The discrete logarithm problem (DLP) in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have a time complexity of O ( N log N ) O\left(\sqrt{N}\log N) and a space complexity of O ( N ) O\left(\sqrt{N}) , where N N is the order of the group. (If N N is unknown, a simple modification would achieve a time complexity of O ( N ( log N ) 2 ) O\left(\sqrt{N}{\left(\log N)}^{2}) .) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this article is a deterministic algorithm for solving the DLP in a semigroup. Specifically, let x x be an element in a semigroup having finite order N x {N}_{x} . The article provides an algorithm, which, given any element y ∈ ⟨ x ⟩ y\in \langle x\rangle , provides all natural numbers m m with x m = y {x}^{m}=y , and has time complexity O ( N x ( log N x ) 2 ) O\left(\sqrt{{N}_{x}}{\left(\log {N}_{x})}^{2}) steps. The article also gives an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely.
LDPC codes constructed from permutation matrices have recently attracted the interest of many researchers. A crucial point when dealing with such codes is trying to avoid cycles of short length in the associated Tanner graph, i.e. obtaining a possibly large girth. In this paper, we provide a framework to obtain constructions of such codes. We relate criteria for the existence of cycles of a certain length with some number-theoretic concepts, in particular with the so-called Sidon sets. In this way we obtain examples of LDPC codes with a certain girth. Finally, we extend our constructions to also obtain irregular LDPC codes.
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