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Cryptanalysis of a Provably Secure CRT-RSA Algorithm

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Cryptanalysis of a Provably Secure CRT-RSA Algorithm Cryptanalysis of a Provably Secure CRT-RSA Algorithm ∗ David Wagner University of California at Berkeley ABSTRACT study how to build cryptosystems that are robust even in We study a countermeasure proposed to protect Chinese the presence of faults. remainder theorem (CRT) computations for RSA against One such countermeasure was proposed by Bl¨omer, Otto, fault attacks. The scheme was claimed to be provably se- and Seifert in ACM CCS 2003 [4]. Their scheme is intended cure. However, we demonstrate that the proposal is in fact to protect Chinese Remainder Theorem computations, as insecure: it can be broken with a simple and practical fault used in many RSA implementations, against fault attacks. attack. We conclude that the proposed countermeasure is They combine several novel and clever ideas into a counter- not safe for use in its present form. measure that, on the surface, looks quite promising. Dis- appointingly, we show in this paper that their proposal is not secure. We review their scheme (Section 2). Then, we Categories and Subject Descriptors exhibit a simple fault attack against their proposal (Sec- E.3 [Data]: Encryption tion 3), we outline a few other attacks (Section 4), and we conclude with a discussion of the implications of our results General Terms (Section 5{6). Security 2. REVIEW Keywords Notation. Let N = pq denote a RSA modulus, e a pub- Fault attacks, RSA, Chinese remainder theorem, cryptanal- lic exponent, and d the corresponding private exponent, so ysis that ed ≡ 1 (mod '(N)). Let CRT(x; y) mod N denote the number z 2 Z=NZ such that z ≡ x (mod p) and z ≡ y 1. INTRODUCTION (mod q); note that this number is uniquely determined mod- For decades, cryptographers have analyzed the security ulo N. of cryptosystems by treating them as mathematical entities We adopt the notation of Bl¨omer, Otto, and Seifert for and analyzing the properties of the underlying cryptographic describing faults [4]. If x is a value produced during some algorithm. Recently, though, the community has begun to computation, we write f(x) to represent a faulty value that realize that this approach overlooks some important con- replaces x during a fault attack. Also, we write f(x) = x + siderations: to ensure security in practice, one must also e(x) so that e(x) denotes the error introduced by the fault. consider the properties of the implementations of these al- We write l(x) for the number of bits needed to represent the gorithms. Strikingly, Boneh, DeMillo, and Lipton showed variable x, so that x 2 f0; 1; : : : ; 2l(x) − 1g. that when cryptographic computations are faulty, the nom- inal security of the cryptosystem may evaporate [5]. More- RSA, CRT, and fault attacks. Many RSA implementa- over, practical experience has shown that, in some impor- tions use the Chinese Remainder Theorem (CRT) to speed tant cases, attackers may be able to induce faults at will up private-key computations. In a standard CRT-RSA al- at arbitrary points in a computation [1, 2, 3], putting secu- gorithm, we compute md mod N as follows: first, compute rity at risk. Results like this have motivated researchers to def def dp dq Sp = m mod p; Sq = m mod q; ∗[email protected]. This research was supported in part by NSF ITR CCR-0113941. def def where dp = d mod p − 1 and dq = d mod q − 1; then, com- pute def S = CRT(Sp; Sq) mod N Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are and output S. This provides a fourfold speedup over straight- not made or distributed for profit or commercial advantage and that copies forward exponentiation. bear this notice and the full citation on the first page. To copy otherwise, to However, Lenstra showed that this use of CRT puts im- republish, to post on servers or to redistribute to lists, requires prior specific plementations at great risk, if faults can occur during the permission and/or a fee. CCS'04, October 25-29, 2004, Washington, DC, USA. computation [7]. Suppose that the exponentiation modulo Copyright 2004 ACM 1-58113-961-6/04/0010 ...$5.00. p is erroneous, so that instead of receiving the correct value dp Sp = m mod p, the device instead receives some faulty ition is that if there are any errors in the computation, then value f(Sp). In this case, the device will compute the erro- we will most likely have c1 6= 1 or c2 6= 1, which renders Sig neous output f(S) = CRT(f(Sp); Sq) mod N. Notice that irreversibly altered and (we hope) useless to an attacker. f(S) is correct modulo q but incorrect modulo p. Since e S ≡ m (mod N) for a correct computation, we see that Threat models. The authors of the BOS algorithm pro- e f(S) mod N is ≡ m (mod q) but 6≡ m (mod p). Conse- posed four different threat models under which a candidate e quently, gcd(m − f(S) mod N; N) reveals p and thereby algorithm could be evaluated: allows factoring N. This shows that the standard use of CRT-RSA is totally insecure in the presence of faults. Fault Model #1: The attacker can cause a fault in a sin- This motivates the search for some way to secure CRT- gle bit. The attacker has full control over the timing RSA against fault attacks, so that we can retain the CRT's and location of the fault. speed advantages without its susceptibility to fault attacks. Fault Model #2: The attacker can cause a fault in a sin- The first countermeasure that might come to mind is to gle byte. The attacker has full control over the timing, check the correctness of the result of the computation before but may have only partial control over the location outputting it, so that erroneous values are not revealed to of the fault (e.g., which byte is affected), and cannot the attacker. This defeats Lenstra's attack, but it turns out predict the new faulty value that is introduced. that checking the output for correctness is not enough to stop certain other fault attacks [8]. Consequently, we need Fault Model #3: The attacker can cause a fault in a sin- a stronger defense, if fault attacks are a realistic threat. gle byte. The attacker has only partial control over In response, Yen, Kim, Lin, and Moon proposed the ap- the timing and location of the fault, and cannot pre- pealing notion of infective computation [9]. An infective dict the new faulty value. computation is one where any error introduced by a fault propagates throughout the computation, ultimately ensur- Fault Model #4: The attacker can cause a fault in a sin- ing that the final result is randomized in such a way that gle variable. The attacker has only partial control over no useful information can be obtained from any erroneous the timing and no control over the location; the target- output of the computation. Consequently, with an infective ted variable will be replaced by an entirely new random algorithm, there is no need to check the output for errors; value not known to the attacker. any error will not be useful to the attacker. Yen, Kim, Lin, and Moon proposed a candidate scheme claimed to achieve Note that these models are listed in terms of decreasing this property, but it was later shown to be insecure [4]. Con- attacker power, so a scheme that is secure against one fault sequently, it remained a challenging open problem to find model will be secure against all higher-numbered models as some infective implementation of CRT-RSA. well. At ACM CCS 2003, Bl¨omer, Otto, and Seifert proposed a CRT-RSA scheme based on the ideas of infective compu- Provable security. In the ACM CCS 2003 paper propos- tation. Since they did not give their proposal a name, for ing this construction, the BOS algorithm was claimed to be this paper we will call it the BOS algorithm. We describe provably secure against fault attacks: \we rigorously ana- the BOS algorithm next. lyze the success probability of an adversary . we prove that our new algorithm is secure against the Bellcore attack" [4, The BOS algorithm. In the BOS algorithm, we first gen- p.311]. In particular, the paper claims security against Fault Models #2, #3, and #4. erate random 80-bit primes t1; t2. Next, we set Unfortunately, although the scheme is claimed to be prov- def dp = d mod '(p · t1); ably secure, there is no statement of a security theorem and def dq = d mod '(q · t2); no security reduction or mathematical proof. Instead, the def −1 paper systematically enumerates many possible attacks and et1 = d mod t1; def −1 shows that these attacks do not work. Such results are of et2 = d mod t2: considerable interest, but they fall short of a proof of se- These values are kept secret from the attacker and never curity. There is no assurance that the attacks examined disclosed; they will be used to check for faults. exhaust the space of all possible attacks, so the risk is that With these definitions, the algorithm is as follows [4]: other methods not anticipated by the designers of the cryp- tosystem might be able to defeat the scheme. In the re- Input. Z Z ∗ A message m 2 ( =N ) . mainder of the paper, we show that further analysis in fact def d Output. Sig = m mod N.
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