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

Well-posedness and discretization for a class of models for mixed-dimensional problems with high-dimensional gap

In this work, we show the underlying mathematical structure of mixed-dimensional models arising from the composition of graphs and continuous domains. Such models are becoming popular in applications, in particular, to model the human vasculature. We first discuss the model equations in the strong form, which describes the conservation of mass and Darcy's law in the continuum and network as well as the coupling between them. By introducing proper scaling, we propose a weak form that avoids degeneracy. Well-posedness of the weak form is shown through standard Babu\v ska--Brezzi theory. We also develop the mixed formulation finite-element method and prove its well-posedness. A mass-lumping technique is introduced to derive the two-point flux approximation (TPFA) type discretization as well, due to its importance in applications. Based on the Babu\v ska--Brezzi theory, error estimates can be obtained for both the finite-element scheme and the TPFA scheme. We also discuss efficient linear solvers for discrete problems. Finally, we present some numerical examples to verify the theoretical results and demonstrate the robustness of our proposed discretization schemes.acceptedVersio

Obfuscated Fuzzy Hamming Distance and Conjunctions from Subset Product Problems

We consider the problem of obfuscating programs for fuzzy matching (in other words, testing whether the Hamming distance between an $n$-bit input and a fixed $n$-bit target vector is smaller than some predetermined threshold). This problem arises in biometric matching and other contexts. We present a virtual-black-box (VBB) secure and input-hiding obfuscator for fuzzy matching for Hamming distance, based on certain natural number-theoretic computational assumptions. In contrast to schemes based on coding theory, our obfuscator is based on computational hardness rather than information-theoretic hardness, and can be implemented for a much wider range of parameters. The Hamming distance obfuscator can also be applied to obfuscation of matching under the $\ell_1$ norm on $\mathbb{Z}^n$. We also consider obfuscating conjunctions. Conjunctions are equivalent to pattern matching with wildcards, which can be reduced in some cases to fuzzy matching. Our approach does not cover as general a range of parameters as other solutions, but it is much more compact. We study the relation between our obfuscation schemes and other obfuscators and give some advantages of our solution