Nowadays sparse systems of equations occur frequently in science and engineering. In this contrib... more Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems. It turns out that a combinatorial MaxMinMax problem provides bounds on the average computational complexity of sparse systems. In this paper we initiate a study of a linear algebra variation of this MaxMinMax problem.
In 1930, Keller conjectured that every tiling of RnRn by unit cubes contains a pair of cubes shar... more In 1930, Keller conjectured that every tiling of RnRn by unit cubes contains a pair of cubes sharing a complete (n−1)(n−1)-dimensional face. Only 50 years later, Lagarias and Shor found a counterexample for all n≥10n≥10. In this note we show that neither a modification of Keller's conjecture to tiles of more complex shape is true.
In this paper, we investigate the maximum size of a minimal dictionary of a binary prefix-code st... more In this paper, we investigate the maximum size of a minimal dictionary of a binary prefix-code string. We develop exact formulas for the maximum number of codewords of a minimal dictionary, which belongs to a binary string of some length. Further, we elaborate on the computational complexity of our approach and its relation to the Lambert function. We also present a way, how this information enables us to efficiently construct a Huffman code in the case of uniform probability distribution of codewords.The paper is of mathematical nature, i.e. all the methodology used in the paper is based on mathematical proofs.
Tiling problems belong to the oldest problems in mathematics. They attracted attention of many fa... more Tiling problems belong to the oldest problems in mathematics. They attracted attention of many famous mathematicians. Even one of the Hilbert's problems is devoted to the topic. The interest in tilings by unit cubes originated with a conjecture raised by Minkowski in 1907. We discuss this conjecture, its history and variations, and then we describe some problems that Minkowski's conjecture, in turn, suggested. We will focus on tilings of R n by the n-cross; a cluster of 2n + 1 unit cubes comprising a cube and its reflections in all faces. Some " unexpected " tilings by n-crosses will be presented.
The first and the third authors recently introduced a spectral construction of plateaued and of 5... more The first and the third authors recently introduced a spectral construction of plateaued and of 5-value spectrum functions. In particular, the design of the latter class requires a specification of integers {W (u) : }, so that the sequence {W (u) : u ∈ F n 2 } is a valid spectrum of a Boolean function (recovered using the inverse Walsh transform). Technically, this is done by allocating a suitable Walsh support . In addition, two dual functions g [i] : S [i] → F2 (with #S [i] = 2 λ i ) are employed to specify the signs through W (u) = 2 n+s i 2
In the past when only “paper and pencil” ciphers were in use, a mnemonic Bellaso’s method was app... more In the past when only “paper and pencil” ciphers were in use, a mnemonic Bellaso’s method was applied to obtain a permutation from a meaningful phrase. It turns out that even nowadays this method and its variations are employed and provide a simple but at the same time efficient tool for some ciphers. To the best of our knowledge the security of such ciphers has not been evaluated yet in all respects. Therefore, in this paper we initiate a study of their security with respect to the probability distribution of permutations obtained by Bellaso’s method.
Payan, On the existence of three-dimensional tiling in the Lee metric, European J. Combin. 19 (19... more Payan, On the existence of three-dimensional tiling in the Lee metric, European J. Combin. 19 (1998) 567-572] that there is no tiling of the three-dimensional space R 3 with Lee spheres of radius at least 2. In particular, this verifies the Golomb-Welch conjecture for n = 3. Špacapan, [S. Špacapan, Non-existence of face-to-face four-dimensional tiling in the Lee metric, European J. Combin. 28 (2007) 127-133], using a computer-based proof, showed that the statement is true for R 4 as well. In this paper we introduce a new method that will allow us not only to provide a short proof for the four-dimensional case but also to extend the result to R 5 . In addition, we provide a new proof for the three-dimensional case, just to show the power of our method, although the original one is more elegant. The main ingredient of our proof is the non-existence of the perfect Lee 2-error correcting code over Z of block size n = 3, 4, 5.
We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete gr... more We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on Z p , p an odd prime. Let K p be the complete graph on p vertices. We will usually take Z p , the cyclic group of order p, as the set of vertices of K p . The length of the edge uv, u, v ∈ K p (or the distance of u and v) is given by d(u, v) = min (|u-v|, p-|u-v|). Given a path P = (v 1 , v 2 , . . . , v m ), we denote the multiset of edge-lengths of P by d(P ): d(P ) = {d(v i , v i+1 ) : i = 1, 2, . . . , m-1}. Marco Buratti [1] formulated the following problem: Let p = 2n + 1 be a prime, let L be any list of 2n elements, each from the set {1, 2, . . . , n}. Does there exist a Hamiltonian path H in K p with V (K p ) = Z p such that the set of edge-lengths of H comprises L? (That is, such that d(H) = L?) He conjectured that the answer is yes for every list L. A realization of a list L is a Hamiltonian path (x 0 , x 1 , . . . , x 2n ) on vertices of Z p such that the (multi)-set of edge-lengths {d(x i , x i+1 ) : 0, 1, . . . , 2n -1} equals L. In other words, Buratti's conjecture says that every such list L has a realization (or is realizable). If a list L consists of a 1 1's, a 2 2's, . . . , a n n's, where a 1 + a 2 + . . . a n = 2n, we will use exponential notation for L and write L = 1 a 1 2 a 2 . . . n an or, alternatively, we will say that L is of type [a 1 , a 2 , . . . , a n ], or for the sake of brevity, we will write simply L = [a 1 , a 2 , . . . , a n ]. To best of our knowledge, no results on Buratti's conjecture have been published so far. However, using a computer, Mariusz Meszka [3] has verified the validity of Buratti's conjecture for all primes p ≤ 23. The problem does not appear to be easy. The purpose of this article is to present some initial ideas, approaches and results towards the complete solution of Buratti's conjecture. One such approach is outlined in Section 2 where certain graphs having lists as vertices, arranged in lexicographic order, are considered. Some properties of the smallest list the electronic journal of combinatorics 16 (2009), #R20 without a realization, if such exists, are derived. In Section 3 we prove that certain classes of lists are realizable. In particular, we show that any list where one of the distances occurs "sufficiently many times" is realizable. We also show that any list consisting of just two distinct distances is realizable. The general case of lists with only two distances, that is, when p is any positive integer, not just a prime, is characterized in Section 4. This characterization is a clear indication of the complexity of the problem.
In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert'... more In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.
The "Gluing Algorithm" of Semaev [Des. Codes Cryptogr. 49 (2008), 47-60] -that finds all solution... more The "Gluing Algorithm" of Semaev [Des. Codes Cryptogr. 49 (2008), 47-60] -that finds all solutions of a sparse system of linear equations over the Galois field GF (q) -has average running time O(mq max|∪ k 1 X j |-k ), where m is the total number of equations, and ∪ k 1 Xj is the set of all unknowns actively occurring in the first k equations. Our goal here is to minimize the exponent of q in the case where every equation contains at most three unknowns. The main result states that if the total number |∪ m 1 Xj | of unknowns is equal to m, then the best achievable exponent is between c1m and c2m for some positive constants c1 and c2.
Nowadays sparse systems of equations occur frequently in science and engineering. In this contrib... more Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems. It turns out that a combinatorial MaxMinMax problem provides bounds on the average computational complexity of sparse systems. In this paper we initiate a study of a linear algebra variation of this MaxMinMax problem.
In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert'... more In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.
It is proved that for n ≥ 6, the number of perfect matchings in a simple connected cubic graph on... more It is proved that for n ≥ 6, the number of perfect matchings in a simple connected cubic graph on 2n vertices is at most 4fn-1, with fn being the n-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph G equals the expected value of a random variable defined on all 2-colorings of edges of G. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided.
In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert’... more In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert’s Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.
A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class indu... more A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. Despite recent progress for large $\Delta$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $\Delta = 3$, leaving the need for new approaches to verify the conjecture for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.
Since 1968, when the Golomb-Welch conjecture was raised, it has become the main motive power behi... more Since 1968, when the Golomb-Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb-Welch conjecture. Further, new results on Golomb-Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.
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