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A multi-set (ms) is a set where an element can occur more than once. ms hash functions (mshfs) map mss of arbitrary cardinality to fixed-length strings. This paper introduces a new rsa-based mshf. The new function is efficient and produces small hashes. We prove that the proposed mshf is collision-resistant under the assumption of unforgeability of deterministic rsa signatures. In many...
In previous work we showed how to compress certain prime-order subgroups of the cyclotomic subgroups of orders 22m + 1 of the multiplicative groups of by a factor of 4. We also showed that single-exponentiation can be efficiently performed using compressed representations. In this paper we show that double-exponentiation can be efficiently performed using factor-4...
This paper extends Joux-Naccache-Thomé’s e-th root algorithm to the static Diffie-Hellman problem (sdhp). The new algorithm can be adapted to diverse finite fields by customizing it with an nfs-like core or an ffs-like core. In both cases, after a number of non-adaptive sdhp oracle queries, the attacker builds-up the ability to solve new sdhp instances unknown before the query phase...
Gaudry and Schost gave a low-memory algorithm for solving the 2-dimensional discrete logarithm problem. We present an improvement to their algorithm and extend this improvement to the general multidimensional DLP. An important component of the algorithm is a multidimensional pseudorandom walk which we analyse thoroughly in the 1 and 2 dimensional cases as well as giving some discussion for higher...
Let $f:{\mathbb{F}_2^{n}}\to {\mathbb{F}_2^{n}}$ be an almost perfect nonlinear function (APN). The set $D_f:=\{(a,b)\: :\: f(x+a)-f(x)=b\mbox{\ has two solutions}\}$ can be used to distinguish APN functions up to equivalence. We investigate the multiplier groups of theses sets Df. This extends earlier work done by the authors [1].
Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. (Hyper)-bent functions are not classified. A complete...
The Rayleigh quotient of a bent function is an invariant under the action of the orthogonal group, and it measures the distance of the function to its dual. An efficient algorithm is derived that generates all bent functions of given Rayleigh quotient. The Rayleigh quotient of some bent functions obtained by primary (Maiorana McFarland, Dillon) or secondary (direct and indirect sum) constructions...
This paper is a survey of bounds and constructions for subspace codes designed for the injection metric, a distance measure that arises in the context of correcting adversarial packet insertions in linear network coding. The construction of lifted rank-metric codes is reviewed, along with improved constructions leading to codes with strictly more codewords. Algorithms for encoding and decoding are...
In this work, we consider the pairwise error probability (PEP) of a linear programming (LP) decoder for a general binary linear code as formulated by Feldman et al. (IEEE Trans. Inf. Theory, March 2005) on a quantized additive white Gaussian noise (AWGN) channel. With a quantized AWGN (QAWGN) channel, we mean a channel where we first compute log-likelihood ratios as for an AWGN channel and then quantize...
In previous works we considered codes defined as ideals of quotients of skew polynomial rings, so called Ore rings of automorphism type. In this paper we consider codes defined as modules over skew polynomial rings, removing therefore some of the constraints on the length of the skew codes defined as ideals. The notion of BCH codes can be extended to this new approach and the skew codes whose duals...
Several open questions in coding theory relate to non- existence or construction of certain optimal codes. Many previous problems of this kind have been solved by studying possible weight enumerators. A couple of authors in this decade have proposed using higher weights (generalised Hamming weights) to a similar effect. In this paper we suggest one approach based on the weight hierarchy, and it allows...
In this paper, we establish a mass formula for even codes over . In particular, a formula giving the total number of distinct Type II self-dual codes over of length n is established for each positive integer n divisible by 8.
A classification of self-dual -codes of modest lengths is known for small k. For k = 4,6,8,9 and 10, the classification of self-dual -codes is extended to lengths 19,12,12,12 and 10, respectively, by considering k-frames of unimodular lattices.
Some linear codes associated to maximal algebraic curves via Feng-Rao construction are investigated. In several case, these codes have better minimum distance with respect to the previously known linear codes with same length and dimension.
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