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Advances in Mathematics of Communications

2007 , Volume 1 , Issue 4

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The partial order of perfect codes associated to a perfect code
Olof Heden
2007, 1(4): 399-412 doi: 10.3934/amc.2007.1.399 +[Abstract](238) +[PDF](171.9KB)
Abstract:
It is clarified whether or not ''full rank perfect 1-error correcting binary codes act like primes in the family of all perfect 1-error correcting binary codes''. Thereby the well known connection between perfect 1-error correcting binary codes and tilings will be discussed and used.
On the security of generalized Jacobian cryptosystems
Isabelle Déchène
2007, 1(4): 413-426 doi: 10.3934/amc.2007.1.413 +[Abstract](323) +[PDF](227.8KB)
Abstract:
Generalized Jacobians are natural candidates to use in discrete logarithm (DL) based cryptography since they include the multiplicative group of finite fields, algebraic tori, elliptic curves as well as all Jacobians of curves. This thus led to the study of the simplest nontrivial generalized Jacobians of an elliptic curve, for which an efficient group law algorithm was recently obtained. With these explicit equations at hand, it is now possible to concretely study the corresponding discrete logarithm problem (DLP); this is what we undertake in this paper. In short, our results highlight the close links between the DLP in these generalized Jacobians and the ones in the underlying elliptic curve and finite field.
Additive cyclic codes over $\mathbb F_4$
W. Cary Huffman
2007, 1(4): 427-459 doi: 10.3934/amc.2007.1.427 +[Abstract](236) +[PDF](390.1KB)
Abstract:
In this paper we find a canonical form decomposition for additive cyclic codes of odd length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also reprove that each code is the $\mathbb F_2$-span of at most two codewords and their cyclic shifts, a fact first proved in [2]. A count is given for the number of codes that are the $\mathbb F_2$-span of one codeword and its cyclic shifts. We can examine this decomposition to see precisely when the code is self-orthogonal or self-dual under the trace inner product. Using this, a count is presented for the number of self-orthogonal and self-dual additive cyclic codes of odd length. We also provide a count of the additive cyclic and additive cyclic self-orthogonal codes as a function of their $\mathbb F_2$-dimension.
Bounds on the growth rate of the peak sidelobe level of binary sequences
Denis Dmitriev and Jonathan Jedwab
2007, 1(4): 461-475 doi: 10.3934/amc.2007.1.461 +[Abstract](269) +[PDF](317.7KB)
Abstract:
The peak sidelobe level (PSL) of a binary sequence is the largest absolute value of all its nontrivial aperiodic autocorrelations. A classical problem of digital sequence design is to determine how slowly the PSL of a length $n$ binary sequence can grow, as $n$ becomes large. Moon and Moser showed in 1968 that the growth rate of the PSL of almost all length $n$ binary sequences lies between order $\sqrt{n\log n}$ and $\sqrt{n}$, but since then no theoretical improvement to these bounds has been found.
   We present the first numerical evidence on the tightness of these bounds, showing that the PSL of almost all binary sequences of length $n$ appears to grow exactly like order $\sqrt{n\log n}$, and that the PSL of almost all $m$-sequences of length $n$ appears to grow exactly like order $\sqrt{n}$. In the case of $m$-sequences, a key algorithmic insight reveals behaviour that was previously well beyond the range of computation.
Decoding the Mathieu group M12
Robert F. Bailey and John N. Bray
2007, 1(4): 477-487 doi: 10.3934/amc.2007.1.477 +[Abstract](207) +[PDF](164.1KB)
Abstract:
The sporadic Mathieu group M12 can be viewed as an error-correcting code, where the codewords are the group's elements written as permutations in list form, and with the usual Hamming distance. We investigate the properties of this group as a code, in particular determining completely the probabilities of successful and ambiguous decoding of words with more than 3 errors (which is the number that can be guaranteed to be corrected).
Public key cryptography based on semigroup actions
Gérard Maze, Chris Monico and Joachim Rosenthal
2007, 1(4): 489-507 doi: 10.3934/amc.2007.1.489 +[Abstract](551) +[PDF](248.6KB)
Abstract:
A generalization of the original Diffie-Hellman key exchange in $(\mathbb Z$∕$p\mathbb Z)$* found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general.
   In Section 5 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.
On a classification of finite statistical tests
David Lubicz
2007, 1(4): 509-524 doi: 10.3934/amc.2007.1.509 +[Abstract](242) +[PDF](213.8KB)
Abstract:
Statistical tests of random sequences are often used in cryptography in order to perform some routine checks for random and pseudo-random number generators. Most of the test suites available are based on the theory of hypothesis testing which allows one to decide whether a sample has been drawn following a certain distribution. In this article, we develop a theoretical foundation of statistical tests of random sequences and hypothesis testing with a focus on cryptographic applications and we draw some interesting practical consequences.

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