long;” “encoding error”
Steps:
1. EMSA-PSS encoding: Apply the EMSA-PSS encoding operation (Section 9.1.1)
to the message M to produce an encoded message EM of length (modBits – 1)/8
octets such that the bit length of the integer OS2IP (EM) (see Section 4.2) is at
most modBits – 1, where modBits is the length in bits of the RSA modulus n:
EM = EMSA-PSS-ENCODE (M, modBits – 1) .
Note that the octet length of EM will be one less than k if modBits – 1 is divisible
by 8 and equal to k otherwise. If the encoding operation outputs “message too
long,” output “message too long” and stop. If the encoding operation outputs
“encoding error,” output “encoding error” and stop.
2. RSA signature:
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 28
Copyright © 2002 RSA Security Inc.
a. Convert the encoded message EM to an integer message representative m
(see Section 4.2):
m = OS2IP (EM) .
b. Apply the RSASP1 signature primitive (Section 5.2.1) to the RSA private
key K and the message representative m to produce an integer signature
representative s:
s = RSASP1 (K, m) .
c. Convert the signature representative s to a signature S of length k octets
(see Section 4.1):
S = I2OSP (s, k) .
3. Output the signature S.
8.1.2 Signature verification operation
RSASSA-PSS-VERIFY ((n, e), M, S)
Input: (n, e) signer’s RSA public key
M message whose signature is to be verified, an octet string
S signature to be verified, an octet string of length k, where k is the
length in octets of the RSA modulus n
Output: “valid signature” or “invalid signature”
Steps:
1. Length checking: If the length of the signature S is not k octets, output “invalid
signature” and stop.
2. RSA verification:
a. Convert the signature S to an integer signature representative s (see
Section 4.2):
s = OS2IP (S) .
b. Apply the RSAVP1 verification primitive (Section 5.2.2) to the RSA
public key (n, e) and the signature representative s to produce an integer
message representative m:
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 29
Copyright © 2002 RSA Security Inc.
m = RSAVP1 ((n, e), s) .
If RSAVP1 output “signature representative out of range,” output “invalid
signature” and stop.
c. Convert the message representative m to an encoded message EM of
length emLen = (modBits – 1)/8 octets, where modBits is the length in
bits of the RSA modulus n (see Section 4.1):
EM = I2OSP (m, emLen) .
Note that emLen will be one less than k if modBits – 1 is divisible by 8 and
equal to k otherwise. If I2OSP outputs “integer too large,” output “invalid
signature” and stop.
3. EMSA-PSS verification: Apply the EMSA-PSS verification operation (Section
9.1.2) to the message M and the encoded message EM to determine whether they
are consistent:
Result = EMSA-PSS-VERIFY (M, EM, modBits – 1) .
4. If Result = “consistent,” output “valid signature.” Otherwise, output “invalid
signature.”
8.2 RSASSA-PKCS1-v1_5
RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the EMSAPKCS1-v1_5
encoding method. It is compatible with the IFSSA scheme defined in IEEE
Std 1363-2000 [26], where the signature and verification primitives are IFSP-RSA1 and
IFVP-RSA1 and the message encoding method is EMSA-PKCS1-v1_5 (which is not
defined in IEEE Std 1363-2000, but is in the IEEE P1363a draft [27]).
The length of messages on which RSASSA-PKCS1-v1_5 can operate is either
unrestricted or constrained by a very large number, depending on the hash function
underlying the EMSA-PKCS1-v1_5 method.
Assuming that computing eth
roots modulo n is infeasible and the hash function in
EMSA-PKCS1-v1_5 has appropriate properties, RSASSA-PKCS1-v1_5 is conjectured to
provide secure signatures. More precisely, forging signatures without knowing the RSA
private key is conjectured to be computationally infeasible. Also, in the encoding method
EMSA-PKCS1-v1_5, a hash function identifier is embedded in the encoding. Because of
this feature, an adversary trying to find a message with the same signature as a previously
signed message must find collisions of the particular hash function being used; attacking
a different hash function than the one selected by the signer is not useful to the adversary.
See [34] for further discussion.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 30
Copyright © 2002 RSA Security Inc.
Note. As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method has the property that the
encoded message, converted to an integer message representative, is guaranteed to be large and at least
somewhat “random”. This prevents attacks of the kind proposed by Desmedt and Odlyzko [16] where
multiplicative relationships between message representatives are developed by factoring the message
representatives into a set of small values (e.g., a set of small primes). Coron, Naccache, and Stern [15]
showed that a stronger form of this type of attack could be quite effective against some instances of the
ISO/IEC 9796-2 signature scheme. They also analyzed the complexity of this type of attack against the
EMSA-PKCS1-v1_5 encoding method and concluded that an attack would be impractical, requiring more
operations than a collision search on the underlying hash function (i.e., more than 2
80
operations).
Coppersmith, Halevi, and Jutla [11] subsequently extended Coron et al.’s attack to break the ISO/IEC 9796-
1 signature scheme with message recovery. The various attacks illustrate the importance of carefully
constructing the input to the RSA signature primitive, particularly in a signature scheme with message
recovery. Accordingly, the EMSA-PKCS-v1_5 encoding method explicitly includes a hash operation and is
not intended for signature schemes with message recovery. Moreover, while no attack is known against the
EMSA-PKCS-v1_5 encoding method, a gradual transition to EMSA-PSS is recommended as a precaution
against future developments.
8.2.1 Signature generation operation
RSASSA-PKCS1-V1_5-SIGN (K, M)
Input: K signer’s RSA private key
M message to be signed, an octet string
Output: S signature, an octet string of length k, where k is the length in octets
of the RSA modulus n
Errors: “message too long”; “RSA modulus too short”
Steps:
1. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
operation (Section 9.2) to the message M to produce an encoded message EM of
length k octets:
EM = EMSA-PKCS1-V1_5-ENCODE (M, k) .
If the encoding operation outputs “message too long,” output “message too long”
and stop. If the encoding operation outputs “intended encoded message length too
short,” output “RSA modulus too short” and stop.
2. RSA signature:
a. Convert the encoded message EM to an integer message representative m
(see Section 4.2):
m = OS2IP (EM) .
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 31
Copyright © 2002 RSA Security Inc.
b. Apply the RSASP1 signature primitive (Section 5.2.1) to the RSA private
key K and the message representative m to produce an integer signature
representative s:
s = RSASP1 (K, m) .
c. Convert the signature representative s to a signature S of length k octets
(see Section 4.1):
S = I2OSP (s, k) .
3. Output the signature S.
8.2.2 Signature verification operation
RSASSA-PKCS1-V1_5-VERIFY ((n, e), M, S)
Input: (n, e) signer’s RSA public key
M message whose signature is to be verified, an octet string
S signature to be verified, an octet string of length k, where k is the
length in octets of the RSA modulus n
Output: “valid signature” or “invalid signature”
Errors: “message too long”; “RSA modulus too short”
Steps:
1. Length checking: If the length of the signature S is not k octets, output “invalid
signature” and stop.
2. RSA verification:
a. Convert the signature S to an integer signature representative s (see
Section 4.2):
s = OS2IP (S) .
b. Apply the RSAVP1 verification primitive (Section 5.2.2) to the RSA
public key (n, e) and the signature representative s to produce an integer
message representative m:
m = RSAVP1 ((n, e), s) .
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 32
Copyright © 2002 RSA Security Inc.
If RSAVP1 outputs “signature representative out of range,” output
“invalid signature” and stop.
c. Convert the message representative m to an encoded message EM of
length k octets (see Section 4.1):
EM’ = I2OSP (m, k) .
If I2OSP outputs “integer too large,” output “invalid signature” and stop.
3. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
operation (Section 9.2) to the message M to produce a second encoded message
EM’ of length k octets:
EM’ = EMSA-PKCS1-V1_5-ENCODE (M, k) .
If the encoding operation outputs “message too long,” output “message too long”
and stop. If the encoding operation outputs “intended encoded message length too
short,” output “RSA modulus too short” and stop.
4. Compare the encoded message EM and the second encoded message EM’. If they
are the same, output “valid signature”; otherwise, output “invalid signature.”
Note. Another way to implement the signature verification operation is to apply a “decoding” operation (not
specified in this document) to the encoded message to recover the underlying hash value, and then to
compare it to a newly computed hash value. This has the advantage that it requires less intermediate storage
(two hash values rather than two encoded messages), but the disadvantage that it requires additional code.
9 Encoding methods for signatures with appendix
Encoding methods consist of operations that map between octet string messages and octet
string encoded messages, which are converted to and from integer message
representatives in the schemes. The integer message representatives are processed via the
primitives. The encoding methods thus provide the connection between the schemes,
which process messages, and the primitives.
An encoding method for signatures with appendix, for the purposes of this document,
consists of an encoding operation and optionally a verification operation. An encoding
operation maps a message M to an encoded message EM of a specified length. A
verification operation determines whether a message M and an encoded message EM are
consistent, i.e., whether the encoded message EM is a valid encoding of the message M.
The encoding operation may introduce some randomness, so that different applications of
the encoding operation to the same message will produce different encoded messages,
which has benefits for provable security. For such an encoding method, both an encoding
and a verification operation are needed unless the verifier can reproduce the randomness
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 33
Copyright © 2002 RSA Security Inc.
(e.g., by obtaining the salt value from the signer). For a deterministic encoding method
only an encoding operation is needed.
Two encoding methods for signatures with appendix are employed in the signature
schemes and are specified here: EMSA-PSS and EMSA-PKCS1-v1_5.
9.1 EMSA-PSS
This encoding method is parameterized by the choice of hash function, mask generation
function, and salt length. These options should be fixed for a given RSA key, except that
the salt length can be variable (see [31] for discussion). Suggested hash and mask
generation functions are given in Appendix B. The encoding method is based on Bellare
and Rogaway’s Probabilistic Signature Scheme (PSS) [4][5]. It is randomized and has an
encoding operation and a verification operation.
Figure 2 illustrates the encoding operation.
bc
padding2
maskedDB
M
padding1 mHash salt
salt
H
’
MGF
M’ =
DB =
EM =
a
Hash
Hash
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 34
Copyright © 2002 RSA Security Inc.
Figure 2: EMSA-PSS encoding operation. Verification operation follows reverse steps
to recover salt, then forward steps to recompute and compare H.
Notes.
1. The encoding method defined here differs from the one in Bellare and Rogaway’s submission to IEEE
P1363a [5] in three respects:
• It applies a hash function rather than a mask generation function to the message. Even though the
mask generation function is based on a hash function, it seems more natural to apply a hash
function directly.
• The value that is hashed together with the salt value is the string (0x)00 00 00 00 00 00 00 00 ||
mHash rather than the message M itself. Here, mHash is the hash of M. Note that the hash function
is the same in both steps. See Note 3 below for further discussion. (Also, the name “salt” is used
instead of “seed”, as it is more reflective of the value’s role.)
• The encoded message in EMSA-PSS has nine fixed bits; the first bit is 0 and the last eight bits
form a “trailer field”, the octet 0xbc. In the original scheme, only the first bit is fixed. The rationale
for the trailer field is for compatibility with the Rabin-Williams IFSP-RW signature primitive in
IEEE Std 1363-2000 [26] and the corresponding primitive in the draft ISO/IEC 9796-2 [29].
2. Assuming that the mask generation function is based on a hash function, it is recommended that the
hash function be the same as the one that is applied to the message; see Section 8.1 for further
discussion.
3. Without compromising the security proof for RSASSA-PSS, one may perform steps 1 and 2 of EMSAPSS-ENCODE
and EMSA-PSS-VERIFY (the application of the hash function to the message) outside the
module that computes the rest of the signature operation, so that mHash rather than the message M
itself is input to the module. In other words, the security proof for RSASSA-PSS still holds even if an
opponent can control the value of mHash. This is convenient if the module has limited I/O bandwidth,
e.g., a smart card. Note that previous versions of PSS [4][5] did not have this property. Of course, it
may be desirable for other security reasons to have the module process the full message. For instance,
the module may need to “see” what it is signing if it does not trust the component that computes the
hash value.
4. Typical salt lengths in octets are hLen (the length of the output of the hash function Hash) and 0. In
both cases the security of RSASSA-PSS can be closely related to the hardness of inverting RSAVP1.
Bellare and Rogaway [4] give a tight lower bound for the security of the original RSA-PSS scheme,
which corresponds roughly to the former case, while Coron [12] gives a lower bound for the related
Full Domain Hashing scheme, which corresponds roughly to the latter case. In [13] Coron provides a
general treatment with various salt lengths ranging from 0 to hLen; see [27] for discussion. See also
[31], which adapts the security proofs in [4][13] to address the differences between the original and the
present version of RSA-PSS as listed in Note 1 above.
5. As noted in IEEE P1363a [27], the use of randomization in signature schemes – such as the salt value
in EMSA-PSS – may provide a “covert channel” for transmitting information other than the message
being signed. For more on covert channels, see [50].
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 35
Copyright © 2002 RSA Security Inc.
9.1.1 Encoding operation
EMSA-PSS-ENCODE (M, emBits)
Options: Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
sLen intended length in octets of the salt
Input: M message to be encoded, an octet string
emBits maximal bit length of the integer OS2IP (EM) (see Section 4.2), at
least 8hLen + 8sLen + 9
Output: EM encoded message, an octet string of length emLen = emBits/8
Errors: “encoding error”; “message too long”
Steps:
1. If the length of M is greater than the input limitation for the hash function (261
– 1
octets for SHA-1), output “message too long” and stop.
2. Let mHash = Hash (M), an octet string of length hLen.
3. If emLen < hLen + sLen + 2, output “encoding error” and stop.
4. Generate a random octet string salt of length sLen; if sLen = 0, then salt is the
empty string.
5. Let
M’ = (0x)00 00 00 00 00 00 00 00 || mHash || salt;
M’ is an octet string of length 8 + hLen + sLen with eight initial zero octets.
6. Let H = Hash (M’), an octet string of length hLen.
7. Generate an octet string PS consisting of emLen – sLen – hLen – 2 zero octets.
The length of PS may be 0.
8. Let DB = PS || 0x01 || salt; DB is an octet string of length emLen – hLen – 1.
9. Let dbMask = MGF (H, emLen – hLen – 1).
10. Let maskedDB = DB ⊕ dbMask.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 36
Copyright © 2002 RSA Security Inc.
11. Set the leftmost 8emLen – emBits bits of the leftmost octet in maskedDB to zero.
12. Let EM = maskedDB || H || 0xbc.
13. Output EM.
9.1.2 Verification operation
EMSA-PSS-VERIFY (M, EM, emBits)
Options: Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
sLen intended length in octets of the salt
Input: M message to be verified, an octet string
EM encoded message, an octet string of length emLen = emBits/8
emBits maximal bit length of the integer OS2IP (EM) (see Section 4.2), at
least 8hLen + 8sLen + 9
Output: “consistent” or “inconsistent”
Steps:
1. If the length of M is greater than the input limitation for the hash function (261
– 1
octets for SHA-1), output “inconsistent” and stop.
2. Let mHash = Hash (M), an octet string of length hLen.
3. If emLen < hLen + sLen + 2, output “inconsistent” and stop.
4. If the rightmost octet of EM does not have hexadecimal value 0xbc, output
“inconsistent” and stop.
5. Let maskedDB be the leftmost emLen – hLen – 1 octets of EM, and let H be the
next hLen octets.
6. If the leftmost 8emLen – emBits bits of the leftmost octet in maskedDB are not all
equal to zero, output “inconsistent” and stop.
7. Let dbMask = MGF (H, emLen – hLen – 1).
8. Let DB = maskedDB ⊕ dbMask.
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Copyright © 2002 RSA Security Inc.
9. Set the leftmost 8emLen – emBits bits of the leftmost octet in DB to zero.
10. If the emLen – hLen – sLen – 2 leftmost octets of DB are not zero or if the octet at
position emLen – hLen – sLen – 1 (the leftmost position is “position 1”) does not
have hexadecimal value 0x01, output “inconsistent” and stop.
11. Let salt be the last sLen octets of DB.
12. Let
M’ = (0x)00 00 00 00 00 00 00 00 || mHash || salt ;
M’ is an octet string of length 8 + hLen + sLen with eight initial zero octets.
13. Let H’ = Hash (M’), an octet string of length hLen.
14. If H = H’, output “consistent.” Otherwise, output “inconsistent.”
9.2 EMSA-PKCS1-v1_5
This encoding method is deterministic and only has an encoding operation.
EMSA-PKCS1-v1_5-ENCODE (M, emLen)
Option: Hash hash function (hLen denotes the length in octets of the hash
function output)
Input: M message to be encoded
emLen intended length in octets of the encoded message, at least tLen +
11, where tLen is the octet length of the DER encoding T of a
certain value computed during the encoding operation
Output: EM encoded message, an octet string of length emLen
Errors: “message too long”; “intended encoded message length too short”
Steps:
1. Apply the hash function to the message M to produce a hash value H:
H = Hash (M) .
If the hash function outputs “message too long,” output “message too long” and
stop.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 38
Copyright © 2002 RSA Security Inc.
2. Encode the algorithm ID for the hash function and the hash value into an ASN.1
value of type DigestInfo (see Appendix A.2.4) with the Distinguished Encoding
Rules (DER), where the type DigestInfo has the syntax
DigestInfo ::= SEQUENCE {
digestAlgorithm AlgorithmIdentifier,
digest OCTET STRING
}
The first field identifies the hash function and the second contains the hash value.
Let T be the DER encoding of the DigestInfo value (see the notes below) and let
tLen be the length in octets of T.
3. If emLen < tLen + 11, output “intended encoded message length too short” and
stop.
4. Generate an octet string PS consisting of emLen – tLen – 3 octets with
hexadecimal value 0xff. The length of PS will be at least 8 octets.
5. Concatenate PS, the DER encoding T, and other padding to form the encoded
message EM as
EM = 0x00 || 0x01 || PS || 0x00 || T .
6. Output EM.
Notes.
1. For the six hash functions mentioned in Appendix B.1, the DER encoding T of the DigestInfo value is
equal to the following:
MD2: (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04 10 || H.
MD5: (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04 10 || H.
SHA-1: (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14 || H.
SHA-256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00 04 20 || H.
SHA-384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00 04 30 || H.
SHA-512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00 04 40 || H.
2. In version 1.5 of this document, T was defined as the BER encoding, rather than the DER encoding, of
the DigestInfo value. In particular, it is possible – at least in theory – that the verification operation
defined in this document (as well as in version 2.0) rejects a signature that is valid with respect to the
specification given in PKCS #1 v1.5. This occurs if other rules than DER are applied to DigestInfo (e.g.,
an indefinite length encoding of the underlying SEQUENCE type). While this is unlikely to be a concern
in practice, a cautious implementer may choose to employ a verification operation based on a BER
decoding operation as specified in PKCS #1 v1.5. In this manner, compatibility with any valid
implementation based on PKCS #1 v1.5 is obtained. Such a verification operation should indicate
whether the underlying BER encoding is a DER encoding and hence whether the signature is valid with
respect to the specification given in this document.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 39
Copyright © 2002 RSA Security Inc.
A. ASN.1 syntax
A.1 RSA key representation
This section defines ASN.1 object identifiers for RSA public and private keys, and
defines the types RSAPublicKey and RSAPrivateKey. The intended application of these
definitions includes X.509 certificates, PKCS #8 [46], and PKCS #12 [47].
The object identifier rsaEncryption identifies RSA public and private keys as defined in
Appendices A.1.1 and A.1.2. The parameters field associated with this OID in a value of
type AlgorithmIdentifier shall have a value of type NULL.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
The definitions in this section have been extended to support multi-prime RSA, but are
backward compatible with previous versions.
A.1.1 RSA public key syntax
An RSA public key should be represented with the ASN.1 type RSAPublicKey:
RSAPublicKey ::= SEQUENCE {
modulus INTEGER, -- n
publicExponent INTEGER -- e
}
The fields of type RSAPublicKey have the following meanings:
• modulus is the RSA modulus n.
• publicExponent is the RSA public exponent e.
A.1.2 RSA private key syntax
An RSA private key should be represented with the ASN.1 type RSAPrivateKey:
RSAPrivateKey ::= SEQUENCE {
version Version,
modulus INTEGER, -- n
publicExponent INTEGER, -- e
privateExponent INTEGER, -- d
prime1 INTEGER, -- p
prime2 INTEGER, -- q
exponent1 INTEGER, -- d mod (p-1)
exponent2 INTEGER, -- d mod (q-1)
coefficient INTEGER, -- (inverse of q) mod p
otherPrimeInfos OtherPrimeInfos OPTIONAL
}
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 40
Copyright © 2002 RSA Security Inc.
The fields of type RSAPrivateKey have the following meanings:
• version is the version number, for compatibility with future revisions of this
document. It shall be 0 for this version of the document, unless multi-prime is used, in
which case it shall be 1.
Version ::= INTEGER { two-prime(0), multi(1) }
(CONSTRAINED BY {-- version must be multi if otherPrimeInfos present --})
• modulus is the RSA modulus n.
• publicExponent is the RSA public exponent e.
• privateExponent is the RSA private exponent d.
• prime1 is the prime factor p of n.
• prime2 is the prime factor q of n.
• exponent1 is d mod (p − 1).
• exponent2 is d mod (q − 1).
• coefficient is the CRT coefficient q–1 mod p.
• otherPrimeInfos contains the information for the additional primes r3, …, ru, in order. It
shall be omitted if version is 0 and shall contain at least one instance of OtherPrimeInfo
if version is 1.
OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
OtherPrimeInfo ::= SEQUENCE {
prime INTEGER, -- ri
exponent INTEGER, -- di
coefficient INTEGER -- ti
}
The fields of type OtherPrimeInfo have the following meanings:
• prime is a prime factor ri of n, where i ≥ 3.
• exponent is di = d mod (ri − 1).
• coefficient is the CRT coefficient ti = (r1 · r2 · … · ri–1)–1
mod ri.
Note. It is important to protect the RSA private key against both disclosure and modification. Techniques
for such protection are outside the scope of this document. Methods for storing and distributing private keys
and other cryptographic data are described in PKCS #12 and #15.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 41
Copyright © 2002 RSA Security Inc.
A.2 Scheme identification
This section defines object identifiers for the encryption and signature schemes. The
schemes compatible with PKCS #1 v1.5 have the same definitions as in PKCS #1 v1.5.
The intended application of these definitions includes X.509 certificates and PKCS #7.
Here are type identifier definitions for the PKCS #1 OIDs:
PKCS1Algorithms ALGORITHM-IDENTIFIER ::= {
{ OID rsaEncryption PARAMETERS NULL } |
{ OID md2WithRSAEncryption PARAMETERS NULL } |
{ OID md5WithRSAEncryption PARAMETERS NULL } |
{ OID sha1WithRSAEncryption PARAMETERS NULL } |
{ OID sha256WithRSAEncryption PARAMETERS NULL } |
{ OID sha384WithRSAEncryption PARAMETERS NULL } |
{ OID sha512WithRSAEncryption PARAMETERS NULL } |
{ OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } |
PKCS1PSourceAlgorithms |
{ OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } ,
... -- Allows for future expansion --
}
A.2.1 RSAES-OAEP
The object identifier id-RSAES-OAEP identifies the RSAES-OAEP encryption scheme.
id-RSAES-OAEP OBJECT IDENTIFIER ::= { pkcs-1 7 }
The parameters field associated with this OID in a value of type AlgorithmIdentifier shall
have a value of type RSAES-OAEP-params:
RSAES-OAEP-params ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty
}
The fields of type RSAES-OAEP-params have the following meanings:
• hashAlgorithm identifies the hash function. It shall be an algorithm ID with an OID in
the set OAEP-PSSDigestAlgorithms. For a discussion of supported hash functions, see
Appendix B.1.
HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }
OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-sha1 PARAMETERS NULL }|
{ OID id-sha256 PARAMETERS NULL }|
{ OID id-sha384 PARAMETERS NULL }|
{ OID id-sha512 PARAMETERS NULL },
... -- Allows for future expansion --
}
The default hash function is SHA-1:
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Copyright © 2002 RSA Security Inc.
sha1 HashAlgorithm ::= {
algorithm id-sha1,
parameters SHA1Parameters : NULL
}
SHA1Parameters ::= NULL
• maskGenAlgorithm identifies the mask generation function. It shall be an algorithm ID
with an OID in the set PKCS1MGFAlgorithms, which for this version shall consist of idmgf1,
identifying the MGF1 mask generation function (see Appendix B.2.1). The
parameters field associated with id-mgf1 shall be an algorithm ID with an OID in the set
OAEP-PSSDigestAlgorithms, identifying the hash function on which MGF1 is based.
MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-mgf1 PARAMETERS HashAlgorithm },
... -- Allows for future expansion --
}
The default mask generation function is MGF1 with SHA-1:
mgf1SHA1 MaskGenAlgorithm ::= {
algorithm id-mgf1,
parameters HashAlgorithm : sha1
}
• pSourceAlgorithm identifies the source (and possibly the value) of the label L. It shall
be an algorithm ID with an OID in the set PKCS1PSourceAlgorithms, which for this
version shall consist of id-pSpecified, indicating that the label is specified explicitly.
The parameters field associated with id-pSpecified shall have a value of type OCTET
STRING, containing the label. In previous versions of this specification, the term
“encoding parameters” was used rather than “label”, hence the name of the type
below.
PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }
PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-pSpecified PARAMETERS EncodingParameters },
... -- Allows for future expansion --
}
id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 }
EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
The default label is an empty string (so that lHash will contain the hash of the empty
string):
pSpecifiedEmpty PSourceAlgorithm ::= {
algorithm id-pSpecified,
parameters EncodingParameters : emptyString
}
emptyString EncodingParameters ::= ’’H
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Copyright © 2002 RSA Security Inc.
If all of the default values of the fields in RSAES-OAEP-params are used, then the algorithm
identifier will have the following value:
rSAES-OAEP-Default-Identifier RSAES-AlgorithmIdentifier ::= {
algorithm id-RSAES-OAEP,
parameters RSAES-OAEP-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
pSourceAlgorithm pSpecifiedEmpty
}
}
RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} }
A.2.2 RSAES-PKCS1-v1_5
The object identifier rsaEncryption (see Appendix A.1) identifies the RSAES-PKCS1v1_5
encryption scheme. The parameters field associated with this OID in a value of type
AlgorithmIdentifier shall have a value of type NULL. This is the same as in PKCS #1 v1.5.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
A.2.3 RSASSA-PSS
The object identifier id-RSASSA-PSS identifies the RSASSA-PSS encryption scheme.
id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }
The parameters field associated with this OID in a value of type AlgorithmIdentifier shall
have a value of type RSASSA-PSS-params:
RSASSA-PSS-params ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
saltLength [2] INTEGER DEFAULT 20,
trailerField [3] TrailerField DEFAULT trailerFieldBC
}
The fields of type RSASSA-PSS-params have the following meanings:
• hashAlgorithm identifies the hash function. It shall be an algorithm ID with an OID in
the set OAEP-PSSDigestAlgorithms (see Appendix A.2.1). The default hash function is
SHA-1.
• maskGenAlgorithm identifies the mask generation function. It shall be an algorithm ID
with an OID in the set PKCS1MGFAlgorithms (see Appendix A.2.1). The default mask
generation function is MGF1 with SHA-1. For MGF1 (and more generally, for other
mask generation functions based on a hash function), it is recommended that the
underlying hash function be the same as the one identified by hashAlgorithm; see Note
2 in Section 9.1 for further comments.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 44
Copyright © 2002 RSA Security Inc.
• saltLength is the octet length of the salt. It shall be an integer. For a given
hashAlgorithm, the default value of saltLength is the octet length of the hash value.
Unlike the other fields of type RSASSA-PSS-params, saltLength does not need to be
fixed for a given RSA key pair.
• trailerField is the trailer field number, for compatibility with the draft IEEE P1363a
[27]. It shall be 1 for this version of the document, which represents the trailer field
with hexadecimal value 0xbc. Other trailer fields (including the trailer field HashID ||
0xcc in IEEE P1363a) are not supported in this document.
TrailerField ::= INTEGER { trailerFieldBC(1) }
If the default values of the hashAlgorithm, maskGenAlgorithm, and trailerField fields of
RSASSA-PSS-params are used, then the algorithm identifier will have the following value:
rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= {
algorithm id-RSASSA-PSS,
parameters RSASSA-PSS-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
saltLength 20,
trailerField trailerFieldBC
}
}
RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} }
Note. In some applications, the hash function underlying a signature scheme is identified separately from
the rest of the operations in the signature scheme. For instance, in PKCS #7 [45], a hash function identifier
is placed before the message and a “digest encryption” algorithm identifier (indicating the rest of the
operations) is carried with the signature. In order for PKCS #7 to support the RSASSA-PSS signature
scheme, an object identifier would need to be defined for the operations in RSASSA-PSS after the hash
function (analogous to the RSAEncryption OID for the RSASSA-PKCS1-v1_5 scheme). S/MIME CMS [25]
takes a different approach. Although a hash function identifier is placed before the message, an algorithm
identifier for the full signature scheme may be carried with a CMS signature (this is done for DSA
signatures). Following this convention, the id-RSASSA-PSS OID can be used to identify RSASSA-PSS
signatures in CMS. Since CMS is considered the successor to PKCS #7 and new developments such as the
addition of support for RSASSA-PSS will be pursued with respect to CMS rather than PKCS #7, an OID for
the “rest of” RSASSA-PSS is not defined in this version of PKCS #1.
A.2.4 RSASSA-PKCS1-v1_5
The object identifier for RSASSA-PKCS1-v1_5 shall be one of the following. The choice
of OID depends on the choice of hash algorithm: MD2, MD5, SHA-1, SHA-256, SHA-
384, or SHA-512. Note that if either MD2 or MD5 is used, then the OID is just as in
PKCS #1 v1.5. For each OID, the parameters field associated with this OID in a value of
type AlgorithmIdentifier shall have a value of type NULL. The OID should be chosen in
accordance with the following table:
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 45
Copyright © 2002 RSA Security Inc.
Hash algorithm OID
MD2 md2WithRSAEncryption ::= {pkcs-1 2}
MD5 md5WithRSAEncryption ::= {pkcs-1 4}
SHA-1 sha1WithRSAEncryption ::= {pkcs-1 5}
SHA-256 sha256WithRSAEncryption ::= {pkcs-1 11}
SHA-384 sha384WithRSAEncryption ::= {pkcs-1 12}
SHA-512 sha512WithRSAEncryption ::= {pkcs-1 13}
The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type DigestInfo,
where the type DigestInfo has the syntax
DigestInfo ::= SEQUENCE {
digestAlgorithm DigestAlgorithm,
digest OCTET STRING
}
digestAlgorithm identifies the hash function and shall be an algorithm ID with an OID in
the set PKCS1-v1-5DigestAlgorithms. For a discussion of supported hash functions, see
Appendix B.1.
DigestAlgorithm ::= AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} }
PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-md2 PARAMETERS NULL }|
{ OID id-md5 PARAMETERS NULL }|
{ OID id-sha1 PARAMETERS NULL }|
{ OID id-sha256 PARAMETERS NULL }|
{ OID id-sha384 PARAMETERS NULL }|
{ OID id-sha512 PARAMETERS NULL }
}
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 46
Copyright © 2002 RSA Security Inc.
B. Supporting techniques
This section gives several examples of underlying functions supporting the encryption
schemes in Section 7 and the encoding methods in Section 9. A range of techniques is
given here to allow compatibility with existing applications as well as migration to new
techniques. While these supporting techniques are appropriate for applications to
implement, none of them is required to be implemented. It is expected that profiles for
PKCS #1 v2.1 will be developed that specify particular supporting techniques.
This section also gives object identifiers for the supporting techniques.
B.1 Hash functions
Hash functions are used in the operations contained in Sections 7 and 9. Hash functions
are deterministic, meaning that the output is completely determined by the input. Hash
functions take octet strings of variable length, and generate fixed length octet strings. The
hash functions used in the operations contained in Sections 7 and 9 should generally be
collision-resistant. This means that it is infeasible to find two distinct inputs to the hash
function that produce the same output. A collision-resistant hash function also has the
desirable property of being one-way; this means that given an output, it is infeasible to
find an input whose hash is the specified output. In addition to the requirements, the hash
function should yield a mask generation function (Appendix B.2) with pseudorandom
output.
Six hash functions are given as examples for the encoding methods in this document:
MD2 [33], MD5 [41], SHA-1 [38], and the proposed algorithms SHA-256, SHA-384,
and SHA-512 [39]. For the RSAES-OAEP encryption scheme and EMSA-PSS encoding
method, only SHA-1 and SHA-256/384/512 are recommended. For the EMSA-PKCS1v1_5
encoding method, SHA-1 and SHA-256/384/512 are recommended for new
applications. MD2 and MD5 are recommended only for compatibility with existing
applications based on PKCS #1 v1.5.
The object identifiers id-md2, id-md5, id-sha1, id-sha256, id-sha384, and id-sha512, identify
the respective hash functions:
id-md2 OBJECT IDENTIFIER ::= {
iso (1) member-body (2) us (840) rsadsi (113549) digestAlgorithm (2) 2
}
id-md5 OBJECT IDENTIFIER ::= {
iso (1) member-body (2) us (840) rsadsi (113549) digestAlgorithm (2) 5
}
id-sha1 OBJECT IDENTIFIER ::= {
iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2) 26
}
id-sha256 OBJECT IDENTIFIER ::= {
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 47
Copyright © 2002 RSA Security Inc.
joint-iso-itu-t (2) country (16) us (840) organization (1) gov (101)
csor (3) nistalgorithm (4) hashalgs (2) 1
}
id-sha384 OBJECT IDENTIFIER ::= {
joint-iso-itu-t (2) country (16) us (840) organization (1) gov (101)
csor (3) nistalgorithm (4) hashalgs (2) 2
}
id-sha512 OBJECT IDENTIFIER ::= {
joint-iso-itu-t (2) country (16) us (840) organization (1) gov (101)
csor (3) nistalgorithm (4) hashalgs (2) 3
}
The parameters field associated with these OIDs in a value of type AlgorithmIdentifier shall
have a value of type NULL.
Note. Version 1.5 of PKCS #1 also allowed for the use of MD4 in signature schemes. The cryptanalysis of
MD4 has progressed significantly in the intervening years. For example, Dobbertin [18] demonstrated how
to find collisions for MD4 and that the first two rounds of MD4 are not one-way [20]. Because of these
results and others (e.g. [8]), MD4 is no longer recommended. There have also been advances in the
cryptanalysis of MD2 and MD5, although not enough to warrant removal from existing applications. Rogier
and Chauvaud [43] demonstrated how to find collisions in a modified version of MD2. No one has
demonstrated how to find collisions for the full MD5 algorithm, although partial results have been found
(e.g. [9][19]).
To address these concerns, SHA-1, SHA-256, SHA-384, or SHA-512 are recommended for new
applications. As of today, the best (known) collision attacks against these hash functions are generic attacks
with complexity 2
L/2
, where L is the bit length of the hash output. For the signature schemes in this
document, a collision attack is easily translated into a signature forgery. Therefore, the value L / 2 should be
at least equal to the desired security level in bits of the signature scheme (a security level of B bits means
that the best attack has complexity 2
B
). The same rule of thumb can be applied to RSAES-OAEP; it is
recommended that the bit length of the seed (which is equal to the bit length of the hash output) be twice the
desired security level in bits.
B.2 Mask generation functions
A mask generation function takes an octet string of variable length and a desired output
length as input, and outputs an octet string of the desired length. There may be restrictions
on the length of the input and output octet strings, but such bounds are generally very
large. Mask generation functions are deterministic; the octet string output is completely
determined by the input octet string. The output of a mask generation function should be
pseudorandom: Given one part of the output but not the input, it should be infeasible to
predict another part of the output. The provable security of RSAES-OAEP and RSASSAPSS
relies on the random nature of the output of the mask generation function, which in
turn relies on the random nature of the underlying hash.
One mask generation function is given here: MGF1, which is based on a hash function.
MGF1 coincides with the mask generation functions defined in IEEE Std 1363-2000 [26]
and the draft ANSI X9.44 [1]. Future versions of this document may define other mask
generation functions.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 48
Copyright © 2002 RSA Security Inc.
B.2.1 MGF1
MGF1 is a Mask Generation Function based on a hash function.
MGF1 (mgfSeed, maskLen)
Options: Hash hash function (hLen denotes the length in octets of the hash
function output)
Input: mgfSeed seed from which mask is generated, an octet string
maskLen intended length in octets of the mask, at most 232
hLen
Output: mask mask, an octet string of length maskLen
Error: “mask too long”
Steps:
1. If maskLen > 232
hLen, output “mask too long” and stop.
2. Let T be the empty octet string.
3. For counter from 0 to  maskLen / hLen  – 1, do the following:
a. Convert counter to an octet string C of length 4 octets (see Section 4.1):
C = I2OSP (counter, 4) .
b. Concatenate the hash of the seed mgfSeed and C to the octet string T:
T = T || Hash (mgfSeed || C) .
4. Output the leading maskLen octets of T as the octet string mask.
The object identifier id-mgf1 identifies the MGF1 mask generation function:
id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 }
The parameters field associated with this OID in a value of type AlgorithmIdentifier shall
have a value of type hashAlgorithm, identifying the hash function on which MGF1 is
based.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 49
Copyright © 2002 RSA Security Inc.
C. ASN.1 module
PKCS-1 {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1) modules(0)
pkcs-1(1)
}
-- $ Revision: 2.1 $
-- This module has been checked for conformance with the ASN.1 standard by
-- the OSS ASN.1 Tools
DEFINITIONS EXPLICIT TAGS ::=
BEGIN
-- EXPORTS ALL
-- All types and values defined in this module are exported for use in other
-- ASN.1 modules.
IMPORTS
id-sha256, id-sha384, id-sha512
FROM NIST-SHA2 {
joint-iso-itu-t(2) country(16) us(840) organization(1) gov(101)
csor(3) nistalgorithm(4) modules(0) sha2(1)
};
-- ============================
-- Basic object identifiers
-- ============================
-- The DER encoding of this in hexadecimal is:
-- (0x)06 08
-- 2A 86 48 86 F7 0D 01 01
--
pkcs-1 OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1
}
--
-- When rsaEncryption is used in an AlgorithmIdentifier the parameters
-- MUST be present and MUST be NULL.
--
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
--
-- When id-RSAES-OAEP is used in an AlgorithmIdentifier the parameters MUST
-- be present and MUST be RSAES-OAEP-params.
--
id-RSAES-OAEP OBJECT IDENTIFIER ::= { pkcs-1 7 }
--
-- When id-pSpecified is used in an AlgorithmIdentifier the parameters MUST
-- be an OCTET STRING.
--
id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 }
--
-- When id-RSASSA-PSS is used in an AlgorithmIdentifier the parameters MUST
-- be present and MUST be RSASSA-PSS-params.
--
id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 50
Copyright © 2002 RSA Security Inc.
--
-- When the following OIDs are used in an AlgorithmIdentifier the parameters
-- MUST be present and MUST be NULL.
--
md2WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 2 }
md5WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 4 }
sha1WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 5 }
sha256WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 11 }
sha384WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 12 }
sha512WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 13 }
--
-- This OID really belongs in a module with the secsig OIDs.
--
id-sha1 OBJECT IDENTIFIER ::= {
iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2) 26
}
--
-- OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5.
--
id-md2 OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2
}
id-md5 OBJECT IDENTIFIER ::= {
iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5
}
--
-- When id-mgf1 is used in an AlgorithmIdentifier the parameters MUST be
-- present and MUST be a HashAlgorithm, for example sha1.
--
id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 }
-- ================
-- Useful types
-- ================
ALGORITHM-IDENTIFIER ::= CLASS {
&id OBJECT IDENTIFIER UNIQUE,
&Type OPTIONAL
}
WITH SYNTAX { OID &id [PARAMETERS &Type] }
-- Note: the parameter InfoObjectSet in the following definitions allows a
-- distinct information object set to be specified for sets of algorithms
-- such as:
-- DigestAlgorithms ALGORITHM-IDENTIFIER ::= {
-- { OID id-md2 PARAMETERS NULL }|
-- { OID id-md5 PARAMETERS NULL }|
-- { OID id-sha1 PARAMETERS NULL }
-- }
--
AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::= SEQUENCE {
algorithm
ALGORITHM-IDENTIFIER.&id({InfoObjectSet}),
parameters
ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm}) OPTIONAL
}
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 51
Copyright © 2002 RSA Security Inc.
-- ==============
-- Algorithms
-- ==============
--
-- Allowed EME-OAEP and EMSA-PSS digest algorithms.
--
OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-sha1 PARAMETERS NULL }|
{ OID id-sha256 PARAMETERS NULL }|
{ OID id-sha384 PARAMETERS NULL }|
{ OID id-sha512 PARAMETERS NULL },
... -- Allows for future expansion --
}
--
-- Allowed EMSA-PKCS1-v1_5 digest algorithms.
--
PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-md2 PARAMETERS NULL }|
{ OID id-md5 PARAMETERS NULL }|
{ OID id-sha1 PARAMETERS NULL }|
{ OID id-sha256 PARAMETERS NULL }|
{ OID id-sha384 PARAMETERS NULL }|
{ OID id-sha512 PARAMETERS NULL }
}
sha1 HashAlgorithm ::= {
algorithm id-sha1,
parameters SHA1Parameters : NULL
}
HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} }
SHA1Parameters ::= NULL
--
-- Allowed mask generation function algorithms.
-- If the identifier is id-mgf1, the parameters are a HashAlgorithm.
--
PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-mgf1 PARAMETERS HashAlgorithm },
... -- Allows for future expansion --
}
--
-- Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and
-- id-RSASSA-PSS.maskGenAlgorithm.
--
mgf1SHA1 MaskGenAlgorithm ::= {
algorithm id-mgf1,
parameters HashAlgorithm : sha1
}
MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
--
-- Allowed algorithms for pSourceAlgorithm.
--
PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= {
{ OID id-pSpecified PARAMETERS EncodingParameters },
... -- Allows for future expansion --
}
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 52
Copyright © 2002 RSA Security Inc.
EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
--
-- This identifier means that the label L is an empty string, so the digest
-- of the empty string appears in the RSA block before masking.
--
pSpecifiedEmpty PSourceAlgorithm ::= {
algorithm id-pSpecified,
parameters EncodingParameters : emptyString
}
PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }
emptyString EncodingParameters ::= ’’H
--
-- Type identifier definitions for the PKCS #1 OIDs.
--
PKCS1Algorithms ALGORITHM-IDENTIFIER ::= {
{ OID rsaEncryption PARAMETERS NULL } |
{ OID md2WithRSAEncryption PARAMETERS NULL } |
{ OID md5WithRSAEncryption PARAMETERS NULL } |
{ OID sha1WithRSAEncryption PARAMETERS NULL } |
{ OID sha256WithRSAEncryption PARAMETERS NULL } |
{ OID sha384WithRSAEncryption PARAMETERS NULL } |
{ OID sha512WithRSAEncryption PARAMETERS NULL } |
{ OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } |
PKCS1PSourceAlgorithms |
{ OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } ,
... -- Allows for future expansion --
}
-- ===================
-- Main structures
-- ===================
RSAPublicKey ::= SEQUENCE {
modulus INTEGER, -- n
publicExponent INTEGER -- e
}
--
-- Representation of RSA private key with information for the CRT algorithm.
--
RSAPrivateKey ::= SEQUENCE {
version Version,
modulus INTEGER, -- n
publicExponent INTEGER, -- e
privateExponent INTEGER, -- d
prime1 INTEGER, -- p
prime2 INTEGER, -- q
exponent1 INTEGER, -- d mod (p-1)
exponent2 INTEGER, -- d mod (q-1)
coefficient INTEGER, -- (inverse of q) mod p
otherPrimeInfos OtherPrimeInfos OPTIONAL
}
Version ::= INTEGER { two-prime(0), multi(1) }
(CONSTRAINED BY {-- version must be multi if otherPrimeInfos present --})
OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 53
Copyright © 2002 RSA Security Inc.
OtherPrimeInfo ::= SEQUENCE {
prime INTEGER, -- ri
exponent INTEGER, -- di
coefficient INTEGER -- ti
}
--
-- AlgorithmIdentifier.parameters for id-RSAES-OAEP.
-- Note that the tags in this Sequence are explicit.
--
RSAES-OAEP-params ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty
}
--
-- Identifier for default RSAES-OAEP algorithm identifier.
-- The DER Encoding of this is in hexadecimal:
-- (0x)30 0D
-- 06 09
-- 2A 86 48 86 F7 0D 01 01 07
-- 30 00
-- Notice that the DER encoding of default values is "empty".
--
rSAES-OAEP-Default-Identifier RSAES-AlgorithmIdentifier ::= {
algorithm id-RSAES-OAEP,
parameters RSAES-OAEP-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
pSourceAlgorithm pSpecifiedEmpty
}
}
RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} }
--
-- AlgorithmIdentifier.parameters for id-RSASSA-PSS.
-- Note that the tags in this Sequence are explicit.
--
RSASSA-PSS-params ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
saltLength [2] INTEGER DEFAULT 20,
trailerField [3] TrailerField DEFAULT trailerFieldBC
}
TrailerField ::= INTEGER { trailerFieldBC(1) }
--
-- Identifier for default RSASSA-PSS algorithm identifier
-- The DER Encoding of this is in hexadecimal:
-- (0x)30 0D
-- 06 09
-- 2A 86 48 86 F7 0D 01 01 0A
-- 30 00
-- Notice that the DER encoding of default values is "empty".
--
rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= {
algorithm id-RSASSA-PSS,
parameters RSASSA-PSS-params : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
saltLength 20,
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 54
Copyright © 2002 RSA Security Inc.
trailerField trailerFieldBC
}
}
RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} }
--
-- Syntax for the EMSA-PKCS1-v1_5 hash identifier.
--
DigestInfo ::= SEQUENCE {
digestAlgorithm DigestAlgorithm,
digest OCTET STRING
}
DigestAlgorithm ::= AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} }
END -- PKCS1Definitions
________________________________________________________________________________
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 55
Copyright © 2002 RSA Security Inc.
D. Intellectual property considerations
The RSA public-key cryptosystem is described in U.S. Patent 4,405,829, which expired
on September 20, 2000. RSA Security Inc. makes no other patent claims on the
constructions described in this document, although specific underlying techniques may be
covered.
Multi-prime RSA is described in U.S. Patent 5,848,159.
The University of California has indicated that it has a patent pending on the PSS
signature scheme [5]. It has also provided a letter to the IEEE P1363 working group
stating that if the PSS signature scheme is included in an IEEE standard, “the University
of California will, when that standard is adopted, FREELY license any conforming
implementation of PSS as a technique for achieving a digital signature with appendix”
[23]. The PSS signature scheme is specified in the IEEE P1363a draft [27], which was in
ballot resolution when this document was published.
License to copy this document is granted provided that it is identified as “RSA Security
Inc. Public-Key Cryptography Standards (PKCS)” in all material mentioning or
referencing this document.
RSA Security Inc. makes no other representations regarding intellectual property claims
by other parties. Such determination is the responsibility of the user.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 56
Copyright © 2002 RSA Security Inc.
E. Revision history
Versions 1.0 – 1.3
Versions 1.0 – 1.3 were distributed to participants in RSA Data Security, Inc.’s PublicKey
Cryptography Standards meetings in February and March 1991.
Version 1.4
Version 1.4 was part of the June 3, 1991 initial public release of PKCS. Version 1.4 was
published as NIST/OSI Implementors’ Workshop document SEC-SIG-91-18.
Version 1.5
Version 1.5 incorporated several editorial changes, including updates to the references
and the addition of a revision history. The following substantive changes were made:
 Section 10: “MD4 with RSA” signature and verification processes were added.
 Section 11: md4WithRSAEncryption object identifier was added.
Version 1.5 was republished as IETF RFC 2313.
Version 2.0
Version 2.0 incorporated major editorial changes in terms of the document structure and
introduced the RSAES-OAEP encryption scheme. This version continued to support the
encryption and signature processes in version 1.5, although the hash algorithm MD4 was
no longer allowed due to cryptanalytic advances in the intervening years. Version 2.0 was
republished as IETF RFC 2437 [35].
Version 2.1
Version 2.1 introduces multi-prime RSA and the RSASSA-PSS signature scheme with
appendix along with several editorial improvements. This version continues to support
the schemes in version 2.0.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 57
Copyright © 2002 RSA Security Inc.
F. References
[1] ANSI X9F1 Working Group. ANSI X9.44 Draft D2: Key Establishment Using
Integer Factorization Cryptography. Working Draft, March 2002.
[2] M. Bellare, A. Desai, D. Pointcheval and P. Rogaway. Relations Among Notions
of Security for Public-Key Encryption Schemes. In H. Krawczyk, editor,
Advances in Cryptology – Crypto ’98, volume 1462 of Lecture Notes in Computer
Science, pp. 26 – 45. Springer Verlag, 1998.
[3] M. Bellare and P. Rogaway. Optimal Asymmetric Encryption – How to Encrypt
with RSA. In A. De Santis, editor, Advances in Cryptology – Eurocrypt ’94,
volume 950 of Lecture Notes in Computer Science, pp. 92 – 111. Springer Verlag,
1995.
[4] M. Bellare and P. Rogaway. The Exact Security of Digital Signatures – How to
Sign with RSA and Rabin. In U. Maurer, editor, Advances in Cryptology –
Eurocrypt ’96, volume 1070 of Lecture Notes in Computer Science, pp. 399 –
416. Springer Verlag, 1996.
[5] M. Bellare and P. Rogaway. PSS: Provably Secure Encoding Method for Digital
Signatures. Submission to IEEE P1363 working group, August 1998. Available
from http://grouper.ieee.org/groups/1363/.
[6] D. Bleichenbacher. Chosen Ciphertext Attacks Against Protocols Based on the
RSA Encryption Standard PKCS #1. In H. Krawczyk, editor, Advances in
Cryptology – Crypto ’98, volume 1462 of Lecture Notes in Computer Science, pp.
1 – 12. Springer Verlag, 1998.
[7] D. Bleichenbacher, B. Kaliski and J. Staddon. Recent Results on PKCS #1: RSA
Encryption Standard. RSA Laboratories’ Bulletin No. 7, June 1998.
[8] B. den Boer and A. Bosselaers. An Attack on the Last Two Rounds of MD4. In
J. Feigenbaum, editor, Advances in Cryptology – Crypto ’91, volume 576 of
Lecture Notes in Computer Science, pp. 194 – 203. Springer Verlag, 1992.
[9] B. den Boer and A. Bosselaers. Collisions for the Compression Function of MD5.
In T. Helleseth, editor, Advances in Cryptology – Eurocrypt ’93, volume 765 of
Lecture Notes in Computer Science, pp. 293 – 304. Springer Verlag, 1994.
[10] D. Coppersmith, M. Franklin, J. Patarin and M. Reiter. Low-Exponent RSA with
Related Messages. In U. Maurer, editor, Advances in Cryptology – Eurocrypt ’96,
volume 1070 of Lecture Notes in Computer Science, pp. 1 – 9. Springer Verlag,
1996.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 58
Copyright © 2002 RSA Security Inc.
[11] D. Coppersmith, S. Halevi and C. Jutla. ISO 9796-1 and the New Forgery
Strategy. Presented at the rump session of Crypto ’99, August 1999.
[12] J.-S. Coron. On the Exact Security of Full Domain Hashing. In M. Bellare, editor,
Advances in Cryptology – Crypto 2000, volume 1880 of Lecture Notes in
Computer Science, pp. 229 – 235. Springer Verlag, 2000.
[13] J.-S. Coron. Optimal Security Proofs for PSS and Other Signature Schemes. In L.
Knudsen, editor, Advances in Cryptology – Eurocrypt 2002, volume 2332 of
Lecture Notes in Computer Science, pp. 272 – 287. Springer Verlag, 2002.
[14] J.-S. Coron, M. Joye, D. Naccache and P. Paillier. New Attacks on PKCS #1 v1.5
Encryption. In B. Preneel, editor, Advances in Cryptology – Eurocrypt 2000,
volume 1807 of Lecture Notes in Computer Science, pp. 369 – 379. Springer
Verlag, 2000.
[15] J.-S. Coron, D. Naccache and J. P. Stern. On the Security of RSA Padding. In M.
Wiener, editor, Advances in Cryptology – Crypto ’99, volume 1666 of Lecture
Notes in Computer Science, pp. 1 – 18. Springer Verlag, 1999.
[16] Y. Desmedt and A.M. Odlyzko. A Chosen Text Attack on the RSA Cryptosystem
and Some Discrete Logarithm Schemes. In H.C. Williams, editor, Advances in
Cryptology – Crypto ’85, volume 218 of Lecture Notes in Computer Science, pp.
516 – 522. Springer Verlag, 1986.
[17] T. Dierks and C. Allen. IETF RFC 2246: The TLS Protocol Version 1.0. January
1999.
[18] H. Dobbertin. Cryptanalysis of MD4. In D. Gollmann, editor, Fast Software
Encryption ’96, volume 1039 of Lecture Notes in Computer Science, pp. 55 – 72.
Springer Verlag, 1996.
[19] H. Dobbertin. Cryptanalysis of MD5 Compress. Presented at the rump session of
Eurocrypt ’96, May 1996.
[20] H. Dobbertin. The First Two Rounds of MD4 are Not One-Way. In S. Vaudenay,
editor, Fast Software Encryption ’98, volume 1372 in Lecture Notes in Computer
Science, pp. 284 – 292. Springer Verlag, 1998.
[21] E. Fujisaki, T. Okamoto, D. Pointcheval and J. Stern. RSA-OAEP is Secure under
the RSA Assumption. In J. Kilian, editor, Advances in Cryptology – Crypto 2001,
volume 2139 of Lecture Notes in Computer Science, pp. 260 – 274. Springer
Verlag, 2001.
[22] H. Garner. The Residue Number System. IRE Transactions on Electronic
Computers, EC-8 (6), pp. 140 – 147, June 1959.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 59
Copyright © 2002 RSA Security Inc.
[23] M.L. Grell. Re: Encoding Methods PSS/PSS-R. Letter to IEEE P1363 working
group, University of California, June 15, 1999. Available from
http://grouper.ieee.org/groups/1363/P1363/patents.html.
[24] J. Håstad. Solving Simultaneous Modular Equations of Low Degree. SIAM
Journal of Computing, volume 17, pp. 336 – 341, 1988.
[25] R. Housley. IETF RFC 2630: Cryptographic Message Syntax. June 1999.
[26] IEEE Std 1363-2000: Standard Specifications for Public Key Cryptography.
IEEE, August 2000.
[27] IEEE P1363 working group. IEEE P1363a D10: Standard Specifications for
Public Key Cryptography: Additional Techniques. November 1, 2001. Available
from http://grouper.ieee.org/groups/1363/.
[28] ISO/IEC 9594-8:1997: Information technology – Open Systems Interconnection –
The Directory: Authentication Framework. 1997.
[29] ISO/IEC FDIS 9796-2: Information Technology – Security Techniques – Digital
Signature Schemes Giving Message Recovery – Part 2: Integer Factorization
Based Mechanisms. Final Draft International Standard, December 2001.
[30] ISO/IEC 18033-2: Information Technology – Security Techniques – Encryption
Algorithms – Part 2: Asymmetric Ciphers. V. Shoup, editor, Text for 2nd
Working
Draft, January 2002.
[31] J. Jonsson. Security Proof for the RSA-PSS Signature Scheme (extended
abstract). Second Open NESSIE Workshop. September 2001. Full version
available from http://eprint.iacr.org/2001/053/.
[32] J. Jonsson and B. Kaliski. On the Security of RSA Encryption in TLS. In
Advances in Cryptology – Crypto 2002, to appear.
[33] B. Kaliski. IETF RFC 1319: The MD2 Message-Digest Algorithm. April 1992.
[34] B. Kaliski. On Hash Function Identification in Signature Schemes. In B. Preneel,
editor, RSA Conference 2002, Cryptographers’ Track, volume 2271 of Lecture
Notes in Computer Science, pp. 1 – 16. Springer Verlag, 2002.
[35] B. Kaliski and J. Staddon. IETF RFC 2437: PKCS #1: RSA Cryptography
Specifications Version 2.0. October 1998.
[36] J. Manger. A Chosen Ciphertext Attack on RSA Optimal Asymmetric Encryption
Padding (OAEP) as Standardized in PKCS #1 v2.0. In J. Kilian, editor, Advances
in Cryptology – Crypto 2001, volume 2139 of Lecture Notes in Computer
Science, pp. 260 – 274. Springer Verlag, 2001.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 60
Copyright © 2002 RSA Security Inc.
[37] A. Menezes, P. van Oorschot and S. Vanstone. Handbook of Applied
Cryptography. CRC Press, 1996.
[38] National Institute of Standards and Technology (NIST). FIPS Publication 180-1:
Secure Hash Standard. April 1994.
[39] National Institute of Standards and Technology (NIST). Draft FIPS 180-2: Secure
Hash Standard. Draft, May 2001. Available from http://www.nist.gov/sha/.
[40] J.-J. Quisquater and C. Couvreur. Fast Decipherment Algorithm for RSA PublicKey
Cryptosystem. Electronics Letters, 18 (21), pp. 905 – 907, October 1982
[41] R. Rivest. IETF RFC 1321: The MD5 Message-Digest Algorithm. April 1992.
[42] R. Rivest, A. Shamir and L. Adleman. A Method for Obtaining Digital Signatures
and Public-Key Cryptosystems. Communications of the ACM, 21 (2), pp. 120-126,
February 1978.
[43] N. Rogier and P. Chauvaud. The Compression Function of MD2 is not Collision
Free. Presented at Selected Areas of Cryptography ’95. Carleton University,
Ottawa, Canada. May 1995.
[44] RSA Laboratories. PKCS #1 v2.0: RSA Encryption Standard. October 1998.
[45] RSA Laboratories. PKCS #7 v1.5: Cryptographic Message Syntax Standard.
November 1993.
[46] RSA Laboratories. PKCS #8 v1.2: Private-Key Information Syntax Standard.
November 1993.
[47] RSA Laboratories. PKCS #12 v1.0: Personal Information Exchange Syntax
Standard. June 1999.
[48] V. Shoup. OAEP Reconsidered. In J. Kilian, editor, Advances in Cryptology –
Crypto 2001, volume 2139 of Lecture Notes in Computer Science, pp. 239 – 259.
Springer Verlag, 2001.
[49] R. D. Silverman. A Cost-Based Security Analysis of Symmetric and Asymmetric
Key Lengths. RSA Laboratories Bulletin No. 13, April 2000. Available from
http://www.rsasecurity.com.rsalabs/bulletins/.
[50] G. J. Simmons. Subliminal communication is easy using the DSA. In T.
Helleseth, editor, Advances in Cryptology – Eurocrypt ’93, volume 765 of Lecture
Notes in Computer Science, pp. 218-232. Springer-Verlag, 1993.
PKCS #1 V2.1: RSA CRYPTOGRAPHY STANDARD (June 14, 2002) 61
Copyright © 2002 RSA Security Inc.
G. About PKCS
The Public-Key Cryptography Standards are specifications produced by RSA
Laboratories in cooperation with secure systems developers worldwide for the purpose of
accelerating the deployment of public-key cryptography. First published in 1991 as a
result of meetings with a small group of early adopters of public-key technology, the
PKCS documents have become widely referenced and implemented. Contributions from
the PKCS series have become part of many formal and de facto standards, including
ANSI X9 and IEEE P1363 documents, PKIX, SET, S/MIME, SSL/TLS, and
WAP/WTLS.
Further development of PKCS occurs through mailing list discussions and occasional
workshops, and suggestions for improvement are welcome. For more information,
contact:
PKCS Editor
RSA Laboratories
174 Middlesex Turnpike
Bedford, MA 01730 USA
pkcs-editor@rsasecurity.com
http://www.rsasecurity.com/rsalabs/pkcs