# Digital signatures are one of the most important inventions/applications of modern cryptography.

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1 CHAPTER 7: DIGITAL SIGNATURES Digital signatures are one of the most important inventions/applications of modern cryptography. Part VII Digital signatures The problem is how can a user sign (electronically) an (electronic) message such that everybody (or the intended addressee only) can verify the signature that should be good enough also for legal purposes. Moreover, a properly implemented digital signature should give the receiver a reason to believe that the received message was created by the claimed sender and was not altered in transit. In many countries digital signatures have legal significance. prof. Jozef Gruska IV Digital signatures 2/60 DIGITAL SIGNATURES - OBSERVATION HASH FUNCTIONS - SUMMARY 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. Cryptographically secure hash functions h are such one-way functions that map huge sets S to much smaller sets H (of hashes/fingerprints ) and have the following properties: 1 For no w S it is feasible to find a w 1 S such that h(w) = h(w 1 ). 2 It is infeasible to find two different w, w 1 S such that h(w) = h(w 1 ). prof. Jozef Gruska IV Digital signatures 3/60 prof. Jozef Gruska IV Digital signatures 4/60

2 BASIC IDEAS EXAMPLE BASIC IDEAS Example: Assume that each user A can use a special public-key cryptosystem (e A,d A ). One way to sign a message w by a user A so that any user can verify the signature, is to apply on w (as the signing procedure) the mapping d A : d A (w) signature verification: e A (d A (w)) = w One way to sign a message w by a user A so that only the user B can verify the signature, is to apply on w (as the signing procedure) at first the mapping d A and then, on the outcome, e B : e B (d A (w)) signature verification: e A (d B (e B (d A (w)))) = w A way to send a message w, and a signed message digest of w, obtained by using a hash function h so that any responder can verifier the signature: (w, d A (h(w))) Example Assume Alice succeeds to factor the integer Bob used, as modulus, to sign his will, using RSA, 20 years ago. Even if the key has already expired, Alice can rewrite Bob s will, leaving fortune to her, and date it 20 years ago. Moral: It may pay off to factor a single integers using many very powerful computers for many years. prof. Jozef Gruska IV Digital signatures 5/60 DIGITAL SIGNATURES - BASIC GOALS prof. Jozef Gruska IV Digital signatures 6/60 DIGITAL SIGNATURES - BASIC MODES Basic requirements - I. Digital signatures 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 behalf of any other user. Basic requirements - II A valid digital signature should give the recipient reasons to believe that the message was created by a known sender and that it was not altered in transit. Note 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 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 taken that digital signatures are not misused. This chapter contains some of the main techniques for design and verification of digital signatures (as well as some possible attacks on them). If only signature (but not the encryption of the message) are of importance, then it suffices that Alice sends to Bob Caution: Signing a message w by A for B by (w, d A (w)) e B (d A (w)) is O.K., but the symmetric solution, with encoding first: is not good. c = d A (e B (w)) Indeed, an active enemy, the tamperer, can intercept the message, then can compute d T (e A (c)) = d T (e B (w)) and can send the outcome to Bob, pretending that it is from him/tamperer (without being able to decrypt/know the message). Any public-key cryptosystem in which the plaintext and cryptotext spaces are the same can be used for digital signature. prof. Jozef Gruska IV Digital signatures 7/60 prof. Jozef Gruska IV Digital signatures 8/60

3 WHY TO SIGN HASHES of MESSAGES and not MESSAGES THEMSELVES A SCHEME of DIGITAL SIGNATURE SYSTEMS SIMPLIFIED VERSION There are several reasons why it is better to sign hashes of messages than messages themselves. For efficiency: Hashes are much shorter and so are their signatures - this is a way to save resources (time,...) For compatibility Messages are typically bit strings. Digital signature schemes, such as RSA, operate often on other domains. A hash function can be used to convert an arbitrary input into the proper form. For integrity: If hashing is not used, a message has to be often split into blocks and each block signed separately. However, the receiver may not able to find out whether all blocks have been signed and in the proper order. A digital signature system (DSS) consists of: P - the space of possible plaintexts (messages). S - the space of possible signatures. K - the space of possible keys. For each k K there is a signing algorithm sig k and a corresponding verification algorithm ver k such that and sig k : P S. ver k : P S {true, false} ver k (w, s) = ( true if s = sig k (w);, false otherwise. Algorithms sig k and ver k should be computable in polynomial time. Verification algorithm can be publicly known; signing algorithm (actually only its key) should be kept secret prof. Jozef Gruska IV Digital signatures 9/60 prof. Jozef Gruska IV Digital signatures 10/60 DIGITAL SIGNATURE SCHEMES I DIGITAL SIGNATURES SCHEMES II - conditions Digital signature schemes are basic tools for authentication and non-repudiation 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 - one for each signer K v - a set of public keys for verification - one for each signer 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} M S For each k from K v there exists a single and easy to compute verification mapping ver k : M S {true, false} Correctness: For each message m from M and public key k in K v, it should hold if there is an r from {0, 1} such that ver k (m, s) = true 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 should be 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. such that the following two conditions are satisfied: prof. Jozef Gruska IV Digital signatures 11/60 prof. Jozef Gruska IV Digital signatures 12/60

4 ATTACK MODELS on DIGITAL SIGNATURES LEVELS of BREAKING of DIGITAL SIGNATURES Basic attack models KEY-ONLY ATTACK : The attacker is only given the public verification key. KNOWN SIGNATURES ATTACK : The attacker is given valid signatures for several messages known but not chosen by the attacker. CHOSEN SIGNATURES ATTACK : The attacker is given valid signatures for sever al messages chosen by the attacker. ADAPTIVE CHOSEN SIGNATURES ATTACKS : The attacker is given valid signatures for sever al messages chosen by the attacker where messages chosen may depend on previous signatures given for chosen messages. 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 a priory 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). Observe that to forge a signature scheme means to produce a new signature - it is not forgery to obtain from the signer a valid signature. prof. Jozef Gruska IV Digital signatures 13/60 A DIGITAL SIGNATURE of one BIT Let us start with a very simple, but much illustrative (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 and kept secret by the signer, and three items are made public, where Signature of a bit b: Verification of such a signature f, (0, s 0 ), (1, s 1 ) s 0 = f (k 0 ), s 1 = f (k 1 ) (b, k b ). s b = f (k b ) SECURITY? prof. Jozef Gruska IV Digital signatures 15/60 prof. Jozef Gruska IV Digital signatures 14/60 FROM RSA CRYPTOSYSTEM to RSA SIGNATURES The idea of RSA cryptosystem is simple. Public key: modulus n = pq and encryption exponent e. Secret key: decryption exponent d and primes p, q Encryption of a message w: c = w e Decryption of the cryptotext c: w = c d. Does it has a sense to change the order of these two operations: To do first and then c = w d Is this a crazy idea? No, we just ned to call these operations differently. Indeed, c e? s = w d will be seen as the signature of the message w and as a verification of such the signature w = s e prof. Jozef Gruska IV Digital signatures 16/60

5 RSA SIGNATURES and ATTACKS on them Let us have an RSA cryptosystem with encryption and decryption exponents e and d and modulus n. Signing of a message w: s = (w, σ), where σ = w d mod n ENCRYPTIONS versus SIGNATURES Let each user U use a cryptosystem with encryption and decryption algorithms: e U, d U Let w be a message Verification of a signature s = (w, σ): w = σ e mod n? Attacks Encryption: Decryption: PUBLIC-KEY ENCRYPTIONS e U (w) d U (e U (w)) It might happen that Bob accepts a signature not produced by Alice. Indeed, let Eve, using Alice s public key, compute w e and say that (w e, w) is a message signed by Alice. Everybody verifying such a signature as Alice s signature gets w e = w e. Some new signatures can be produced without knowing the secret key. Indeed, is σ 1 and σ 2 are signatures for w 1 and w 2, then σ 1 σ 2 and σ 1 1 are signatures for w 1 w 2 and w 1 1. prof. Jozef Gruska IV Digital signatures 17/60 FROM PKC to DSS - again Signing: Verification of the signature: PUBLIC-KEY SIGNATURES d U (w) e U (d U (w)) prof. Jozef Gruska IV Digital signatures 18/60 RABIN SIGNATURES 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). Signing of a message w by a user A so that only user B can verify the signature: e B (d A (w)). Sending a message w and a signed message digest of w obtained by using a (standard) hash function h: (w, d A (h(w))). 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)). A collision-resistant hash function h : {0, 1} {0, 1} k is used for some fixed k. Keys generation The signer S chooses primes p, q of size approximately k/2 and computes n = pq. n is the public key the pair (p, q) is the secret key. Signing : To sign a message w, the signer chooses random string U and calculates h(wu); If h(wu) QR(n), the signer picks a new U and repeats the process; Signer solves the equation x 2 = h(wu) mod n; The pair (U, x) is the signature of w. Verification Given a message w and a signature (U, x) the verifier V computes x 2 and h(wu) and verifies that they are equal. prof. Jozef Gruska IV Digital signatures 19/60 prof. Jozef Gruska IV Digital signatures 20/60

6 IMPORTANT FACTS ElGamal SIGNATURES Design of the ElGamal digital signature system: choose: prime p, integers 1 q x p, where q is a primitive element of Z p ; Fact 1 If a b (mod(p 1)) then for any integer x x a x b (modp) Fact 2 If a, n, x, y are integers and gcd(a, n) = 1, then x y (modφ(n)) implies a x a y (modn) Compute: y = q x mod p key K = (p, q, x, y) public key (p, q, y) - secret key: x Signature of a message w: Let r Zp 1 be randomly chosen and kept secret. sig(w, r) = (a, b), where a = q r mod p and b = (w xa)r 1 (mod (p 1)). Verification: accept a signature (a,b) of w as valid if y a a b = q w (mod p) (Indeed: y a a b q ax q rb q ax+w ax+k(p 1) q w (mod p)) prof. Jozef Gruska IV Digital signatures 21/60 ElGamal SIGNATURE - EXAMPLE prof. Jozef Gruska IV Digital signatures 22/60 SECURITY of ElGamal SIGNATURES Example choose: p = 11, q = 2, x = 8 compute: y = q x = 2 8 mod 11 = 3 message w = 5 is signed as (a,b) (where )a = q r mod p, w = xa + rb mod (p 1). If we choose r = 9 (this choice is O.K. because gcd(9, 10) = 1) then we compute a = q r = 2 9 mod 11 = 6 solve equation: b (mod 10) that is 7 9b (mod 10) and so we get b=3 signature is now: (6, 3) 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). 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 q r(w xa)r 1 q w xa q w y a ) what is infeasible. If Eve first chooses b and then tries to find a, she has to solve the equation y a a b q xa q rb q w (mod p). It is not known whether this equation can be solved for any given b efficiently. 2 If Eve chooses a and b and tries to determine w such that (a,b) is signature of w, then she has to compute discrete logarithm lg q y a a b. Hence, Eve can not sign a random message this way. prof. Jozef Gruska IV Digital signatures 23/60 prof. Jozef Gruska IV Digital signatures 24/60

7 From ElGamal to DSA (DIGITAL SIGNATURE STANDARD) DIGITAL SIGNATURE STANDARD I DSA is a digital signature standard, described on the next two slides, that is a modification of ElGamal digital signature scheme. It was proposed in August 1991 and adopted in December Any proposal for digital signature standard has to go through a very careful scrutiny. Why? Encryption of a message is usually done only once and therefore it usually suffices to use a cryptosystem that is secure at the time of the encryption. On the other hand, a signed message could be a contract or a will and it can happen that it will be needed to verify its signature many years after the message is signed. Since ElGamal signature is no more secure than discrete logarithm, it is necessary to use large p, with at least 512 bits. However, with ElGamal this would lead to signatures with at least 1024 bits what is too much for such applications as smart cards. 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. Design of DSA 1 The following global public key components are chosen: p - a random l-bit prime, 512 l 1024, l = 64k. q - a random 160-bit prime dividing p -1. r = h (p 1)/q mod p, where h is a random primitive element of Z p, such that r > 1, r 1 (observe that r is a q-th root of 1 mod p). 2 The following user s private key component is chosen: x - a random integer (once), 0 < x < q, 3 The following value is also made public y = r x mod p. 4 Key is K = (p, q, r, x, y) prof. Jozef Gruska IV Digital signatures 25/60 DIGITAL SIGNATURE STANDARD II Signing and Verification Signing of a 160-bit plaintext w choose random 0 < k < q compute a = (r k mod p) mod q compute b = k 1 (w + xa) mod q where kk 1 1 (mod q) signature: sig(w, k) = (a, b) Verification of signature (a, b) compute z = b 1 mod q compute u 1 = wz mod q, u 2 = az mod q verification: ver K (w, a, b) = true (r u 1 y u 2 mod p) mod q = a prof. Jozef Gruska IV Digital signatures 27/60 prof. Jozef Gruska IV Digital signatures 26/60 Fiat-Shamir SIGNATURE SCHEME Choose primes p, q, compute n = pq andq choose: as a public key integers v 1,..., v k and compute, as a secret key, s 1,..., s k, s i = v 1 i mod n. Protocol for Alice to sign a message w: 1 Alice first chooses (as a security parameter) an integer t, then t random integers 1 r 1,..., r t < n, and computes x i = ri 2 mod n, for 1 i t. 2 Alice uses a publicly known hash function h to compute H = h(wx 1 x 2... x t ) and then uses the first kt bits of H, denoted as b ij, 1 i t, 1 j k as follows. 3 Alice computes y 1,..., y t y i = r i ky j=1 s b ij j mod n 4 Alice sends to Bob w, all b ij, all y i and also h {Bob already knows Alice s public key v 1,..., v k } 5 Bob finally computes z 1,..., z k, where z i = y 2 i ky j=1 v b ij j mod n = ri 2 ky j=1 (v 1 j ) b ij ky j=1 v b ij j = ri 2 = x i and verifies that the first k t bits of h(wx 1 x 2... x t ) are the b ij values that Alice has sent to him. Security of this signature scheme is 2 kt. Advantage over the RSA-based signature scheme: only about 5% of modular multiplications are needed. prof. Jozef Gruska IV Digital signatures 28/60

8 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 their messages to be encrypted. Problem: Can Alice and Bob set up a subliminal channel, a covert communication channel between them, in full view of Walter, even though the messages themselves that they exchange contain no secret information? prof. Jozef Gruska IV Digital signatures 29/60 LAMPORT ONE-TIME SIGNATURES Lamport signature scheme shows how to construct a signature scheme for one use only - from any cryptographically secure 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 Y be chosen randomly and z ij = f (y ij ). The key K consists of 2k y s and z s. y s form the secret key, z s form the public key. Signing of a message x = x 1... x k {0, 1} k and sign(x 1... x k ) = (y 1,x1,..., y k,xk ) = (a 1,..., a k ) - notation verif (x 1... x k, a 1,..., a k ) = true f (a i ) = z i,xi, 1 i k Eve cannot forge a signature because she is unable to invert one-way functions. Important note: Lamport signature scheme can be used to sign only one message. prof. Jozef Gruska IV Digital signatures 31/60 Ong-Schnorr-Shamir SUBLUMINAL 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 communication channel between them, in full view of Walter, even though the messages themselves contain no secret information? Yes. Alice and Bob create first the following communication scheme: They choose a large n and an integer k such that gcd(n, k) = 1. They calculate h = k 2 mod n = (k 1 ) 2 mod n. They make h, n to be public key They keep secret k as trapdoor information. Let w be secret message Alice wants to send (it has to be such that gcd(w, n) =1) Denote a harmless message she uses by w (it has to be such that gcd(w,n) = 1) Signing by Alice: S 1 = 1 2 ( w w S 2 = k 2 ( w w + w) mod n w) mod n Signature: (S 1, S 2 ). Alice then sends to Bob (w, S 1, S 2 ) Signature verification method for Walter: w = S1 2 hs2 2 ( mod n) w Decryption by Bob: w = mod n (S 1 + k 1 S 2 ) prof. Jozef Gruska IV Digital signatures 30/60 LAMPORT SIGNATURE - ANOTHER VIEW A cryptographically secure hash function h : {0, 1} {0, 1} 256 is available as well as a cryptographically secure random number generator G. Keys generation: To create the private key the signer S uses the random number generator G to generate 256 pairs of random 256-bits numbers (in total bits - 16 KB in total).this will be secret key of the signer. To create the public key S, signer hashes each of the 512 random numbers in the private key, creating 512 hashes, each of 256 bits (16 KB in total). The 512 numbers form the public key. Signing: If the signer S wants to sign a message w she computes first h(w). Afterwards, to each of 256 bits in the hash, S picks the corresponding number from its private key - if the i-th bit is 0 (1), S chooses the first (second) number of the i-th pair.this gives the signer 256 random numbers (8 KB in total). These numbers are the signature of the message w. Verification: To verify the Signer s signature of the message w the verifier V first computes h(w).then V uses bits of the hash to pick out 256 of the hashes of Signer s public key.v picks the hashes in the same manner that signer picked the random numbers for the signature.the verifier then computes 256 hashes of the 256 random numbers of the Signer s signature. If these 256 hashes match exactly 256 hashes he just picked from Signer s public key then the signature is valid.if not something went prof. Jozef Gruska IV Digital signatures 32/60

9 Lamport signature - III. MERKLE SIGNATURES - I. An interest in the Lamport signatures steams to a large extend from the fact that it is believed that Lamport signatures with large hash functions will stay secure even when quantum computers are available. Merkle signature scheme with a parameter m = 2 n allows to sign any of the given 2 n messages (and no other). The scheme is based on so-called hash trees and uses a fixed collision resistant hash function h as well as Lamport one-time signatures and its security depends on their security. The main reason why Merkle Signature Scheme is of interest, is that it is believed to be resistant to attacks using quantum computers. prof. Jozef Gruska IV Digital signatures 33/60 MERKLE SIGNATURES - II. Public key generation - a single key for all signings At first one needs to generate public keys PK i and secret keys SK i for all 2 n messages m i and to compute also h(pk i ) for all i < 2 n. As the next step a complete binary tree with 2 n leaves is designed and the value h(pk i ) is stored in the i-the leave, counting from left to right. Moreover, to each internal node the hash of the concatenation of hashes of its two children is stored. The hash assigned to the root is the public key of the Merkle signature scheme and the tree is called Merkle tree. See next figure for a Merkle tree. prof. Jozef Gruska IV Digital signatures 34/60 MERKLE SIGNATURE - III. Signature generation To sign a message m i, this message is firs signed using the one-use signature scheme with keys (PK i, SK i ). This signature plus a sequence of n hashes chosen from all those nodes that are needed to compute the hash of the root, is the Merkle signature, see next figure where hashes assigned tom the gray node and a sequence of black nodes form the signature. The verifier knows the public key - hash assigned to the root and signature created as above. This allows him to compute all hashes assigned to the root from the leave to the root and to verify that the value assigned this way agrees with he public key - hash assigned to the root. hash of concatenation of hashes of children [h] hash of concatenation of hashes of children [h] h(pkk i) in the i th node h(pkk i) in the i th node prof. Jozef Gruska IV Digital signatures 35/60 prof. Jozef Gruska IV Digital signatures 36/60

10 GMR SIGNATURE SCHEME In 1988 Shafi Goldwasser, Silvio Micali and Ronald Rivest were the first to define rigorously security requirements for digital signature schemes. They also presented a new signature scheme, known nowadays as GMR signature scheme. It was the first signature scheme that was proven as being robust against an adaptive chosen message attacks:an adversary who receives signatures of messages of his choice (where each message may be chosen in a way that depends on the signatures of previously chosen messages) cannot later forge the signature even of a single additional message. prof. Jozef Gruska IV Digital signatures 37/60 TIMESTAMPING SIGNING of HASHES/FINGERPRINTS Signature schemes 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 short ones and to sign each block separately. Disadvantages: signing is slow and for long signatures integrity is not protected. The solution is to use a fast public hash function h which maps a message of any length to a fixed length hash. The hash is then signed. Example: message message digest El Gamal signature w z = h (w) y = sig(z) arbitrary length 160bits 320bits If Bob wants to send a signed message w he sends (w, sig(h(w)). prof. Jozef Gruska IV Digital signatures 38/60 BLIND SIGNATURES 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 protocol should provide a proof that a message was signed at a certain time. In the following pub denotes some publicly known information that could not be predicted before the day of the signature (for example, stock-market data). Timestamping by Bob of a signature on a message w, using a hash function h. Bob computes z = h(w); Bob computes z = h(z pub); { } denotes concatenation Bob computes y = sig(z ); Bob publishes (z, pub, y) in the next day newspaper. It is now clear that signature could not be done after the triple (z, pub, y) was published, but also not before the date pub was known. The basic idea is that Alice makes Bob to sign a message m without Bob knowing m, therefore blindly this is needed in e-commerce. Blind signing can be realized by a two party protocol, between the Alice and Bob, that has the following properties. In order to sign (by Bob) a message m, Alice creates, using a blinding procedure, from the message m a new message m from which m can not be obtained without knowing a secret, and sends m to Bob. Bob signs the message m to get a signature s m (of m ) and sends s m to Alice. The signing is to be done in such a way that Alice can afterwards compute, using an unblinding procedure, from Bob s signature s m of m Bob s signature s m of m. prof. Jozef Gruska IV Digital signatures 39/60 prof. Jozef Gruska IV Digital signatures 40/60

11 Chaum s BLIND SIGNATURE SCHEME FAIL-THEN-STOP SIGNATURES This blind signature protocol combines RSA with blinding/unblinding features. Let Bob s RSA public key be (n, e) and his private key be 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 1 s (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. 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 such a signature scheme should not be used any longer. 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 sees a forged signature he is then able to compute his secret key. By submitting this key to the trusted authority he can 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 signature Scheme. prof. Jozef Gruska IV Digital signatures 41/60 DIGITAL SIGNATURES with ENCRYPTION and RESENDING prof. Jozef Gruska IV Digital signatures 42/60 A SURPRISING ATTACK to the PREVIOUS SCHEME let us consider the following communication between Alice and Bob: 1 Alice signs the message: s A (w). 2 Alice encrypts the signed message: e B (s A (w)) and sends it to Bob. 3 Bob decrypts the signed message: d B (e B (s A (w))) = s A (w). 4 Bob verifies the signature and recovers the message v A (s A (w)) = w. Consider now the case of resending the message as a receipt 5 Bob signs and encrypts the message and sends to Alice e A (s B (w)). 6 Alice decrypts the message and verifies the signature. Assume now: v x = e x, s x = d x for all users x. 1 Mallot intercepts e B (s A (w)). 2 Later Mallot sends e B (s A (w)) to Bob pretending it is from him (from Mallot). 3 Bob decrypts and verifies the message by computing e M (d B (e B (d A (w)))) = e M (d A (w)) a garbage. 4 Bob goes on with the protocol and returns to Mallot the receipt: e M (d B (e M (d A (w)))) 5 Mallot can then get w. Indeed, Mallot can compute e A (d M (e B (d M (e M (d B (e M (d A (w)))))))) = w. prof. Jozef Gruska IV Digital signatures 43/60 prof. Jozef Gruska IV Digital signatures 44/60

12 ANOTHER MAN-IN-THE-MIDDLE ATTACK Consider the following protocol: 1 Alice sends the pair (e B (e B (w) A), B) to Bob. 2 Bob uses d B to get A and w, and acknowledges the receipt by sending the pair (e A (e A (w) B), A) to Alice. (Here the function e and d are assumed to operate on strings and identificators A, B,... are strings.) What can an active eavesdropper C do? C can learn (e A (e A (w) B), A) and therefore e A (w ) for w = e A (w) B. C can now send to Alice the pair (e A (e A w ) C), A). Alice, thinking that this is the step 1 of the protocol, acknowledges the receipt by sending the pair (e C (e C (w ) A), C) to C. C is now able to learn w and therefore also e A (w). C now sends to Alice the pair (e A (e A (w) C), A). Alice makes acknowledgment by sending the pair (e C (e C (w) A), C). C is now able to learn w. prof. Jozef Gruska IV Digital signatures 45/60 Diffie-Hellman PUBLIC ESTABLISHMENT of SECRET KEYS - repetition Main problem of the secret-key cryptography: a need to make a secure distribution (establishment) of secret keys ahead of transmissions. Diffie+Hellman solved this problem in 1976 by designing a protocol for secure key establishment (distribution) over public channels. Diffie-Hellman Protocol: If two parties, Alice and Bob, want to create a common secret key, then they first agree, somehow, on a large prime p and a q<p of large order in Z p and then they perform, through a public channel, the following activities. Alice chooses, randomly, a large 1 x < p 1 and computes X = q x mod p. Bob also chooses, again randomly, a large 1 y < p 1 and computes Y = q y mod p. Alice and Bob exchange X and Y, through a public channel, but keep x, y secret. Alice computes Y x mod p and Bob computes X y mod p and then each of them has the key K = q xy mod p. An eavesdropper seems to need, in order to determine x from X, q, p and y from Y, q, p, a capability to compute discrete logarithms, or to compute q xy from q x and q y, what is believed to be infeasible. prof. Jozef Gruska IV Digital signatures 47/60 PROBABILISTIC SIGNATURES SCHEMES - PSS Let us have integers k, l, n such that k + l < n, a trapdoor permutation a pseudorandom bit generator and a hash function f : D D, D {0, 1} n, G : {0, 1} l {0, 1} k {0, 1} n (l+k), w (G 1 (w), G 2 (w)) h : {0, 1} {0, 1} l. The following PSS scheme is applicable to messages of arbitrary length. Signing: of a message w {0, 1}. 1 Choose random r {0, 1} k and compute m = h(w r). 2 Compute G(m) = (G 1 (m), G 2 (m)) and y = m (G 1 (m) r) G 2 (m). 3 Signature of w is σ = f 1 (y). Verification of a signed message (w, σ). Compute f (σ) and decompose f (σ) = m t u, where m = l, t = k and u = n (k + l). Compute r = t G 1 (m). Accept signature σ if h(w r) = m and G 2 (m) = u; otherwise reject it. prof. Jozef Gruska IV Digital signatures 46/60 AUTHENTICATED Diffie-Hellman KEY EXCHANGE Let each user U has 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. 1 Alice and Bob choose large prime p and a generator q Z p. 2 Alice chooses a random x and Bob chooses a random y. 3 Alice computes q x mod p, and Bob computes q y mod p. 4 Alice sends q x to Bob. 5 Bob computes K = q xy mod p. 6 Bob sends q y and e K (s B (q y, q x )) to Alice. 7 Alice computes K = q xy mod p. 8 Alice decrypts e K (s B (q y, q x )) to obtain s B (q y, q x ). 9 Alice verifies, using an authority, that v B is Bob s verification algorithm. 10 Alice uses v B to verify Bob s signature. 11 Alice sends e K (s A (q x, q y )) to Bob. 12 Bob decrypts, verifies v A, and verifies Alice s signature. An enhanced version of the above protocol is known as Station-to-Station protocol. prof. Jozef Gruska IV Digital signatures 48/60

13 SECURITY of DIGITAL SIGNATURES It is very non-trivial to define security of a 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. prof. Jozef Gruska IV Digital signatures 49/60 A signature scheme is secure (or unforgeable) if HISTORY every feasible of DIGITAL chosen SIGNATURES message attack succeeds with at most negligible probability. In 1976 Diffie and Hellman were first to describe the idea of a digital signature scheme. However, they only conjectured that such schemes may exist. In 1977 RSA was invented that could be used to produce a primitive (not secure enough) digital signatures. The first widely marketed software package to offer digital signature was Lotus Notes 1.0, based on RSA and released in 1989 ElGamal digital signatures were invented in In 1988 Goldwasser, Micali and Rivest were first to rigorously define (perfect) security of digital signature schemes. prof. Jozef Gruska IV Digital signatures 51/60 THRESHOLD DIGITAL SIGNATURES The idea of a (t+1, n) threshold signature scheme is to distribute the power of the signing operation to (t+1) parties out of n. A (t+1) threshold 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 threshold RSA signature schemes. There is no proof yet whether Shoup s scheme is provably secure. prof. Jozef Gruska IV Digital signatures 50/60 APPENDIX APPENDIX prof. Jozef Gruska IV Digital signatures 52/60

14 GENERAL OBSERVATIONS - I. GENERAL OBSERVATIONS - II. Digital signatures are often used to implement electronic signatures - this is a broader term that refers to any electronic data that carries the intend of a signature. Not all electronic signatures use digital signatures. In some countries digital signatures have legal significance Properly implemented digital signatures are more difficult to forge than the handwritten ones. Digital signatures can also provide non-repudiation. This means that the signer cannot successfully claim: (a) that he did not signed a message, (b) his private key remain secret. Whitfield Diffie and Martin Hellman were the first, in 1976, to describe the idea of digital signatures, although they only conjectured that such schemes exist. The first broadly marketed software package to offer digital signature was Lotus Notes 1.0, released in 1989, which used RSA algorithm DSA was adopted in US as Federal Information Processing Standard for digital signatures in Adaptation was revised in 1996, 2000, 2009 and 2013 DES is covered by US-patent attributed to David W. Krantz (former NSA employe). Claus P. Schnor claims that his US patent covered DSA. prof. Jozef Gruska IV Digital signatures 53/60 SPECIAL TYPES of DIGITAL SIGNATURES prof. Jozef Gruska IV Digital signatures 54/60 GROUP 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 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 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. 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. Aggregate signatures They are signature schemes that support aggregation in the following sense: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users in such a way that this single signature will convince the verifier that the n users did indeed signed the n original messages. prof. Jozef Gruska IV Digital signatures 55/60 prof. Jozef Gruska IV Digital signatures 56/60

15 UNCONDITIONALLY SECURE DIGITAL SIGNATURES BIRTHDAY PARADOX ATTACK on DIGITAL SIGNATURE Assume Alice uses a hash function that produces 50 bits. Any of the digital signature schemes introduced so far can be forged by anyone having enough computer power. Chaum and Roijakkers (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 participant can convince any other participant of the signature s validity, without interaction with the original signer. Fred, who wants Alice to sign a fraudulent contract, find 30 places in a good document, where he can make change in the document (adding a coma, space,... ) such that Alice would not notice that. By choosing at each place whether to make or not a change, he can produce 2 30 documents essentially identical with the original good document. Similarly, Fred makes 2 30 changes of the fraudulent document. Considering birthday problem with n = 2 50, r = 2 30 we get that r = λn, with λ = 2 10 and therefore with probability 1 e there is a version of the good document that has the same hash as a version of the fraudulent document. Finding a match, Fred can ask Alice to sign a good version and then append the signature to the fraudulent contract. prof. Jozef Gruska IV Digital signatures 57/60 prof. Jozef Gruska IV Digital signatures 58/60 BREAKING CRYPTOSYSTEMS and DIGITAL SIGNATURES We say that an encryption system has been broken if one can determine a plaintext from a cryptotext (often). A digital signature system is considered as broken if one can (often) forge signatures. In both cases, a more ambitious goal is to find the private key. FORGING and MISUSING of ElGamal SIGNATURES There are ways 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 the pair a = q i y j mod p; b = aj 1 mod (p 1); w = aij 1 mod (p 1) (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 not used 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 the system can be broken. prof. Jozef Gruska IV Digital signatures 59/60 prof. Jozef Gruska IV Digital signatures 60/60

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