The most widely used hash functions from MD4 family have been broken, which lead to a public competition on designing new hash functions held by NIST. This paper focuses on one concept called near-collision resistance: computationally difficult to find a pair of messages with hash values differing in only few bits, which new hash functions should satisfy. In this paper, we will give a model of near-collisions on MD4, and apply it to attack protocols including HMAC/NMAC-MD4 and MD4(Password||Challenge). Our new outer-key recovery attacks on HMAC/NMAC-MD4 has a complexity of 272 online queries and 277 MD4 computations, while previous result was 288 online queries and 295 MD4 computations. Our attack on MD4(Password||Challenge) can recover 16 password characters with a complexity of 237 online queries and 221 MD4 computations, which is the first approach to attack such protocols.

In this paper, we present a lattice based method on small secret exponent attack on the RSA scheme. Boneh and Durfee reduced the attack to finding the small roots of the bivariate modular equation: x(N+1+y)+1 ≡ 0 (mod e), where N is an RSA modulus and e is the RSA public key and proposed a lattice based algorithm for solving the problem. When the secret exponent d is less than N0.292, their method breaks the RSA scheme. Since the lattice used in the analysis is not full-rank, the analysis is not easy. Blömer and May proposed an alternative algorithm that uses a full-rank lattice, even though it gives a bound (d≤N0.290) that is worse than Boneh-Durfee. However, the proof for their bound is still complicated. Herrmann and May, however, have given an elementary proof for the Boneh-Durfee's bound: d≤N0.292. In this paper, we first give an elementary proof for achieving Blömer-May's bound: d≤N0.290. Our proof employs the unravelled linearization technique introduced by Herrmann and May and is rather simpler than that of Blömer-May's proof. We then provide a unified framework — which subsumes the two previous methods, the Herrmann-May and the Blömer-May methods, as a special case — for constructing a lattice that can be are used to solve the problem. In addition, we prove that Boneh-Durfee's bound: d≤N0.292 is still optimal in our unified framework.

In this paper, we present practical password recovery attacks against two challenge and response authentication protocols using MD4. For attacks on protocols, the number of queries is one of the most important factors because the opportunity where an attacker can ask queries is very limited in real protocols. When responses are computed as MD4(Password||Challenge), which is called prefix approach, previous work needs to ask 237 queries to recover a password. Asking 237 queries in real protocols is almost impossible. In our attack, to recover up to 8-octet passwords, we only need 1 time the amount of eavesdropping, 17 queries, and 234 MD4 off-line computations. To recover up to 12-octet passwords, we only need 210 times the amount of eavesdropping, 210 queries, and 241 off-line MD4 computations. When responses are computed as MD4(Password||Challenge||Password), which is called hybrid approach, previous work needs to ask 263 queries, while in our attack, up to 8-octet passwords are practically recovered by 28 times the amount of eavesdropping, 28 queries, and 239 off-line MD4 computations. Our idea is guessing a part of passwords so that we can simulate values of intermediate chaining variables from observed hash values. This enables us to use a short local collision that occurs with a very high probability, and thus the number of queries becomes practical.

Secret Handshake protocol allows members of the same group to authenticate each other secretly. That is, two members who belong to the same group can learn counterpart is in the same group, while non-member of the group cannot determine whether the counterpart is a member of the group or not. Yamashita and Tanaka proposed Secret Handshake Scheme with Multiple Groups (SHSMG). They extended a single group setting to a multiple groups setting where two members output "accept" iff both member's affiliations of the multiple groups are identical. In this paper, we first show the flaw of their SHSMG, and we construct a new secure SHSMG. Second, we introduce a new concept of Secret Handshake scheme, "monotone condition Secret Handshake with Multiple Groups (mc-SHSMG)," in order to extend the condition of "accept." In our new setting of handshake protocol, members can authenticate each other in monotone condition (not only both member's affiliations are identical but also the affiliations are not identical). The communication costs and computational costs of our proposed mc-SHSMG are fewer than the trivial construction of mc-SHSMG.

We discuss how to recover RSA secret keys from noisy key bits with erasures and errors. There are two known algorithms recovering original secret keys from noisy keys. At Crypto 2009, Heninger and Shacham proposed a method for the case where an erroneous version of secret keys contains only erasures. Subsequently, Henecka et al. proposed a method for an erroneous version containing only errors at Crypto 2010. For physical attacks such as side-channel and cold boot attacks, we need to study key recovery from a noisy secret key containing both erasures and errors. In this paper, we propose a method to recover a secret key from such an erroneous version and analyze the condition for error and erasure rates so that our algorithm succeeds in finding the correct secret key in polynomial time. We also evaluate a theoretical bound to recover the secret key and discuss to what extent our algorithm achieves this bound.

At Eurocrypt 2011, Kiltz et al. presented two efficient authentication protocols for resource-constrained devices such as radio-frequency identification tags. Kiltz et al. proved that their protocols were provably secure against active attackers. However, they did not refer to the security against man-in-the-middle (MIM) attackers. In this paper, we analyze the security of the protocols against the MIM attacks and reveal the vulnerabilities. More concretely, we propose MIM attacks on them and evaluate authentication rounds required in these attacks precisely. We assume that the tag and reader share a 2l-bit secret key. The expected number of authentication rounds to recover the secret information in the first and second protocol is at most 2l+2 and 4l+4, respectively. These attacks do not contradict the proof of security since the MIM attack is located outside the attack model that Kiltz et al. considered.

The subset sum problem, which is often called as the knapsack problem, is known as an NP-hard problem, and there are several cryptosystems based on the problem. Assuming an oracle for shortest vector problem of lattice, the low-density attack algorithm by Lagarias and Odlyzko and its variants solve the subset sum problem efficiently, when the “density” of the given problem is smaller than some threshold. When we define the density in the context of knapsack-type cryptosystems, weights are usually assumed to be chosen uniformly at random from the same interval. In this paper, we focus on general subset sum problems, where this assumption may not hold. We assume that weights are chosen from different intervals, and make analysis of the effect on the success probability of above algorithms both theoretically and experimentally. Possible application of our result in the context of knapsack cryptosystems is the security analysis when we reduce the data size of public keys.

Many hash-based authentication protocols have been proposed, and proven secure assuming that underlying hash functions are secure. On the other hand, if a hash function compromises, the security of authentication protocols based on this hash function becomes unclear. Therefore, it is significantly important to verify the security of hash-based protocols when a hash function is broken. In this paper, we will re-evaluate the security of two MD5-based authentication protocols based on a fact that MD5 cannot satisfy a required fundamental property named collision resistance. The target protocols are APOP (Authenticated Post Office Protocol) and NMAC (Nested Message Authentication Code), since they or their variants are widely used in real world. For security evaluation of APOP, we will propose a modified password recovery attack procedure, which is twice as fast as previous attacks. Moreover, our attack is more realistic, as the probability of being detected is lower than that of previous attacks. For security evaluation of MD5-based NMAC, we will propose a new key-recovery attack procedure, which has a complexity lower than that of previous attack. The complexity of our attack is 276, while that of previous attack is 2100.** Moreover, our attack has another interesting point. NMAC has two keys: the inner key and the outer key. Our attack can recover the outer key partially without the knowledge of the inner key.

We address the security issue of RSA with implicitly related keys in this paper. Informally, we investigate under what condition is it possible to efficiently factorize RSA moduli in polynomial time given implicit relation of the related private keys that certain portions of bit pattern are the same. We formulate concrete attack scenarios and propose lattice-based cryptanalysis by using lattice reduction algorithms. A subtle lattice technique is adapted to represent an unknown private key with the help of known implicit relation. We analyze a simple case when given two RSA instances with the known amount of shared most significant bits (MSBs) and least significant bits (LSBs) of the private keys. We further extend to a generic lattice-based attack for given more RSA instances with implicitly related keys. Our theoretical results indicate that RSA with implicitly related keys is more insecure and better asymptotic results can be achieved as the number of RSA instances increases. Furthermore, we conduct numerical experiments to verify the validity of the proposed attacks.

CRT-RSA is a variant of RSA, which uses integers dp = d mod (p-1) and dq = d mod (q-1) (CRT-exponents), where d, p, q are the secret keys of RSA. May proposed a method to obtain the secret key in polynomial time if a CRT-exponent is small, moreover Bleichenbacher and May improved this method. On the other hand, Takagi's RSA is a variant of CRT-RSA, whose public key N is of the form prq for a given positive integer r. In this paper, we extend the May's method and the Bleichenbacher-May's method to Takagi's RSA, and we show that we obtain p in polynomial time if by the extended May's method, and if by the extended Bleichenbacher-May's method, when dq is arbitrary small. If r=1, these upper bounds conform to May's and Bleichenbacher-May's results respectively. Moreover, we also show that the upper bound of pr increase with an increase in r. Since these attacks are heuristic algorithms, we provide several experiments which show that we can obtain the secret key in practice.

Super-anomalous elliptic curves over a ring Z/nZ ;(n=Πi=1k piei) are defined by extending anomalous elliptic curves over a prime filed Fp. They have n points over a ring Z/nZ and pi points over Fpi for all pi. We generalize Satoh-Araki-Smart algorithm and Ruck algorithm, which solve a discrete logarithm problem over anomalous elliptic curves. We prove that a "discrete logarithm problem over super-anomalous elliptic curves" can be solved in deterministic polynomial time without knowing prime factors of n.

In this paper, we propose two new chosen-ciphertext (CCA) secure schemes from the computational Diffie-Hellman (CDH) and bilinear computational Diffie-Hellman (BCDH) assumptions. Our first scheme from the CDH assumption is constructed by extending Cash-Kiltz-Shoup scheme. This scheme yields the same ciphertext as that of Hanaoka-Kurosawa scheme (and thus Cramer-Shoup scheme) with cheaper computational cost for encryption. However, key size is still the same as that of Hanaoka-Kurosawa scheme. Our second scheme from the BCDH assumption is constructed by extending Boyen-Mei-Waters scheme. Though this scheme requires a stronger underlying assumption than the CDH assumption, it yields significantly shorter key size for both public and secret keys. Furthermore, ciphertext length of our second scheme is the same as that of the original Boyen-Mei-Waters scheme.

For RSA, May showed a deterministic polynomial time equivalence of computing d to factoring N(=pq). On the other hand, Takagi showed a variant of RSA such that the decryption algorithm is faster than the standard RSA, where N=prq while ed=1 mod(p-1)(q-1). In this paper, we show that a deterministic polynomial time equivalence also holds in this variant. The coefficient matrix T to which LLL algorithm is applied is no longer lower triangular, and hence we develop a new technique to overcome this problem.

We introduce a “generalized small inverse problem (GSIP)” and present an algorithm for solving this problem. GSIP is formulated as finding small solutions of f(x0, x1, ..., xn)=x0 h(x1, ..., xn)+C=0 (mod ; M) for an n-variate polynomial h, non-zero integers C and M. Our algorithm is based on lattice-based Coppersmith technique. We provide a strategy for construction of a lattice basis for solving f=0, which is systematically transformed from a lattice basis for solving h=0. Then, we derive an upper bound such that the target problem can be solved in polynomial time in log M in an explicit form. Since GSIPs include some RSA-related problems, our algorithm is applicable to them. For example, the small key attacks by Boneh and Durfee are re-found automatically.

Subspace membership encryption (SME), a generalization of inner product encryption (IPE), was recently formalized by Boneh, Raghunathan, and Segev in Asiacrypt 2013. The main motivation for SME was that traditional predicate encryptions did not yield function privacy, a security notion introduced by Boneh et al. in Crypto 2013 that captures the privacy of the predicate associated to the secret key. Although they gave a generic construction of SME based on any IPE, we show that their construction of SME for small attribute space was incorrect and provide an attack that breaks the attribute hiding security, a baseline security notion for predicate encryptions that captures the privacy of the attribute associated with the ciphertext. Then, we propose a generalized construction of SME and prove that the attribute hiding security can not be achieved even in the newly defined setting. Finally, we further extend our generalized construction of SME and propose a SME that achieves the attribute hiding property even when the attribute space is small. In exchange our proposed scheme does not yield function privacy and the construction is rather inefficient. Although we did not succeed in constructing a SME both yielding function privacy and attribute hiding security, ours is the first attribute hiding SME scheme whose attribute space is polynomial in the security parameter, and we formalized a richer framework for constructing SMEs and discovered a trade-off like relationship between the two security notions.