Repeated-root cyclic and negacyclic codes over a finite chain ring

AbstractWe show that repeated-root cyclic codes over a finite chain ring are in general not principally generated. Repeated-root negacyclic codes are principally generated if the ring is a Galois ring with characteristic a power of 2. For any other finite chain ring they are in general not principally generated. We also prove results on the structure, cardinality and Hamming distance of repeated-root cyclic and negacyclic codes over a finite chain ring

Higher order differentiation over finite fields with applications to generalising the cube attack

Higher order differentiation was introduced in a cryptographic context by Lai. Several attacks can be viewed in the context of higher order differentiations, amongst them the cube attack of Dinur and Shamir and the AIDA attack of Vielhaber. All of the above have been developed for the binary case. We examine differentiation in larger fields, starting with the field GF(p) of integers modulo a prime p, and apply these techniques to generalising the cube attack to GF(p). The crucial difference is that now the degree in each variable can be higher than one, and our proposed attack will differentiate several times with respect to each variable (unlike the classical cube attack and its larger field version described by Dinur and Shamir, both of which differentiate at most once with respect to each variable). Connections to the Moebius/Reed Muller Transform over GF(p) are also examined. Finally we describe differentiation over finite fields GF(ps) with ps elements and show that it can be reduced to differentiation over GF(p), so a cube attack over GF(ps) would be equivalent to cube attacks over GF(p)

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph

Factoring polynomials over Z4 and over certain Galois

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Further constructions and characterizations of generalized almost perfect nonlinear functions

APN (almost perfect nonlinear) functions over finite fields of even characteristic are interesting and have many applications to the design of symmetric ciphers resistant to differential attacks. This notion was generalized to GAPN (generalized APN) for arbitrary characteristic p by Kuroda and Tsujie. In this paper, we completely classify GAPN monomial functions xd for the case when the exponent d has exactly two non-zero digits when represented in base p; these functions can be viewed as generalizations of the APN Gold functions. In particular, we characterise all the monomial GAPN functions over Fp2 . We also obtain a new characterization for certain GAPN functions over Fpn of algebraic degree p using the multivariate algebraic normal form; this allows us to explicitly construct a family of GAPN functions of algebraic degree p for n= 3 and arbitrary prime p≥ 3

Improving bounds on probabilistic affine tests to estimate the nonlinearity of Boolean functions

17 USC 105 interim-entered record; under temporary embargo.In this paper we want to estimate the nonlinearity of Boolean functions, by probabilistic methods, when it is computationally very expensive, or perhaps not feasible to compute the full Walsh transform (which is the case for almost all functions in a larger number of variables, say more than 30). Firstly, we significantly improve upon the bounds of Zhang and Zheng (1999) on the probabilities of failure of affinity tests based on nonhomomorphicity, in particular, we prove a new lower bound that we have previously conjectured. This new lower bound generalizes the one of Bellare et al. (IEEE Trans. Inf. Theory 42(6), 1781– 1795 1996) to nonhomomorphicity tests of arbitrary order. Secondly, we prove bounds on the probability of failure of a proposed affinity test that uses the BLR linearity test. All these bounds are expressed in terms of the function’s nonlinearity, and we exploit that to provide probabilistic methods for estimating the nonlinearity based upon these affinity tests. We analyze our estimates and conclude that they have reasonably good accuracy, particularly so when the nonlinearity is low.U.S. Government affiliation is unstated in article text

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph