# Chapter 12. Digital signatures Digital signature schemes

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1 Chapter 12 Digital signatures In the public key setting, the primitive used to provide data integrity is a digital signature scheme. In this chapter we look at security notions and constructions for this primitive Digital signature schemes A digital signature scheme is just like a message authentication scheme except for an asymmetry in the key structure. The key sk used to generate signatures (in this setting the tags are often called signatures) is different from the key pk used to verify signatures. Furthermore pk is public, in the sense that the adversary knows it too. So while only a signer in possession of the secret key can generate signatures, anyone in possession of the corresponding public key can verify the signatures. Definition A digital signature scheme DS = (K, Sign, VF) consists of three algorithms, as follows: The randomized key generation algorithm K (takes no inputs and) returns a pair (pk, sk) of keys, the public key and matching secret key, respectively. We write (pk,sk) \$ K for the operation of executing K and letting (pk,sk) be the pair of keys returned. The signing algorithm Sign takes the secret key sk and a message M to return a signature (also sometimes called a tag) σ {0,1} { }. The algorithm may be randomized or stateful. We write σ \$ Sign sk (M) or σ \$ Sign(sk,M) for the operation of running Sign on inputs sk,m and letting σ be the signature returned. The deterministic verification algorithm VF takes a public key pk, a message M, and a candidate signature σ for M to return a bit. We write d VF pk (M,σ) or d VF(pk,M,σ) to denote the operation of running VF on inputs pk,m,σ and letting d be the bit returned. We require that VF pk (M,σ) = 1 for any key-pair (pk,sk) that might be output by K, any message M, and any σ that might be output by Sign sk (M). If Sign is stateless then we associate to each public key a message space Messages(pk) which is the set of all M for which Sign sk (M) never returns. Let S be an entity that wants to have a digital signature capability. The first step is key generation: S runs K to generate a pair of keys (pk,sk) for itself. Note the key generation algorithm is run locally by S. Now, S can produce a digital signature on some document M Messages(pk) by

4 4 DIGITAL SIGNATURES examples is more expensive than just computing on one s own. That is, we would expect q to be smaller than t. That is why q,µ are resources separate from t RSA based signatures The RSA trapdoor permutation is widely used as the basis for digital signature schemes. Let us see how Key generation for RSA systems We will consider various methods for generating digital signatures using the RSA functions. While these methods differ in how the signature and verification algorithms operate, they share a common key-setup. Namely the public key of a user is a modulus N and an encryption exponent e, where N = pq is the product of two distinct primes, and e Z ϕ(n). The corresponding secret contains the decryption exponent d Z ϕ(n) (and possibly other stuff too) where ed 1 (mod ϕ(n)). How are these parameters generated? We refer back to Definition 10.9 where we had introduced the notion of an RSA generator. This is a randomized algorithm having an associated security parameter and returning a pair ((N,e),(N,p,q,d)) satisfying the various conditions listed in the definition. The key-generation algorithm of the digital signature scheme is simply such a generator, meaning the user s public key is (N,e) and its secret key is (N,p,q,d). Note N is not really secret. Still, it turns out to be convenient to put it in the secret key. Also, the descriptions we provide of the signing process will usually depend only on N, d and not p, q, so it may not be clear why p,q are in the secret key. But in practice it is good to keep them there because their use speeds up signing via the Chinese Remainder theorem and algorithm. Recall that the map RSA N,e ( ) = ( ) e mod N is a permutation on ZN with inverse RSA N,d( ) = ( ) d mod N. Below we will consider various signature schemes all of which use the above key generation algorithm and try to build in different ways on the one-wayness of RSA in order to securely sign Trapdoor signatures Trapdoor signatures represent the most direct way in which to attempt to build on the one-wayness of RSA in order to sign. We believe that the signer, being in possession of the secret key N,d, is the only one who can compute the inverse RSA function RSA 1 N,e = RSA N,d. For anyone else, knowing only the public key N, e, this task is computationally infeasible. Accordingly, the signer signs a message by performing on it this hard operation. This requires that the message be a member of ZN, which, for convenience, is assumed. It is possible to verify a signature by performing the easy operation of computing RSA N,e on the claimed signature and seeing if we get back the message. More precisely, let K rsa be an RSA generator with associated security parameter k, as per Definition We consider the digital signature scheme DS = (K rsa,sign,vf) whose signing and verifying algorithms are as follows: Algorithm Sign N,p,q,d (M) If M Z N then return x M d mod N Return x Algorithm VF N,e (M,x) If (M Z N or x Z N ) then return 0 If M = x e mod N then return 1 else return 0 This is a deterministic stateless scheme, and the message space for public key (N,e) is Messages(N,e) = Z N, meaning the only messages that the signer signs are those which are elements of the group Z N.

5 Bellare and Rogaway 5 In this scheme we have denoted the signature of M by x. The signing algorithm simply applies RSA N,d to the message to get the signature, and the verifying algorithm applies RSA N,e to the signature and tests whether the result equals the message. The first thing to check is that signatures generated by the signing algorithm pass the verification test. This is true because of Proposition 10.7 which tells us that if x = M d mod N then x e = M mod N. Now, how secure is this scheme? As we said above, the intuition behind it is that the signing operation should be something only the signer can perform, since computing RSA 1 N,e (M) is hard without knowledge of d. However, what one should remember is that the formal assumed hardness property of RSA, namely one-wayness under known-exponent attack (we call it just one-wayness henceforth) as specified in Definition 10.10, is under a very different model and setting than that of security for signatures. One-wayness tells us that if we select M at random and then feed it to an adversary (who knows N,e but not d) and ask the latter to find x = RSA 1 N,e (M), then the adversary will have a hard time succeeding. But the adversary in a signature scheme is not given a random message M on which to forge a signature. Rather, its goal is to create a pair (M,x) such that VF N,e (M,x) = 1. It does not have to try to imitate the signing algorithm; it must only do something that satisfies the verification algorithm. In particular it is allowed to choose M rather than having to sign a given or random M. It is also allowed to obtain a valid signature on any message other than the M it eventually outputs, via the signing oracle, corresponding in this case to having an oracle for RSA 1 N,e ( ). These features make it easy for an adversary to forge signatures. A couple of simple forging strategies are illustrated below. The first is to simply output the forgery in which the message and signature are both set to 1. The second is to first pick at random a value that will play the role of the signature, and then compute the message based on it: Forger F Sign N,p,q,d ( ) 1 (N, e) Return (1,1) Forger F Sign N,p,q,d ( ) 2 (N, e) x \$ Z N ; M xe mod N Return (M,x) These forgers makes no queries to their signing oracles. We note that 1 e 1 (mod N), and hence the uf-cma-advantage of F 1 is 1. Similarly, the value (M,x) returned by the second forger satisfies x e mod N = M and hence it has uf-cma-advantage 1 too. The time-complexity in both cases is very low. (In the second case, the forger uses the O(k 3 ) time to do its exponentiation modulo N.) So these attacks indicate the scheme is totally insecure. The message M whose signature the above forger managed to forge is random. This is enough to break the scheme as per our definition of security, because we made a very strong definition of security. Actually for this scheme it is possible to even forge the signature of a given message M, but this time one has to use the signing oracle. The attack relies on the multiplicativity of the RSA function. Forger F Sign N,e ( ) (N,e) \$ M 1 ZN {1,M} ; M 2 MM1 1 mod N x 1 Sign N,e (M 1 ) ; x 2 Sign N,e (M 2 ) x x 1 x 2 mod N Return (M,x) Given M the forger wants to compute a valid signature x for M. It creates M 1,M 2 as shown, and obtains their signatures x 1,x 2. It then sets x = x 1 x 2 mod N. Now the verification algorithm will check whether x e mod N = M. But note that x e (x 1 x 2 ) e x e 1x e 2 M 1 M 2 M (mod N).

6 6 DIGITAL SIGNATURES Here we used the multiplicativity of the RSA function and the fact that x i is a valid signature of M i for i = 1,2. This means that x is a valid signature of M. Since M 1 is chosen to not be 1 or M, the same is true of M 2, and thus M was not an oracle query of F. So F succeeds with probability one. These attacks indicate that there is more to signatures than one-wayness of the underlying function The hash-then-invert paradigm Real-world RSA based signature schemes need to surmount the above attacks, and also attend to other impracticalities of the trapdoor setting. In particular, messages are not usually group elements; they are possibly long files, meaning bit strings of arbitrary lengths. Both issues are typically dealt with by pre-processing the given message M via a hash function to yield a point y in the range of RSA N,e, and then applying RSA 1 N,e to y to obtain the signature. The hash function is public, meaning its description is known, and anyone can compute it. To make this more precise, let K rsa be an RSA generator with associated security parameter k and let Keys be the set of all modulli N that have positive probability to be output by K rsa. Let Hash be a family of functions whose key-space is Keys and such that Hash N : {0,1} Z N for every N Keys. Let DS = (K rsa,sign,vf) be the digital signature scheme whose signing and verifying algorithms are as follows: Algorithm Sign N,p,q,d (M) y Hash N (M) x y d mod N Return x Algorithm VF N,e (M,x) y Hash N (M) y x e mod N If y = y then return 1 else return 0 Let us see why this might help resolve the weaknesses of trapdoor signatures, and what requirements security imposes on the hash function. Let us return to the attacks presented on the trapdoor signature scheme above. Begin with the first forger we presented, who simply output (1, 1). Is this an attack on our new scheme? To tell, we see what happens when the above verification algorithm is invoked on input 1, 1. We see that it returns 1 only if Hash N (1) 1 e (mod N). Thus, to prevent this attack it suffices to ensure that Hash N (1) 1. The second forger we had previously set M to x e mod N for some random x ZN. What is the success probability of this strategy under the hash-then-invert scheme? The forger wins if x e mod N = Hash(M) (rather than merely x e mod N = M as before). The hope is that with a good hash function, it is very unlikely that x e mod N = Hash N (M). Consider now the third attack we presented above, which relied on the multiplicativity of the RSA function. For this attack to work under the hash-then-invert scheme, it would have to be true that Hash N (M 1 ) Hash N (M 2 ) Hash N (M) (mod N). (12.1) Again, with a good hash function, we would hope that this is unlikely to be true. The hash function is thus supposed to destroy the algebraic structure that makes attacks like the above possible. How we might find one that does this is something we have not addressed. While the hash function might prevent some attacks that worked on the trapdoor scheme, its use leads to a new line of attack, based on collisions in the hash function. If an adversary can find two distinct messages M 1,M 2 that hash to the same value, meaning Hash N (M 1 ) = Hash N (M 2 ), then it can easily forge signatures, as follows:

7 Bellare and Rogaway 7 Forger F Sign N,p,q,d ( ) (N,e) x 1 Sign N,p,q,d (M 1 ) Return (M 2,x 1 ) This works because M 1,M 2 have the same signature. Namely because x 1 is a valid signature of M 1, and because M 1,M 2 have the same hash value, we have x e 1 Hash N (M 1 ) Hash N (M 2 ) (mod N), and this means the verification procedure will accept x 1 as a signature of M 2. Thus, a necessary requirement on the hash function Hash is that it be CR2-KK, meaning given N it should be computationally infeasible to find distinct values M,M such that Hash N (M) = Hash N (M ). Below we will go on to more concrete instantiations of the hash-then-invert paradigm. But before we do that, it is important to try to assess what we have done so far. Above, we have pin-pointed some features of the hash function that are necessary for the security of the signature scheme. Collision-resistance is one. The other requirement is not so well formulated, but roughly we want to destroy algebraic structure in such a way that Equation (12.1), for example, should fail with high probability. Classical design focuses on these attacks and associated features of the hash function, and aims to implement suitable hash functions. But if you have been understanding the approaches and viewpoints we have been endeavoring to develop in this class and notes, you should have a more critical perspective. The key point to note is that what we need is not really to pin-point necessary features of the hash function to prevent certain attacks, but rather to pin-point sufficient features of the hash function, namely features sufficient to prevent all attacks, even ones that have not yet been conceived. And we have not done this. Of course, pinning down necessary features of the hash function is useful to gather intuition about what sufficient features might be, but it is only that, and we must be careful to not be seduced into thinking that it is enough, that we have identified all the concerns. Practice proves this complacence wrong again and again. How can we hope to do better? Return to the basic philosophy of provable security. We want assurance that the signature scheme is secure under the assumption that its underlying primitives are secure. Thus we must try to tie the security of the signature scheme to the security of RSA as a one-way function, and some security condition on the hash function. With this in mind, let us proceed to examine some suggested solutions The PKCS #1 scheme RSA corporation has been one of the main sources of software and standards for RSA based cryptography. RSA Labs (now a part of Security Dynamics Corporation) has created a set of standards called PKCS (Public Key Cryptography Standards). PKCS #1 is about signature (and encryption) schemes based on the RSA function. This standard is in wide use, and accordingly it will be illustrative to see what they do. The standard uses the hash-then-invert paradigm, instantiating Hash via a particular hash function PKCS-Hash which we now describe. Recall we have already discussed collision-resistant hash functions. Let us fix a function h: {0,1} {0,1} l where l 128 and which is collisionresistant in the sense that nobody knows how to find any pair of distinct points M,M such that h(m) = h(m ). Currently the role tends to be played by SHA-1, so that l = 160. Prior to that it was MD5, which has l = 128. The RSA PKCS #1 standard defines PKCS-Hash N (M) = 0001FFFF FFFF00 h(m). Here denotes concatenation, and enoughff-bytes are inserted that the length of PKCS-Hash N (M) is equal to k bits. Note the the first four bits of the hash output are zero, meaning as an integer it

8 8 DIGITAL SIGNATURES is certainly at most N, and thus most likely in Z N, since most numbers between 1 and N are in Z N. Also note that finding collisions in PKCS-Hash is no easier than finding collisions in h, so if the latter is collision-resistant then so is the former. Recall that the signature scheme is exactly that of the hash-then-invert paradigm. For concreteness, let us rewrite the signing and verifying algorithms: Algorithm Sign N,p,q,d (M) y PKCS-Hash N (M) x y d mod N Return x Algorithm VF N,e (M,x) y PKCS-Hash N (M) y x e mod N If y = y then return 1 else return 0 Now what about the security of this signature scheme? Our first concern is the kinds of algebraic attacks we saw on trapdoor signatures. As discussed in Section , we would like that relations like Equation (12.1) fail. This we appear to get; it is hard to imagine how PKCS-Hash N (M 1 ) PKCS-Hash N (M 2 ) mod N could have the specific structure required to make it look like the PKCShash of some message. This isn t a proof that the attack is impossible, of course, but at least it is not evident. This is the point where our approach departs from the classical attack-based design one. Under the latter, the above scheme is acceptable because known attacks fail. But looking deeper there is cause for concern. The approach we want to take is to see how the desired security of the signature scheme relates to the assumed or understood security of the underlying primitive, in this case the RSA function. We are assuming RSA is one-way, meaning it is computationally infeasible to compute RSA 1 N,e (y) for a randomly chosen point y ZN. On the other hand, the points to which RSA 1 N,e is applied in the signature scheme are those in the set S N = { PKCS-Hash N (M) : M {0,1} }. The size of S N is at most 2 l since h outputs l bits and the other bits of PKCS-Hash N ( ) are fixed. With SHA-1 this means S N This may seem like quite a big set, but within the RSA domain ZN it is tiny. For example when k = 1024, which is a recommended value of the security parameter these days, we have S N Z N = This is the probability with which a point chosen randomly from Z N lands in S N. For all practical purposes, it is zero. So RSA could very well be one-way and still be easy to invert on S N, since the chance of a random point landing in S N is so tiny. So the security of the PKCS scheme cannot be guaranteed solely under the standard one-wayness assumption on RSA. Note this is true no matter how good is the underlying hash function h (in this case SHA-1) which forms the basis for PKCS-Hash. The problem is the design of PKCS-Hash itself, in particular the padding. The security of the PKCS signature scheme would require the assumption that RSA is hard to invert on the set S N, a miniscule fraction of its full range. (And even this would be only a necessary, but not sufficient condition for the security of the signature scheme.) Let us try to clarify and emphasize the view taken here. We are not saying that we know how to attack the PKCS scheme. But we are saying that an absence of known attacks should not be deemed a good reason to be satisfied with the scheme. We can identify design flaws, such as the way the scheme uses RSA, which is not in accordance with our understanding of the security of RSA as a one-way function. And this is cause for concern.

14 14 DIGITAL SIGNATURES Msg[j] The j-th hash query in the experiment Y [j] The reply of the hash oracle simulator to the above, meaning the value playing the role of H(Msg[j]). For j = i it is y. X[j] For j i, the response to sign query Msg[j], meaning it satisfies (X[j]) e Y [j] (mod N). For j = i it is undefined. The code for the inverter is below. Inverter I(N, e, y) Initialize arrays Msg[1... q hash ], X[1... q hash ], Y [1... q hash ] to empty j 0 ; i \$ {1,...,q hash } Run F on input (N,e) If F makes oracle query (hash,m) then h H -Sim(M) ; return h to F as the answer If F makes oracle query (sign,m) then x Sign-Sim(M) ; return x to F as the answer Until F halts with output (M,x) y H -Sim(M) Return x The inverter responds to oracle queries by using the appropriate subroutines. Once it has the claimed forgery, it makes the corresponding hash query and then returns the signature x. We now describe the hash oracle simulator. It makes reference to the global variables instantiated in in the main code of I. It takes as argument a value v which is simply some message whose hash is requested either directly by F or by the sign simulator below when the latter is invoked by F. We will make use of a subroutine Find that given an array A, a value v and index m, returns 0 if v {A[1],...,A[m]}, and else returns the smallest index l such that v = A[l]. Subroutine H -Sim(v) l Find(Msg,v,j) ; j j + 1 ; Msg[j] v If l = 0 then If j = i then Y [j] y \$ Else X[j] ZN ; Y [j] (X[j])e mod N EndIf Return Y [j] Else If j = i then abort Else X[j] X[l] ; Y [j] Y [l] ; Return Y [j] EndIf EndIf The manner in which the hash queries are answered enables the following sign simulator. Subroutine Sign-Sim(M) h H -Sim(M) If j = i then abort

15 Bellare and Rogaway 15 Else return X[j] EndIf Inverter I might abort execution due to the abort instruction in either subroutine. The first such situation is that the hash oracle simulator is unable to return y as the response to the i-th hash query because this query equals a previously replied to query. The second case is that F asks for the signature of the message which is the i-th hash query, and I cannot provide that since it is hoping the i-th message is the one in the forgery and has returned y as the hash oracle response. Now we need to lower bound the ow-kea-advantage of I with respect to K rsa. There are a few observations involved in verifying the bound claimed in Equation (12.2). First that the view of F at any time at which I has not aborted is the same as in Experiment Exp uf-cma DS (F). This means that the answers being returned to F by I are distributed exactly as they would be in the real experiment. Second, F gets no information about the value i that I chooses at random. Now remember that the last hash simulator query made by I is the message M in the forgery, so M is certainly in the array Msg at the end of the execution of I. Let l = Find(Msg,M,q hash ) be the first index at which M occurs, meaning Msg[l] = M but no previous message is M. The random choice of i then means that there is a 1/q hash chance that i = l, which in turn means that Y [i] = y and the hash oracle simulator won t abort. If x is a correct signature of M we will have x e Y [i] (mod N) because Y [i] is H(M) from the point of view of F. So I is successful whenever this happens PSS0: A security improvement The FDH-RSA signature scheme has the attractive security attribute of possessing a proof of security under the assumption that RSA is a one-way function, albeit in the random oracle model. However the quantitative security as given by Theorem could be better. The theorem leaves open the possibility that one could forge signatures with a probability that is q hash times the probability of being able to invert the RSA function at a random point, the two actions being measured with regard to adversaries with comparable execution time. Since q hash could be quite large, say 2 60, there is an appreciable loss in security here. We now present a scheme in which the security relation is much tighter: the probability of signature forgery is not appreciably higher than that of being able to invert RSA in comparable time. The scheme is called PSS0, for probabilistic signature scheme, version 0, to emphasize a key aspect of it, namely that it is randomized: the signing algorithm picks a new random value each time it is invoked and uses that to compute signatures. The scheme DS = (K rsa,sign,vf), like FDH-RSA, makes use of a public hash function H: {0,1} ZN which is modeled as a random oracle. Additonally it has a parameter s which is the length of the random value chosen by the signing algorithm. We write the signing and verifying algorithms as follows: Algorithm Sign H( ) N,p,q,d (M) r \$ {0,1} s y H(r M) x y d mod N Return (r,x) Algorithm VF H( ) N,e (M,σ) Parse σ as (r,x) where r = s y H(r M) If x e mod N = y Then return 1 else return 0 Obvious range checks are for simplicity not written explicitly in the verification code; for example in a real implementation the latter should check that 1 x < N and gcd(x, N) = 1.

17 Bellare and Rogaway 17 it picks a random X[j] in ZN and returns Y [j] = y (X[j])e mod N. The value X[j] is chosen randomly and independently each time. Now the fact that RSA N,e is a permutation means that all the different Y [j] values are randomly and independently distributed. Furthermore, suppose (M,(r,x)) is a forgery for which hash oracle query r M has been made and got the reponse Y [l] = y (X[l]) e mod N. Then we have (x X[l] 1 ) e y (mod N), and thus the inverse of y is x X[l] 1 mod N. The second problem however, cannot be resolved for FDH. That is exactly why PSS0 pre-pends the random value r to the message before hashing. This effectively separates the two kinds of hash queries: the direct queries of F to the hash oracle, and the indirect queries to the hash oracle arising from the sign oracle. The direct hash oracle queries have the form r M for some l-bit string r and some message M. The sign query is just a message M. To answer it, a value r is first chosen at random. But then the value r M has low probability of having been a previous hash query. So at the time any new direct hash query is made, I can assume it will never be an indirect hash query, and thus reply via the above trick. Here now is the full proof. Proof of Theorem : We first decribe I in terms of two subroutines: a hash oracle simulator H -Sim( ) and a sign oracle simulator Sign-Sim( ). It takes input N,e,y where y Z N, and maintains four tables, R, V, X and Y, each an array with index in the range from 1 to q hash. All these are global variables which will be used also be the subroutines. The intended meaning of the array entries is the following, for j = 1,...,q hash V [j] The j-th hash query in the experiment, having the form R[j] Msg[j] R[j] The first l-bits of V [j] Y [j] The value playing the role of H(V [j]), chosen either by the hash simulator or the sign simulator X[j] If V [j] is a direct hash oracle query of F this satisfies Y [j] X[j] e y (mod N). If V [j] is an indirect hash oracle query this satisfies X[j] e Y [j] (mod N), meaning it is a signature of Msg[j]. Note that we don t actually need to store the array Msg; it is only referred to above in the explanation of terms. We will make use of a subroutine Find that given an array A, a value v and index m, returns 0 if v {A[1],...,A[m]}, and else returns the smallest index l such that v = A[l]. Inverter I(N, e, y) Initialize arrays R[1... q hash ], V [1... q hash ], X[1... q hash ], Y [1... q hash ], to empty j 0 Run F on input N,e If F makes oracle query (hash,v) then h H -Sim(v) ; return h to F as the answer If F makes oracle query (sign,m) then σ Sign-Sim(M) ; return σ to F as the answer Until F halts with output (M,(r,x)) y H -Sim(r M) ; l Find(V,r M,q hash ) w x X[l] 1 mod N ; Return w We now describe the hash oracle simulator. It makes reference to the global variables instantiated

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