65 research outputs found

    On weighing matrices with square weights

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    We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of W(n,k²), k = 1,.. ,12, giving new results for k = 8,...12

    Exhaustive Search for Small Dimension Recursive MDS Diffusion Layers for Block Ciphers and Hash Functions

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    This article presents a new algorithm to find MDS matrices that are well suited for use as a diffusion layer in lightweight block ciphers. Using an recursive construction, it is possible to obtain matrices with a very compact description. Classical field multiplications can also be replaced by simple F2-linear transformations (combinations of XORs and shifts) which are much lighter. Using this algorithm, it was possible to design a 16x16 matrix on a 5-bit alphabet, yielding an efficient 80-bit diffusion layer with maximal branch number.Comment: Published at ISIT 201

    Parity of transversals of Latin squares

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    We introduce a notion of parity for transversals, and use it to show that in Latin squares of order 2 mod 42 \bmod 4, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving E1,…,EnE_1,\dots, E_n, where EiE_i is the number of diagonals of a given Latin square that contain exactly ii different symbols. Let A(i∣j)A(i\mid j) denote the matrix obtained by deleting row ii and column jj from a parent matrix AA. Define tijt_{ij} to be the number of transversals in L(i∣j)L(i\mid j), for some fixed Latin square LL. We show that tab≡tcd mod 2t_{ab}\equiv t_{cd}\bmod2 for all a,b,c,da,b,c,d and LL. Also, if LL has odd order then the number of transversals of LL equals tabt_{ab} mod 2. We conjecture that tac+tbc+tad+tbd≡0 mod 4t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4 for all a,b,c,da,b,c,d. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a kk-regular bipartite graph on 2n2n vertices is divisible by 44 when nn is odd and k≡0 mod 4k\equiv0\bmod 4. We also show that per A(a∣c)+per A(b∣c)+per A(a∣d)+per A(b∣d)≡0 mod 4{\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4 for all a,b,c,da,b,c,d, when AA is an integer matrix of odd order with all row and columns sums equal to k≡2 mod 4k\equiv2\bmod4

    On equiangular lines in 17 dimensions and the characteristic polynomial of a Seidel matrix

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    For ee a positive integer, we find restrictions modulo 2e2^e on the coefficients of the characteristic polynomial χS(x)\chi_S(x) of a Seidel matrix SS. We show that, for a Seidel matrix of order nn even (resp. odd), there are at most 2(e−22)2^{\binom{e-2}{2}} (resp. 2(e−22)+12^{\binom{e-2}{2}+1}) possibilities for the congruence class of χS(x)\chi_S(x) modulo 2eZ[x]2^e\mathbb Z[x]. As an application of these results, we obtain an improvement to the upper bound for the number of equiangular lines in R17\mathbb R^{17}, that is, we reduce the known upper bound from 5050 to 4949.Comment: 21 pages, fixed typo in Lemma 2.

    Modal Decomposition of Feedback Delay Networks

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    Feedback delay networks (FDNs) belong to a general class of recursive filters which are widely used in sound synthesis and physical modeling applications. We present a numerical technique to compute the modal decomposition of the FDN transfer function. The proposed pole finding algorithm is based on the Ehrlich-Aberth iteration for matrix polynomials and has improved computational performance of up to three orders of magnitude compared to a scalar polynomial root finder. We demonstrate how explicit knowledge of the FDN's modal behavior facilitates analysis and improvements for artificial reverberation. The statistical distribution of mode frequency and residue magnitudes demonstrate that relatively few modes contribute a large portion of impulse response energy

    Developments on spectral characterizations of graphs

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    AbstractIn [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments

    On the Complexity of the Generalized MinRank Problem

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    We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most rr. A natural algebraic representation of this problem gives rise to a \emph{determinantal ideal}: the ideal generated by all minors of size r+1r+1 of the matrix. We give new complexity bounds for solving this problem using Gr\"obner bases algorithms under genericity assumptions on the input matrix. In particular, these complexity bounds allow us to identify families of generalized MinRank problems for which the arithmetic complexity of the solving process is polynomial in the number of solutions. We also provide an algorithm to compute a rational parametrization of the variety of a 0-dimensional and radical system of bi-degree (D,1)(D,1). We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.Comment: 29 page
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