Fast parallel absolute irreducibility testing

Fast Parallel Absolute Irreducibility Testing

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the cla...

A Fast Algorithm for Testing Irreducibility of Trinomials

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r) bit-operations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bit-operations than the standard algorithm. Using the algorithm, we have found 18 new irreducible trinomials of degree r in the range 100151 ≤ r ≤ 70005...

Fast Irreducibility and Subgroup Membership Testing in XTR

We describe a new general method to perform part of the setup stage of the XTR system introduced at Crypto 2000, namely finding the trace of a generator of the XTR group. Our method is substantially faster than the general method presented at Asiacrypt 2000. As a side result, we obtain an efficient method to test subgroup membership when

Irreducibility of Polynomials with Low Absolute Values

Absolute Irreducibility of Polynomials via Newton Polytopes

A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein’s criterion. Polynomials from these ...