Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's
ECM factorization algorithm. Since then, his curves and the algorithms associated with them …

Abstract Let ℰ∕ 𝔽 q be an elliptic curve, and P a point in ℰ (𝔽 q) of prime order ℓ. Vélu's
formulæ let us compute a quotient curve ℰ′= ℰ∕⟨ P⟩ and rational maps defining a …

The CRC Handbook of Finite Fields (hereafter referred to as the Handbook) is a reference
book for the theory and applications of finite fields. It is not intended to be an introductory …

In this paper, we describe an algorithm to compute chains of (2, 2)-isogenies between
products of elliptic curves in the theta model. The description of the algorithm is split into …

We introduce Four Q, a high-security, high-performance elliptic curve that targets the 128-bit
security level. At the highest arithmetic level, cryptographic scalar multiplications on Four Q …

In this paper, we highlight the benefits of using genus 2 curves in public-key cryptography.
Compared to the standardized genus 1 curves, or elliptic curves, arithmetic on genus 2 …

Abstract Let (A, L, Θ n) be a dimension g abelian variety together with a level n theta
structure over a field k of odd characteristic. We thus denote by (θ i Θ L)(Z/n Z) g∈ Γ (A, L) …

For counting points of Jacobians of genus 2 curves over a large prime field, the best known
approach is essentially an extension of Schoof's genus 1 algorithm. We propose various …

This paper sets new speed records for high-security constant-time variable-base-point Diffie–
Hellman software: 305395 Cortex-A8-slow cycles; 273349 Cortex-A8-fast cycles; 88916 …

This paper introduces EECM-MPFQ, a fast implementation of the elliptic-curve method of
factoring integers. EECM-MPFQ uses fewer modular multiplications than the well-known …