Classical modular polynomials over GF(2)

Classical modular polynomials over GF(2)

For a prime number l, the classical modular polynomial Phi_l (sometimes called the modular equation of level l) is a polynomial with integer coefficients such that Phi_l(j(t),j(lt))=0 where j(t) is the j-invariant well known from the theory of elliptic functions. While it is difficult to compute Phi_l over the integers due to the rapid growth of its coefficients, there is a beautiful structure emerging if one considers Phi_l over GF(2), where it becomes a sparse polynomial. Even though the modular polynomials considered over GF(2) have important computational applications in elliptic curves cryptography, it appears that the amazing structure of their non-vanishing terms has not been considered in the literature. By considering the power series expansion of j(q), we prove some necessary conditions that a monomial of Phi_l must satisfy in order to have coefficient 1, and we conjecture many more such conditions based on results from our computational investigations.