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We develop a public key cryptosystem whose security is based on the intractability of the ideal membership problem for a noncommutative algebra over a finite field. We show that this system, which is the noncommutative analogue of the Polly Cracker cryptosystem, is more secure than the commutative version. This is due to the fact that there are a number of(More)
1. INTRODUCTION. The groups of invertible matrices over finite fields are among the first groups we meet in a beginning course in modern algebra. Eventually, we find out about simple groups and that the unique simple group of order 168 has two representations as a group of matrices. And this is where we learn that the group of 2 × 2 unimodular matrices over(More)
1. Introduction. Let a and n > 0 be integers, and define G(a, n) to be the directed graph with vertex set V = {0, 1,. .. , n − 1} such that there is an arc from x to y if and only if y ≡ ax (mod n). Recently, Ehrlich [1] studied these graphs in the special case a = 2 and n odd. He proved that if n is odd, then the number of cycles in G(2, n) is odd or even(More)
Gauss's Cyclotomic Formula [3, pp.425-428, p.467] is a neglected mathematical wonder. Theorem 1.1. (Gauss) Let p be an odd prime and set p ′ = (−1) (p−1)/2 p. Then there exist integer polynomials R(x, y) and S(x, y) such that 4(x p + y p) x + y = R(x, y) 2 − p ′ S(x, y) 2 .
Combinatorial designs are collections of subsets of a finite set that satisfy specified conditions, usually involving regularity or symmetry. As the scope of the 984-page Handbook of Combinatorial Designs [7] suggests, this field of study is vast and far reaching. Here is a picture of the very first design to appear in " Opening the Door, " the first of the(More)