CHAPTER 9: User identification and message authentication, Secret sharing and E-commerce Most of today's applications of cryptography ask for authentic data rather than secret data. A practically very important problem is therefore how to protect data and communication against an active attacker (and noise). Main related problems to deal with are: • User identification (authentication): How can a person prove his (her) identity? USER IDENTIFICATION (AUTHENTICATION) User identification (authentication) is a process at which one party (often referred to as a Prover or Alice) convinces a second party often referred to as a Verifier or Bob) of Prover’s identity (that the Prover has actually participated in the identication process. In other words that the Prover has been active in the time the confirmative evidence of identity has been recquired). The purpose of any identification (authentication) process is to preclude (vylucit) some impersonation (zosobnenie) of one person (the Prover) by someone else. Identication usually serves to control access to a resource (often a resource should be accessed only by privileged users). OBJECTIVES of IDENTICATIONS User identification process has to satisfy the following objectives: • The Verifier has to accept Prover’s identity if both parties are honest; • The Verifier cannot later, after a successful identication, pose as the Prover and identicate himself (as the Prover) to another Verifier; • A dishonest party that would claim to be the other party has only negligible chance to identicate itself successfully; • Each of the above conditions remains true even if an attacker has observed or has participated in several identification protocols. USER IDENTIFICATION PROTOCOLS Identification protocols have to satisfy two security conditions: • If one party, say Bob (a verifier), gets a message from the other party, say Alice (a prover), then Bob is able to verify that the user is indeed Alice. • There is no way to pretend, for a third party, say Charles, when communicating with Bob, that he is Alice without Bob having a large chance to find out that. Identification system based on a PKC • Alice chooses a random r and sends e [B][ ](r) to Bob. • Alice identifies a communicating person as Bob if he can send her back r. • Bob identifies a communicating person as Alice if she can send him r. A misuse of the above system We show that (any non-honest) Alice could misuse the above identification scheme. Indeed, Alice could intercept a communication of a Jane ( a new “player'') with Bob, and get a cryptotext e [B][ ](w), the one Jana has been sending to Bob, and then Alice could send e [B][ ](w) to Bob. Honest Bob, who follows fully the protocol, would then return w to Alice and she would get this way the plaintext w. ELEMENTARY AUTHENTICATION PROTOCOLS USER IDENTIFICATION Static means of identification: People can be identified by their attributes (fingerprints), possessions (passports), or knowledge. Dynamic means of identification: Challenge and respond protocols. Both Alice and Bob share a key k and a one-way function f [k]. • Bob sends Alice a random number or string RAND. • Alice sends Bob PI = f [k][ ](RAND). • If Bob gets PI, then he verifies whether PI = f [k][ ](RAND). If yes, he starts to believe that the person he has communicated with is Alice. The process can be repeated to increase probability of a correct identification. Three-way authentication and key agreement A PKC will be used with encryption/decryption algorithms (e, d) and DSS with pairs (s, v). Alice and Bob will have their identity strings I[A] and I[B]. 1. Alice chooses a random r[A], sets t = (I[B], r[A]), signs sig[sA](t) and sends m[1] = (t, sig[sA](t)) to Bob. 2. Bob verifies Alice’s signature, chooses random r[B] and a random session key k. He encrypts k with Alice’s public key, E[eA](k) = c, sets t[1] = (I[A], r[A], r[B], c), signs it with sig[sB](t[1]). Then he sends m[2] = (t[1], sig[sB](t[1])) to Alice. Three-way authentication and key agreement 3. Alice verifies Bob’s signature, and checks that the r[A] she just got matches the one she generated in Step 1. Once verified, she is convinced that she is communicating with Bob. She gets k via D[dA](c) = D[dA](E[eA](k)) = k, sets t[2]= (I[B], r[B]) and signs it with sig[sA](t[2]). Then she sends m[3] = (t[2], sig[sA](t[2])) to Bob. 4. Bob verifies Alice’s signature and checks that r[B] he just got matches his choice in Step 2. If both verifications pass, Alice and Bob have mutually authenticated each other identity and have agreed upon a session key k. DATA AUTHENTICATION The goal of data authentication schemes (protocols) is to handle the case that data are sent through insecure channels. By creating so-called Message Authentication Code (MAC) a sending this MAC, together with a message through an insecure channel, one can create possibility to verify whether data were not changed in the channel. The price to pay is that communicating parties need to share a secret random key that need to be transmitted through a very secure channel.l Schemes for Data Authentication Basic difference between MACs and digital signatures is that MACs are symmetric. Anyone who is able to verify MAC of a message is also able to generate the same MAC, and vice versa. A scheme (M, T, K) for a data authentication is given by: – M is a set of possible messages (data) – T is a set of possible MACs – K is a set of possible keys Moreover, it is required that – to each k from K there is a single and easy to compute authentication mapping auth[k]: {0,1}* x M ® T – and a single easy to compute verification mapping ver[k]: M x T ® {true, false} Two conditions should be satisfied for such a scheme: Correctness: For each m from M and k from K it holds ver[k](m, c) = true, if there exists an r from {0, 1}* such that c = aut[k](r, m) Security: For any m from M and k from K it is computationally unfeasible, without a knowledge of k, to find c from T such that ver[k](m, c) = true FROM BLOCK CIPHERS to MAC – CBC-MAC Let C be an encryption algorithm that maps kbit strings into kbit strings. If a message m = m[1]m[2]...m[l] is divided into blocks of length k, then socalled CBCmode of encryption assumes a choice (random) of a special block y[0] of length k, and performs the following computations for i = 1, . . . ,l y[i] = C(y[i-1] Å m[i]) and then y[1]||y[2] || . . . ||y[l] is the encryption of m and y[l] is MAC for m. A modification of this method is to use another cryptoalgoritm to encrypt the last block m[l]. WEAKNESS of CBS-MAC METHOD Let us have three pairs: a message and its MAC (m[1], c[1]), (m[2], c[2]), (m[3], c[3]) Where m[1] and m[3][ ]have the same length and m[2] = m[1]||B||m’[2]. and let the length of B be also k. The encryption of the block B within m[2] is C(B Å c[1]). If we now define B’ = B Å c[1] Å c[3] , m[4] = m[3]||B’||m’[2] , then, during the encryption of m[4], we get C(B’ Å c[3]) = C(B Å c[1]), This implies that MAC's for m[4] and m[2] are the same. One can therefore forge a new valid pair (m[4], c[2]). ANALYSIS of CBCMAC – a view Theorem Given are two independent random permutations C[1] and C[2] on the set of message blocks M of cardinality n. Let us define MAC(m[1], m[2], . . . , m[l]) = C[2](C[1](...C[1](C[1](m[1]) Å m[2]) Å... Å m[l-1 ]Å m[l]). Let us assume that the MAC function be implemented by an oracle, and consider an adversary who can send queries to the oracle with a limited total length of q. If m[1], ..., m[d] denote the finite block sequences on M which are sent by the adversary to the oracle and let the total number of blocks be less than q. Let the purpose of the adversary be to output a message m which is different from all m[i] together with its MAC value c. Then the probability of success of the adversary (i.e. the probability that the MAC value is correct) is smaller than When q = qn^1/2, this is approximately a = q^2/2 (which is greater than 1 – e^-a ) Implication: if the total length of all authenticated messages is negligible against # n, then there is no better way than the brute force attack to get collisions on the CBCMAC. FROM HASH FUNCTIONS TO MAC So called HMAC was published as the internet standard RFC2104. Let a hash function h processes messages by blocks of b bytes and produces a digest of l bytes and let t be the size of MAC, in bytes. HMAC of a message m with a key k is computed as follows: • If k has more than b bytes replace k with h(k). • Append zero bytes to k to have exactly b bytes. • Compute (using strings opad and ipad defined later) h(k Å opad||h(k Å ipad||m)). and truncate the results to its t leftmost bytes to get HMAX[k](m). In HMAX ipad (opad) consists of b bytes equal to 0x36 (0x5c) hexadecimal. SECURITY of HMAC It can be shown that if • h(k Å ipad||m) defines a secure MAC on fixed length messages, and • h is collision free, then HMAC is a secure MAC on variable length messages with two independent keys. More precisely: Theorem Let h be a hash function which hashes into l bits. Given k[1], k[2] from {0, 1}^l consider the following MAC algorithm MAC[k1,k2](m) = h(k[2]||h(k[1]||m)) If h is collision free and m ® h(k[2]||m) is a secure MAC algorithm for messages m of the fixed length l, then the MAC is a secure MAC algorithm for messages of arbitrary length. Disadvantage of static user identification schemes Everybody who knows your password or PIN can impersonate you. Using so called zero-knowledge identification schemes, discussed in next chapter, you can identify yourself without giving to the identificator the ability to impersonate you. Simplified Fiat-Shamir identification scheme A trusted authority (TA) chooses: large random primes p,q , computes n = pq; and chooses a quadratic residue v Î QR [n], and s such that s ^2^ = v (mod n). public-key: v private-key: s (that Alice knows, but not Bob) Identification protocol (1) Alice chooses a random r < n, computes x = r ^2^ mod n and sends x to Bob. (2) Bob sends to Alice a random bit (a challenge) b. (3) Alice sends Bob (a response) y = rs ^b mod n (4) Bob identifies the sender as Alice if and only if y ^2^ = xv ^b^ mod n, what is taken as a proof that the sender knows square roots of x and of v. Analysis of Fiat-Shamir identification I public-key: v private-key: s (of Alice) such that s ^2^ = v. Protocol • Alice chooses a random r < n, computes x = r ^2 mod n and sends x (her commitment) to Bob. Analysis of Fiat-Shamir identification II Analysis • The first message is a commitment by Alice that she knows square root of x. • The second message is a challenge by Bob. • If Bob sends b = 0, then Alice has to open her commitment and reveals r. • If Bob sends b = 1, the Alice shows her secret s in an “encrypted form''. • The third message is Alice's response to the challenge of Bob. HOW CAN A BAD EVE CHEAT? Eve can send, to fool Bob, as her commitment, either for a random r or In the first case Eve can respond correctly to the Bob’s challenge b=0, by sending r; but cannot respond correctly to the challenge b = 1. In the second case Eve can respond correctly to Bob’s challenge b = 1, by sending r again; but cannot respond correctly to the challenge b = 0. Eve has therefore a 50% chance to cheat. Fiat-Shamir identification scheme parallel version In the following parallel version of Fiat-Shamir idenitification scheme the probability of false identification is decreased. Choose primes p,q, compute n = pq. Choose quadratic residues v [1],…,v [k][ ]Î QR [n]. Compute s [1],…,s [k] such that public-key: v [1],…,v [k][ ] secret-key: s [1],…,s [k] of Alice (1) Alice chooses a random r < n, computes a = r ^2^ mod n and sends a to Bob. (2) Bob sends Alice a random k-bit string b [1]… b [k]. (3) Alice sends to Bob (4) Bob accepts if and only if Alice and Bob repeat this protocol t times, until Bob is convinced that Alice knows s[1],…,s[k] . The chance that Alice fools Bob is 2 ^-kt, a decrease comparing with the chance 1/2 of the previous version of the identification scheme. The Schnorr identification scheme - setting This is a practically attractive and computationally efficient (in time, space + communication) scheme which minimizes storage + computations performed by Alice (to be a smart card). Scheme requires also a trusted authority (TA) which (1) chooses: a large prime p < 2 ^512, a large prime q dividing p -1 and q £ 2 ^140, an a Î Z [p]* of order q, a security parameter t such that 2 ^t^ < q, p, q, a, t are made public. (2) establishes: a secure digital signature scheme with a secret signing algorithm sig [TA] and a public verification algorithm ver [TA]. Schnorr identification scheme 1. Alice chooses a random 0 £ k < q and computes g = a ^k[ ]mod p. Okamoto identification scheme The disadvantage of the Schnorr identification scheme is that there is no proof of its security. For the modification of the Schnorr identification scheme presented below, a proof of security exists. Basic setting: To set up the scheme the TA chooses: • a large prime p £ 2 ^512, • a large prime q ³ 2 ^140 dividing p -1; • two elements a [1], a [2] Î Z [p]* of order q. TA makes public p, q, a [1], a [2] and keeps secret (also before Alice and Bob) c = lg[a][1] a [2]. Finally, TA chooses a signature scheme and a hash function. Okamoto identification scheme – basics once more Basic setting TA chooses: a large prime p £ 2 ^512,large prime q ³ 2 ^140 dividing p -1; two elements a [1], a [2]Î Z [p]* of order q. TA keep secret (also from Alice and Bob) c = lg[a][1] a [2]. Issuing a certificate to Alice • TA establishes Alice's identity and issues an identification string ID(Alice). • Alice randomly chooses 0 £ a [1], a [2][ ]£ q -1 and sends to TA. v = a[1 ]^-a1a [2 ]^-a^2 mod p. • TA generates a signature s = sig [TA](ID(Alice), v) and sends to Alice the certificate C (Alice) = (ID(Alice), v, s). Okamoto identification scheme Okamoto identification scheme • Alice chooses random 0 £ k[1], k[2][ ]£ q -1 and computes g = a[1 ]^k^1a [2 ]^k2 mod p. Authentication codes They provide methods of ensuring integrity of messages - that a message has not been tampered/changed, and that message originated with the presumed sender. The goal is to achieve authentication even in the presence of Mallot, a man in the middle, who can observe transmitted messages and replace them by messages of his own choise. Formally, an authentication code consists: • A set M of possible messages. • A set T of possible authentication tags. • A set K of possible keys. • A set R of authentication algorithms a [k]: M ® T, one for each k Î K Attacks and deception probabilities There are two basic types of attacks Mallot, the man in the middle,can do. Impersonation. Mallot introduces a message (w, t) into the channel expecting that message will be received as being sent by Alice. Substitution. Mallot replaces a message (w, t) in the channel by a new one, (w', t'), expecting that message will be accepted as being sent by Alice. With any impersonation (substitution) attack a probability P [i] (P [s]) is associated that Mallot will deceive Bob, if Mallot follows an optimal strategy. In order to determine such probabilities we need to know probability distributions p [m] on messages and p [k] on keys. In the following so called |K| ´ |M| authentication matrices will tabulate all authenticated tags. The item in a row corresponding to a key k and in a column corresponding to a message w will contain the authentication tag t [k][ ](w). The goal of authentication codes, to be discussed next, is to decrease probabilities that Mallot performs successfully impersonation or substitution. Example Let M = T = Z[3], K = Z[3] ´ Z[3]. For (i, j) Î K and w Î M, let t[ij](w) = (iw + j) mod 3. The matrix key x message of authentication tags has the form Computation of deception probabilities I Probability of impersonation: For w Î M, t Î T, let us define payoff(w, t) to be the probability that Bob accepts the message (w, t) as authentic. Then (4) (5) In other words, payoff(w, t) is computed by selecting the rows of the authentication matrix that have entry t in column w and summing probabilities of the corresponding keys. Therefore P [I][ ]= max {payoff (w, t), | w Î M, t Î A}. Computation of deception probabilities II Since Mallot wants to maximize his chance of deceiving Bob, he needs to compute p [w,t] = max {payoff(w', t', w, t) | w‘Î M, w ¹ w', t' Î A}. p [w,t] therefore denotes the probability that Mallot can deceive Bob with a substitution in the case (w, t) is the message observed. If Pr[Ma](w, t) is the probability of observing a message (w, t) in the channel, then and The next problem is to show how to construct an authentication code such that the deception probabilities are as low as possible. The concept of orthogonal arrays, introduced next, serves well such a purpose. Orthogonal arrays Definition An orthogonal array OA(n, k, l) is a ln ^2 ´ k array of n symbols, such that in any two columns of the array every one of the possible n ^2 pairs of symbols occurs in exactly l rows. Example OA(3,3,1) obtained from the authentication matrix presented before; Construction and bounds for OAs In an orthogonal array OA(n, k, l) • n determines the number of authenticators (security of the code); • k is the number of messages the code can accommodate; • l relates to the number of keys - ln ^2. The following holds for orthogonal arrays. • If p is prime, then OA(p, p, 1) exits. • Suppose there exists an OA(n, k, l). Then • Suppose that p is a prime and d £ 2 an integer. Then there is an orthogonal array OA(p, (p ^d^ -1)/(p -1), p ^d-2). • Let us have an authentication code with |A| = n and P [i] = P [s] = 1/n.Then |K| ³ n ^2. Moreover, |K| = n ^2 if and only if there is an orthogonal array OA(n, k,1), where |M| = k and P [K] (k) = 1/n ^2 for every key k Î K. The last claim shows that there are no much better approaches to authentication codes with deception probabilities as small as possible than orthogonal arrays. Secret sharing between two parties A moderator distributes a binary-string secret s, between two parties P[1] and P[2] by choosing a random binary string b, of the same length as s, and • by sending b to P[1] and • by sending s Å b to P[2]. This way, none of the parties P[1] and P[2] alone has a slightest idea about s, but both together easily recover s by computing b Å (s Å b) = s. Threshold secret sharing schemes Secret sharing schemes distribute a “secret'' among several users in such a way that only predefined sets of users can “assemble'' the secret. For example, a vault in the bank can be opened only if at least two out of three responsible employees use their knowledge and tools to open the vault. An important special simple case of secret sharing schemes are threshold secret sharing schemes at which a certain threshold of participant is needed and sufficient to assemble the secret. Shamir's (n,t)-threshold scheme Initiation phase: Dealer D chooses a prime p, n distinct x [i], 1 £ i £ n and D gives the values x [i] to the user P [i]. The values x [i] are public. Shamir's scheme - technicalities Shamir's scheme uses the following result concerning polynomials over fields Z[p], where p is prime. Theorem Let be a polynomial of degree t -1 and let P = {(x [i], f(x [i])) | x [i] Î Z[p], i =1,…,t, x [i] ¹ x [J], i ¹ j }. For Q Í P, let P [Q][ ]= { g Î Z [p][ ][x] | deg(g) = t -1, g(x) = y for all (x,y) Î Q}. Then it holds: • P[P][ ]= {f(x)}, i.e. f is the only polynomial of degree t -1, whose graph contains all t points in P. • If Q Ì P is a proper subset of P and x ¹ 0 for all (x, y) Î Q, then each a Î Z [p] appears with the same frequency as the constant coefficient of polynomials in P[Q]. Shamir's (n,t)-threshold scheme - summary To distributes n shares of a secret S among users P [1],…, P [n] a trusted authority TA proceeds as follows: • TA chooses a prime p > max{S, n} and sets a [0][ ]= S. • TA selects randomly a [1],…, a [t-1] Î Z [p] and creates polynomial • TA computes s [i ]= f (i), i = 1,…, n and transfers (i, s [i]) to the user P [i] in a secure way. Any group J of t or more users can compute the secret. Indeed, from the previous corollary we have In case |J| < t, then each a € Z [p] is likely to be the secret. SECRET SHARING – GENERAL CASE A serious limitation of the threshold secret sharing schemes is that all groups of users with the same number of users have the same access to secret. Practical situations usually require that some (sets of) users are more important than others. Let P be a set of users. To deal with above situation such concepts as authorized set of user and access structure are used. An authorized set of users is a set of users who can together construct the secret. An unauthorized set of users is a set of users who alone cannot learn anything about the secret. Let P be a set of users. The access structure is a set such that for all authorized sets A and for all unauthorized sets U. Theorem: For any access structure there exists a secret sharing scheme realizing this access structure. Secret Sharing Schemes with Verification • Secret sharing protocols increase security of a secret information by sharing it between several subjects. • Some secret sharing scheme are such that they work even in case some participants behave incorrectly. • A secret sharing scheme with verification is such a secret sharing scheme that: – Each P[i] is capable to verify correctness of his/her [– ]share s[i] – No participant P[i] is able to provide incorrect information and to convince others about its correctness Feldman’s (k,n)-Protocol Feldman’s protocol is an example of the secret sharing scheme with verification. The protocol is a generalization of Shamir's protocol. It is assumed that all participants can broadcast messages to all others and each of them can determine all senders.. Given are large primes p, q, q|(p - 1), q > n and h < p a generator of Z*[p] . All these numbers, and also the number g = h^(p-1)/q mod p, are public. As in Shamir's scheme, the dealer assigns to each P[i] a specific x[i] from {1, . . . , q – 1}, generates a random polynomial f(x) = (1) such that f(0) = s and sends to each P[i] value y[i] = f(x[i]). In addition, using a broadcasting scheme, the dealer sends to all P[i] all values v[j] = g^aj mod p. Feldman’s (k,n)-Protocol (cont.) Each P[i] verifies that If (1) does not hold, P[i] asks, using the broadcasting scheme, the dealer to broadcast correct value of y[i]. If there are at least k such requests, or some of the new values of y[i] does not satisfies (1), the dealer is considered as not reliable. One can easily verify that if the dealer works correctly, then all relations (1) hold E-COMMERCE Very important is to ensure security of e-money transactions needed for e-commerce. In addition to providing security and privacy, the task is to prevent alterations of purchase orders and forgery of credit card information. DUAL SIGNATURE PROTOCOL We present a protocol to solve the following security and privacy problem in e-commerce: shoppers banks should not know what cardholders are ordering and shops should not learn credit cards numbers. Participants of our e-commerce protocol: a bank, a cardholder, a shop The cardholder uses the following information: • GSO - Goods and Service Order (cardholder's name, shop's name, items being ordered, their quantity,...) • PI - Payment instructions (shop's name, card number, total price,...) Protocol uses a public hash function h. RSA cryptosystem is used and • e [C], e [S] and e [B] are public keys of cardholder, shop, bank and • d [C], d [S] and d [B] are their secret keys. CARDHOLDER and SHOP ACTIONS A cardholder performs the following procedure--GSO-goods and service order • Computes HEGSO = h (e [S](GSO)) - hash value of the encryption of GSO. • Computes HEPI = h (e [B](PI)) - hash value of the encryption of the payment instructions. • Computes HPO = h (HEPI || HEGSO) - Hash values of the Payment Order. • Signs HPO by computing “Dual Signature'' DS = d [C](HPO). • Sends e [S](GSO), DS, HEPI, and e [B](PI) to shop. BANK and SHOP ACTIONS Bank has received HEPI, HEGSO, e [B](PI), and DS and performs the following actions. • Computes h (e [B](PI)) - what should be equal to HEPI. • Computes h (h (e [B](PI)) || HEGSO) what should be equal to e [C](DS) = HPO. • Computes d [B](e [B](PI)) to obtain PI; • Returns an encrypted (with e [S]) digitally signed authorization to shop, guaranteeing the payment. Shop completes the procedure by encrypting, with e [C,] the receipt to the cardholder, indicating that transaction has been completed. It is easy to verify that the above protocol fulfils basic requirements concerning security, privacy and integrity. DIGITAL MONEY Is it possible to have electronic (digital) money? It seems that not, because copies of digital information are indistinguishable from their origin and one could therefore hardly prevent double spending,.... T. Okamoto and K. Ohia formulated six properties digital money systems should have. • One should be able to send e-money through e-networks. • It should not be possible to copy and reuse e-money. • Transactions using e-money should be done off-line - that is no communication with central bank should be needed during translation. • One should be able to sent e-money to anybody. • An e-coin could be divided into e-coins of smaller values. Several system of e-money have been created that satisfy all or at least some of the above requirements. BLIND SIGNATURES - applications Blind digital signatures allow the signer (bank) to sign a message without seeing its content. Scenario: Customer Bob would like to give e-money to Shop. E-money have to be signed by a Bank. Shop must be able to verify Bank's signature. Later, when Shop sends e-money to Bank, Bank should not be able to recognize that it signed these e-money for Bob. Bank has therefore to sign money blindly. Bob can obtain a blind signature for a message m from Bank by executing the Shnorr blind signature protocol described on the next slide. BLIND SIGNATURES - protocols 1. Shnorr's simplified identification protocol in which Bank proves its identity by proving that it knows x. • Bank chooses a random r Î {0,…,q -1} and send a = g ^ ^r to Bob. {By that Bank ``commits’’ itself to r}. • Bob sends to Bank a random c Î {0,…,q -1} {a challenge}. • Bank sends to Bob b = r – cx {a response}. • Bob accepts the proof that bank knows x if a = g ^b y ^c . {because y=g^x