On the Geometry of Cyclic Lattices

Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded cyclic lattices, indicating that well-rounded lattices are more common among cyclic lattices than generically. We also show that SVP is equivalent to SIVP on a positive proportion of Minkowskian well-rounded cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of such lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of ZN closed under the action of subgroups of the permutation group SN, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any N-cycle

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

Asymptotically Efficient Lattice-Based Digital Signatures

We present a general framework that converts certain types of linear collision-resistant hash functions into one-time signatures. Our generic construction can be instantiated based on both general and ideal (e.g. cyclic) lattices, and the resulting signature schemes are provably secure based on the worst-case hardness of approximating the shortest vector (and other standard lattice problems) in the corresponding class of lattices to within a polynomial factor. When instantiated with ideal lattices, the time complexity of the signing and verification algorithms, as well as key and signature size is almost linear (up to poly-logarithmic factors) in the dimension n of the underlying lattice. Since no sub-exponential (in n) time algorithm is known to solve lattice problems in the worst case, even when restricted to ideal lattices, our construction gives a digital signature scheme with an essentially optimal performance/security trade-off

SPRING: Fast Pseudorandom Functions from Rounded Ring Products

International audienceRecently, Banerjee, Peikert and Rosen (EUROCRYPT 2012) proposed new theoretical pseudorandom function candidates based on "rounded products" in certain polynomial rings, which have rigorously provable security based on worst-case lattice problems. The functions also enjoy algebraic properties that make them highly parallelizable and attractive for modern applications, such as evaluation under homomorphic encryption schemes. However, the parameters required by BPR's security proofs are too large for practical use, and many other practical aspects of the design were left unexplored in that work. In this work we give two concrete and practically efficient instantiations of the BPR design, which we call SPRING, for "subset-product with rounding over a ring." One instantiation uses a generator matrix of a binary BCH error-correcting code to "determinstically extract" nearly random bits from a (biased) rounded subset-product. The second instan-tiation eliminates bias by working over suitable moduli and decomposing the computation into "Chinese remainder" components. We analyze the concrete security of these instantiations, and provide initial software implementations whose throughputs are within small factors (as small as 4.5) of those of AES