Let V be a finitely generated multiplicative semi-group with r generators in the ring of integers ℤK of an algebraic number field K of degree n over ℚ. We use various bounds for character sums to obtain results on the distribution of the residues of elements of V modulo an integer ideal q. In the simplest case, when K = ℚ and r = 1 this is a classical question on the distribution of residues of an exponential function, which may be interpreted as concerning the quality of the linear congruential pseudo-random number generator. Besides this well known application we consider several other problems from algebraic number theory, the theory of function fields over a finite field, complexity theory, cryptography, and coding theory where results on the distribution of some group V modulo q play a central role.
CITATION STYLE
Shparlinski, I. E. (1994). On some applications of finitely generated semi-groups. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 877 LNCS, pp. 265–279). Springer Verlag. https://doi.org/10.1007/3-540-58691-1_66
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