We propose a new space efficient operator to multiply elements lying in a binary field GF(2^k). Our approach is based on a novel system of representation called "Double Polynomial System" which set elements as a bivariate polynomials over GF(2). Thanks to this system of representation, we are able to use a Lagrange representation of the polynomials and then get a logarithmic time multiplier with a space complexity of O(k^1.31) improving previous best known method.