Cryptographic Hash Functions in Groups and Provable Properties

We consider several "provably secure" hash functions that compute simple sums in a well chosen group (G,*). Security properties of such functions provably translate in a natural way to computational problems in G that are simple to define and possibly also hard to solve. Given k disjoint lists Li of group elements, the k-sum problem asks for gi ∊ Li such that g1 * g2 *...* gk = 1G. Hardness of the problem in the respective groups follows from some "standard" assumptions used in public-key cryptology such as hardness of integer factoring, discrete logarithms, lattice reduction and syndrome decoding. We point out evidence that the k-sum problem may even be harder than the above problems. Two hash functions based on the group k-sum problem, SWIFFTX and FSB, were submitted to NIST as candidates for the future SHA-3 standard. Both submissions were supported by some sort of a security proof. We show that the assessment of security levels provided in the proposals is not related to the proofs included. The main claims on security are supported exclusively by considerations about available attacks. By introducing "second-order" bounds on bounds on security, we expose the limits of such an approach to provable security. A problem with the way security is quantified does not necessarily mean a problem with security itself. Although FSB does have a history of failures, recent versions of the two above functions have resisted cryptanalytic efforts well. This evidence, as well as the several connections to more standard problems, suggests that the k-sum problem in some groups may be considered hard on its own, and possibly lead to provable bounds on security. Complexity of the non-trivial tree algorithm is becoming a standard tool for measuring the associated hardness. We propose modifications to the multiplicative Very Smooth Hash and derive security from multiplicative k-sums in contrast to the original reductions that related to factoring or discrete logarithms. Although the original reductions remain valid, we measure security in a new, more aggressive way. This allows us to relax the parameters and hash faster. We obtain a function that is only three times slower compared to SHA-256 and is estimated to offer at least equivalent collision resistance. The speed can be doubled by the use of a special modulus, such a modified function is supported exclusively by the hardness of multiplicative k-sums modulo a power of two. Our efforts culminate in a new multiplicative k-sum function in finite fields that further generalizes the design of Very Smooth Hash. In contrast to the previous variants, the memory requirements of the new function are negligible. The fastest instance of the function expected to offer 128-bit collision resistance runs at 24 cycles per byte on an Intel Core i7 processor and approaches the 17.4 figure of SHA-256. The new functions proposed in this thesis do not provably achieve a usual security property such as preimage or collision resistance from a well-established assumption. They do however enjoy unconditional provable separation of inputs that collide. Changes in input that are small with respect to a well defined measure never lead to identical output in the compression function

Cryptanalysis of Dedicated Cryptographic Hash Functions

In this thesis we study the security of a number of dedicated cryptographic hash functions against cryptanalytic attacks. We begin with an introduction to what cryptographic hash functions are and what they are used for. This is followed by strict definitions of the security properties often required from cryptographic hash functions. FSB hashes are a class of hash functions derived from a coding theory problem. We attack FSB by modeling the compression function of the hash by a matrix in GF(2). We show that collisions and preimages can easily be found in FSB with the proposed security parameters. We describe a meet-in-the-middle attack against the FORK-256 hash function. The attack requires 2^112.8 operations to find a collision, which is a 38000-fold improvement over the expected 2^128 operations. We then present a method for finding slid pairs for the compression function of SHA-1; pairs of inputs and messages that produce closely related outputs in the compression function. We also cryptanalyse two block ciphers based on the compression function of MD5, MDC-MD5 and the Kaliski-Robshaw "Crab" encryption algorithm. VSH is a hash function based on problems in number theory that are believed to be hard. The original proposal only claims collision resistance; we demonstrate that VSH does not meet the other hash function requirements of preimage resistance, one-wayness, and collision resistance of truncated variants. To explore more general cryptanalytic attacks, we discuss the d-Monomial test, a statistical test that has been found to be effective in distinguishing iterated Boolean circuits from real random functions. The test is applied to the SHA and MD5 hash functions. We present a new hash function proposal, LASH, and its initial cryptanalysis.The LASH design is based on a simple underlying primitive, and some of its security can be shown to be related to lattice problems

Cryptanalysis of Dedicated Cryptographic Hash Functions

In this thesis we study the security of a number of dedicated cryptographic hash functions against cryptanalytic attacks. We begin with an introduction to what cryptographic hash functions are and what they are used for. This is followed by strict definitions of the security properties often required from cryptographic hash functions. FSB hashes are a class of hash functions derived from a coding theory problem. We attack FSB by modeling the compression function of the hash by a matrix in GF(2). We show that collisions and preimages can easily be found in FSB with the proposed security parameters. We describe a meet-in-the-middle attack against the FORK-256 hash function. The attack requires 2^112.8 operations to find a collision, which is a 38000-fold improvement over the expected 2^128 operations. We then present a method for finding slid pairs for the compression function of SHA-1; pairs of inputs and messages that produce closely related outputs in the compression function. We also cryptanalyse two block ciphers based on the compression function of MD5, MDC-MD5 and the Kaliski-Robshaw "Crab" encryption algorithm. VSH is a hash function based on problems in number theory that are believed to be hard. The original proposal only claims collision resistance; we demonstrate that VSH does not meet the other hash function requirements of preimage resistance, one-wayness, and collision resistance of truncated variants. To explore more general cryptanalytic attacks, we discuss the d-Monomial test, a statistical test that has been found to be effective in distinguishing iterated Boolean circuits from real random functions. The test is applied to the SHA and MD5 hash functions. We present a new hash function proposal, LASH, and its initial cryptanalysis.The LASH design is based on a simple underlying primitive, and some of its security can be shown to be related to lattice problems

VSH and multiplicative modular relations between small primes with polynomial exponents

8 page(s