Fast 4 way vectorized ladder for the complete set of Montgomery curves

This paper introduces 4 way vectorization of Montgomery ladder on any Montgomery form elliptic curve. Our algorithm takes 2M^4+1S^4 (M^4: A vector of four field multiplications, S^4: A vector of four field squarings) per ladder step for variable-scalar variable-point multiplication. This paper also introduces new formulas for doing arithmetic over GF(2^255-19)

High-Performance Scalar Multiplication using 8-Dimensional GLV/GLS Decomposition

This paper explores the potential for using genus~2 curves over quadratic extension fields in cryptography, motivated by the fact that they allow for an 8-dimensional scalar decomposition when using a combination of the GLV/GLS algorithms. Besides lowering the number of doublings required in a scalar multiplication, this approach has the advantage of performing arithmetic operations in a 64-bit ground field, making it an attractive candidate for embedded devices. We found cryptographically secure genus 2 curves which, although susceptible to index calculus attacks, aim for the standardized 112-bit security level. Our implementation results on both high-end architectures (Ivy Bridge) and low-end ARM platforms (Cortex-A8) highlight the practical benefits of this approach

Fast Cryptography in Genus 2

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 curves is typically more involved but allows us to work with moduli of half the size. We give a taxonomy of the best known techniques to realize genus 2 based cryptography, which includes fast formulas on the Kummer surface and efficient 4-dimensional GLV decompositions. By studying different modular arithmetic approaches on these curves, we present a range of genus 2 implementations. On a single core of an Intel Core i7-3520M (Ivy Bridge), our implementation on the Kummer surface breaks the 125 thousand cycle barrier which sets a new software speed record at the 128-bit security level for constant-time scalar multiplications compared to all previous genus 1 and genus 2 implementations

Faster group operations on elliptic curves

This paper improves implementation techniques of Elliptic Curve Cryptography. We introduce new formulae and algorithms for the group law on Jacobi quartic, Jacobi intersection, Edwards, and Hessian curves. The proposed formulae and algorithms can save time in suitable point representations. To support our claims, a cost comparison is made with classic scalar multiplication algorithms using previous and current operation counts. Most notably, the best speeds are obtained from Jacobi quartic curves which provide the fastest timings for most scalar multiplication strategies benefiting from the proposed 12M + 5S + 1D point doubling and 7M + 3S + 1D point addition algorithms. Furthermore, the new addition algorithm provides an efficient way to protect against side channel attacks which are based on simple power analysis (SPA). Keywords: Efficient elliptic curve arithmetic,unified addition, side channel attack

On Kummer Lines with Full Rational 2-torsion and Their Usage in Cryptography

Contains fulltext : 219228.pdf (publisher's version ) (Closed access) Contains fulltext : 219228.pdf (preprint version ) (Closed access

Faster Group Operations on Special Elliptic Curves

This paper is on efficient implementation techniques of Elliptic Curve Cryptography. We improve group operation timings 1 for Hessian and Jacobi-intersection forms of elliptic curves. In this study, traditional coordinates of these forms are modified to speed up the addition operations. For the completeness of our study, we also recall the modified Jacobiquartic coordinates which benefits from similar optimizations. The operation counts on the modified coordinates of these forms are as follows