Chapter 7: Digital signatures Digital signatures are one of the most important inventions/applications of modern cryptography. The problem is how can a user sign a message such that everybody (or the intended addressee only) can verify the digital signature and the signature is good enough also for legal purposes. Digital signatures – basic goals Digital sigantures should be such that each user should be able to verify signatures of other users, but that should give him/her no information how to sign a message on behind of other users. An important difference from a handwritten signature is that digital signature of a message is always intimately connected with the message, and for different messages is different, whereas the handwritten signature is adjoined to the message and always looks the same. Technically, a digital signature signing is performed by a signing algorithm and a digital signature it is verified by a verification algorithm. A copy of a digital (classical) signature is identical (usually distinguishable) to (from) the origin. A care has therefore to be made that a classical signature is not misused. This chapter contains some of the main techniques for design and verification of digital signatures (as well as some attacks to them). Digital signatures If only signature (but not the encryption of the message) are of importance, then it suffices that Alice sends to Bob (w, d[A](w)) Caution: Signing a message w by A for B by e[B](d[A](w)) is O.K., but the symmetric solution, with encoding first: c = d[A](e[B](w)) is not good. Digital Signature Schemes I Digital signatures are basic tools for authentication and nonreputation of messages. A digital signature scheme allows anyone to verify signature of any sender S without providing any information how to generate signatures of S. A Digital Signature Scheme (M, S, K[s], K[v]) is given by: – M a set of messages to be signed – S a set of possible signatures – K[s] a set of private keys for signing – K[v] a set of public keys for verification Moreover, it is required that: – For each k from K[s,] there exists a single and easy to compute signing mapping sig[k]: {0,1}* x M ® S – For each k from K[v] there exists a single and easy to compute verification mapping ver[k]: M x S ® {true, false} such that the following two conditions are satisfied: Digital Signature Schemes II Correctness: For each message m from M and public key k in K[v], it holds ver[k](m, s) = true if there is an r from {0, 1}* such that s = sig[l](r, m) for a private key l from K[s] corresponding to the public key k . Security: For any w from M and k in K[v] , it is computationally infeasible, without the knowledge of the private key corresponding to k, to find a signature s from S such that ver[k](w, s) = true. Attacks on digital signatures Total break of a signature scheme: The adversary manages to recover the secret key from the public key. Universal forgery: The adversary can derive from the public key an algorithm which allows to forge the signature of any message. Selective forgery: The adversary can derive from the public key a method to forge signatures of selected messages (where selection was made prior the knowledge of the public key). Existential forgery: The adversary is able to create from the public key a valid signature of a message m (but has no control for which m). A digital signature of one bit Let us start with a very simple but much illustrating (though non-practical) example how to sign a single bit. Design of the signature scheme: A one-way function f(x) is chosen. Two integers k[0] and k[1] are chosen, by signer, kept secret, and items f, (0, s[0]), (1, s[1]) are made public, where s[0] = f (k[0]), s[1] = f (k[1]) RSA signatures and their attacks Let us have an RSA cryptosystem with encryption and decryption exponents e and d and modulus n. Signing of a message w: Verification of a signature ENCRYPTION versus SIGNATURE Let each user U uses a cryptosystem with encryption and decryption algorithms: e[U], d[U] Let w be a message PUBLIC-KEY ENCRYPTIONS Encryption: e[U][ ](w) Decryption: d[U][ ](e[U][ ](w)) DIGITAL SIGNATURE SYSTEMS – simplified version A digital signature system (DSS) consists: • P - the space of possible plaintexts (messages). • S - the space of possible signatures. • K - the space of possible keys. • For each k Î K there is a signing algorithm sig[k] Î S[a] and a corresponding verification algorithm ver[k] Î V such that - sig[k] : P ® S. - ver[k] : P Ä S ® {true, false} and ver[k][ ](w,s) = true, if s = sig (w); false, otherwise. Algorithms sig[k] and ver[k] should be computable in polynomial time. Verification algorithm can be publically known; signing algorithm (actually only its key) should be kept secret. FROM PKC to DSS - again Any public-key cryptosystem in which the plaintext and cryptotext space are the same, can be used for digital signature. Signing of a message w by a user A so that any user can verify the signature: d[A][ ](w). ElGamal signatures Design of the ElGamal digital siganture system: choose: prime p, integers 1 £ q £ x £ p, where q is a primitive element of Z[p]*; Compute: y = q ^x mod p key K = (p, q, x, y) public key (p, q, y) - trapdoor: x ElGamal signatures - example Example choose: p = 11, q = 2, x = 8 compute: y = 2^8 mod 11 = 3 Signing of w = 5 as (a,b), where a = q^r mod p, w=xa+rb mod (p-1) choose r = 9 – (this choice is O.K. because gcd(9, 10) = 1) compute a = 2^9^ mod 11 = 6 solve equation: 5 º 8 · 6 + 9b (mod 10) that is 7 º 9b (mod 10) Þ b=3 signature: (6, 3) Note: equation that has to be solved: w= xa+rb mod (p-1). Security of ElGamal signatures Let us analyze several ways an eavesdropper Eve can try to forge ElGamal signature (with x - secret; p, q and y = q ^x mod p - public): sig(w, r) = (a, b); where r is random and a = q ^r mod p; b = (w - xa)r ^–1 (mod p –1). • First suppose Eve tries to forge signature for a new message w, without knowing x. • If Eve first chooses a value a and tries to find the corresponding b, it has to compute the discrete logarithm lg [a] q ^w^ y ^-a, because a ^b º q ^r^ ^(w^ ^-^ ^xa)^ ^r^^(^-1^) º q ^w^ ^-^ ^xa º q ^w^ y ^-a. • If Eve first chooses b and then tries to find a, she has to solve the equation y ^a^ a ^b º q ^xa^ q ^rb º q ^w (mod p). It is not known whether this equation can be solved for any given b efficiently. Forging and misusing of ElGamal signatures There are ways how to produce, using ElGamal signature scheme, some valid forged signatures, but they do not allow an opponent to forge signatures on messages of his/her choice. For example, if 0 £ i, j £ p -2 and gcd(j, p -1) = 1, then for a = q ^i y ^j mod p; b = -aj ^-1 mod (p -1); w = -aij ^-1 mod (p -1) the pair (a, b) is a valid signature of the message w. This can be easily shown by checking the verification condition. There are several ways ElGamal signatures can be broken if they are used not carefully enough. For example, the random r used in the signature should be kept secret. Otherwise the system can be broken and signatures forged. Indeed, if r is known, then x can be computed by x = (w - rb) a ^-1 mod (p -1) and once x is known Eve can forge signatures at will. Another misuse of the ElGamal signature system is to use the same r to sign two messages. In such a case x can be computed and system can be broken. Digital Signature Standard In December 1994, on the proposal of the National Institute of Standards and Technology, the following Digital Signature Algorithm (DSA) was accepted as a standard. Digital Signature Standard Signing and Verification Signing of a 160-bit plaintext w • choose random 0 < k < q such that gcd(k, q) = 1 • compute a = (r ^k mod p) mod q • compute b = k ^-1(w + xa) mod q where kk ^-1 º 1 (mod q) • signature: sig(w, k) = (a, b) From ElGamal to DSA DSA is a modification of ElGamal digital signature scheme. It was proposed in August 1991 and adopted in December 1994. Fiat-Shamir signature scheme Choose primes p, q, compute n = pq and choose: as public key v[1],…,v[k] and compute secret key Protocol for Alice to sign a message w: (1) Alice chooses t random integers 1 £ r[1],…,r[t][ ]< n, computes x [i]= r[i]^2^ mod n, 1 £ i £ t. Sad story Alice and Bob got to jail – and, unfortunately, to different jails. Walter, the warden, allows them to communicate by network, but he will not allow that their messages are encrypted. Problem: Can Alice and Bob set up a subliminal channel, a covert communications channel between them, in full view of Walter, even though the messages themselves that they exchange contain no secret information? Ong-Schnorr-Shamir subliminal channel scheme Story Alice and Bob are in different jails. Walter, the warden, allows them to communicate by network, but he will not allow messages to be encrypted. Can they set up a subliminal channel, a covert communications channel between them, in full view of Walter, even though the messages themselves contain no secret information? One-time signatures Lamport signature scheme shows how to construct a signature scheme for one use only from any one-way function. Let k be a positive integer and let P = {0,1}^k be the set of messages. Let f:Y ® Z be a one-way function where Y is a set of` ”signatures''. For 1 £ i £ k, j = 0,1 let y[ij]ÎY be chosen randomly and z[ij][ ]= f (y[ij]). The key K consists of 2k y's and z's. y's are secret, z's are public. Undeniable signatures I Undeniable signatures are signatures that have two properties: • A signature can be verified only at the cooperation with the signer – by means of a challenge-and-response protocol. • The signer cannot deny a correct signature. To achieve that, steps are a part of the protocol that force the signer to cooperate – by means of a disavowal protocol – this protocol makes possible to prove the invalidity of a signature and to show that it is a forgery. (If the signer refuses to take part in the disavowal protocol, then the signature is considered to be genuine.) Undeniable signature protocol of Chaum and van Antwerpen (1989), discussed next, is again based on infeasibility of the computation of the discrete logarithm. Undeniable signatures II Undeniable signatures consist: • Signing algorithm • Verification protocol, that is a challenge-and-response protocol. In this case it is required that a signature cannot be verified without a cooperation of the signer (Bob). This protects Bob against the possibility that documents signed by him are duplicated and distributed without his approval. • Disavowal protocol, by which Bob can prove that a signature is a forgery. This is to prevent Bob from disavowing a signature he made at an earlier time. Fooling and Disallowed protocol Since it holds: Theorem If s ¹ w ^x mod p, then Alice will accept s as a valid signature for w with probability 1/r. Bob cannot fool Alice except with very small probability and security is unconditional (that is, it does not depend on any computational assumption). Fooling and Disallowed protocol • Alice verifies that D ¹ w ^f1q ^f2 (mod p). • Alice concludes that s is a forgery iff (dq ^-e2)^ ^f1 º (Dq ^-^f^2)^ e1 (mod p). Signing of fingerprints Signatures scheme presented so far allow to sign only "short" messages. For example, DSS is used to sign 160 bit messages (with 320-bit signatures). A naive solution is to break long message into a sequence of shortones and to sign each block separately. Disadvantages: signing is slow and for long signatures integrity is not protected. The solution is to use fast public hash functions h which maps a message of any length to a fixed length hash. The hash is then signed. Example: message w arbitrary length message digest z = h (w) 160bits El Gamal signature y = sig(z) 320bits If Bob wants to send a signed message w he sends (w, sig(h(w)). Collision-free hash functions revisited For a hash function it is necessary to be good enough for creating fingerprints that do not allow various forgeries of signatures. Example 1, Eve starts with a valid signature (w, sig(h(w))), computes h(w) and tries to find w ' such that h(w) = h(w '). Would she succeed, then (w ', sig(h(w))) would be a valid signature, a forgery. In order to prevent the above type of attacks, and some other, it is required that a hash function h satisfies the following collision-free property. Timestamping There are various ways that a digital signature can be compromised. For example: if Eve determines the secret key of Bob, then she can forge signatures of any Bob’s message she likes. If this happens, authenticity of all messages signed by Bob before Eve got the secret key is to be questioned. The key problem is that there is no way to determine when a message was signed. A timestamping should provide proof that a message was signed at a certain time. Blind signatures The basic idea is that Sender makes Signer to sign a message m without Signer knowing m, therefore blindly – this is needed in e-commerce. Blind signing can be realized by a two party protocol, between the Sender and the Signer, that has the following properties. • In order to sign (by a Signer) a message m, the Sender computes, using a blinding procedure, from m an m* from which m can not be obtained without knowing a secret, and sends m* to the Signer. • The Signer signs m* to get a signature s[m*] (of m*) and sends s[m*] to the Sender. Signing is done in such a way that the Sender can afterwards compute, using an unblinding procedure, from Signer’s signature s[m*] of m* -- the signer signature s[m] of m. Chum’s blind signatures This blind signature protocol combines RSA with blinding/unblinding features. Bob’s RSA public key is (n,e) and his private key is d. Let m be a message, 0 < m < n, PROTOCOL: • Alice chooses a random 0 < k < n with gcd(n,k)=1. • Alice computes m* = mk^e (mod n) and sends it to Bob (this way Alice blinds the message m). • Bob computed s* = (m*)^d(mod n) and sends s* to Alice (this way Bob signs the blinded message m*). • Alice computes s =k^-1s*(mod n) to obtain Bob’s signature m^d of m (Alice performs unblinding of m*). Verification is equivalent to that of the RSA signature scheme. Fail-then-stop signatures They are signatures schemes that use a trusted authority and provide ways to prove, if it is the case, that a powerful enough adversary is around who could break the signature scheme and therefore its use should be stopped. The scheme is maintained by a trusted authority that chooses a secret key for each signer, keeps them secret, even from the signers themselves, and announces only the related public keys. An important idea is that signing and verification algorithms are enhanced by a so-called proof-of-forgery algorithm. When the signer see a forged signature he is able to compute his secret key and by submitting it to the trusted authority to prove the existence of a forgery and this way to achieve that any further use of the signature scheme is stopped. So called Heyst-Pedersen Scheme is an example of a Fail-Then-Stop siganture Scheme. Digital signatures with encryption and resending 1. Alice signs the message: s[A](w). A surprising attack to the previous scheme 1. Mallot intercept e[B](s[A](w)). A MAN-IN-THE-MIDDLE attack Consider the following protocol: 1. Alice sends Bob the pair (e[B](e[B](w)A), B) to B. 2. Bob uses d[B] to get A and w, and acknowledges by sending the pair (e[A](e[A](w)B), A) to Alice. (Here the function e and d are assumed to operate on numbers, names A,B,… are sequences of digits and e[B](w)A is a sequence of digitals obtained by concatenating e[B](w) and A.) Probabilistic signature schemes - PSS Let us have integers k, l, n such that k+l< n, a permutation a pseudorandom bit generator and a hash function h: {0,1}* ® {0,1} ^l. The following PSS scheme is applicable to messages of arbitrary length. Authenticated Diffie-Hellman key exchange Let each user U have a signature algorithm s[U] and a verification algorithm v[U]. The following protocol allows Alice and Bob to establish a key K to use with an encryption function e[K] and to avoid the man-in-the-middle attack. • Alice and Bob choose large prime p and a generator q Î Z[p]*. Security of digital signatures It is very non-trivial to define security of digital signature. Definition A chosen message attack is a process by which on an input of a verification key one can obtain a signature (corresponding to the given key) to a message of its choice. A chosen message attack is considered to be successful (in so called existential forgery) if it outputs a valid signature for a message for which it has not requested a signature during the attack. A signature scheme is secure (or unforgeable) if every feasible chosen message attack succeeds with at most negligible probability. Treshold Signature Schemes The idea of a (t+1, n) treshold signature scheme is to distribute the power of the signing operation to (t+1) parties out of n. A (t+1) treshold signature scheme should satisfy two conditions. Unforgeability means that even if an adversary corrupts t parties, he still cannot generate a valid signature. Robustness means that corrupted parties cannot prevent uncorrupted parties to generate signatures. Shoup (2000) presented an efficient, non-interactive, robust and unforgeable treshold RSA signature schemes. There is no proof yet whether Shoup’s scheme is provably secure. Digital Signatures - Observation Can we make digital signatures by digitalizing our usual signature and attaching them to the messages (documents) that need to be signed? No, because such signatures could be easily removed and attached to some other documents or messages. Key observation: Digital signatures have to depend not only on the signer, but also on the message that is being signed. SPECIAL TYPES of DIGITAL SIGNATURES • Append-Only Signatures (AOS) have the property that any party given an AOS signature sig[M[1]] on message M[1 ]can compute sig[M[1]II M[2]] for any message M[2]. (Such signatures are of importance in network applications, where users need to delegate their shares of resources to other users). • Identity-Based signatures (IBS) at which the identity of the signer (i.e. her email address) plays the role of her public key. (Such schemes assume the existence of a TA holding a master public-private key pair used to assign secret keys to users based on their identity.) • Hierarchically Identity-Based Signatures are such IBS in which users are arranged in a hierarchy and a user at any level at the hierarchy can delegate secret keys to her descendants based on their identities and her own secret keys. GROUP SIGNATURES • At Group Signatures (GS) a group member can compute a signature that reveals nothing about the signer’s identity, except that he is a member of the group. On the other hand, the group manager can always reveal the identity of the signer. • Hierarchical Group Signatures (HGS) are a generalization of GS that allow multiple group managers to be organized in a tree with the signers as leaves. When verifying a signature, a group manager only learns to which of its subtrees, if any, the signer belongs. Unconditionally secure digital signatures Any of the digital signature schemes introduced so far can be forged by anyone having enough computer power. Caum and Rojakkers (2001) developed, for any fixed set of users, an unconditionally secure signature scheme with the following properties: • Any participant can convince (except with exponentially small probability) any other participant that his signature is valid. • A convinced partipant can convince any other participant of the signature’s validity, without interaction with the original signer.