Abstract. We present a new method to evaluate large degree isogenies between elliptic curves over finite fields. Previous approaches all have exponential running time in the logarithm of the degree. If the endomorphism ring of the elliptic curve is ‘small ’ we can do much better, and we present an algorithm with a running time that is polynomial in the logarithm of the degree. We give several applications of our techniques to pairing based cryptography. 1

Abstract. We develop a new p-adic algorithm to compute the minimal polynomial of a class invariant. Our approach works for virtually any modular function yielding class invariants. The main algorithmic tool is modular polynomials, a concept which we generalize to functions of higher level. 1.

Abstract. Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions. 1.

Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8

Solutions by radicals at singular values kN from new class invariants for N ≡ 3 mod 8