Pascal polynomials over GF(2)

Abstract

The Discrete Logarithm Problem (DLP) is a fundamental cryptographic primitive. The DLP is defined for any cyclic group, specifically finite fields, whether the integers modulo a prime p or a polynomial field of characteristic p modulo some irreducible polynomial f(x). For polynomial fields over a finite field, also known as Galois fields, the DLP can be viewed as finding a solution to the equation 1 + x(i) = x(j) for arbitrary values of i (modulo some primitive polynomial). Solutions are (relatively) easy to find for trinomials and these would be the easiest polynomials to implement in hardware. However, primitive trinomials do not exist for all degrees. Primitive polynomials are irreducible polynomials with an associated primitive root alpha that is a generator of the multiplicative group. Thus the generator alpha generates all nonzero 2(n) - 1 elements of a Galois field whose base field is the integers modulo two. Primitive polynomials over the field of two elements, or GF(2), have important applications in cryptology and coding theory. This thesis investigates properties of polynomials with more than three terms where all but one term is a row of Pascal's triangle modulo two. In other words we define a certain class of polynomials by f(x) + x(n) + p(x) is a row of Pascal's triangle modulo two. This thesis shows that some of these polynomials, which are not trinomials, also have "easy" solutions. We observe that for a polynomial to have an associated primitive element, there are definite restrictions on the degree of the polynomial using particular rows of Pascal's triangle.