Symmetric digit sets for elliptic curve scalar multiplication without precomputation

Abstract

We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely E:y2=x3+Ax in prime characteristic p≡1normalmod0.2em4, and E:y2=x3+B in prime characteristic p≡1normalmod0.2em3. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we consider a width-w-NAF (Non-Adjacent Form) digit expansion of positive integers to the complex base of τ, where τ is a zero of the characteristic polynomial x2−tx+p of the Frobenius endomorphism associated to the curve. We provide a precomputationless algorithm by means of a convenient factorisation of the unit group of residue classes modulo τ in the endomorphism ring, whereby we construct a digit set consisting of powers of subgroup generators, which are chosen as efficient endomorphisms of the curve.

Cross-platform Identity-based Cryptography using WebAssembly

“…With the implementation of better-performing arithmetic functions, we can optimize CryptID. Our main goal is to improve the elliptic-curve arithmetic layer, with the implementation of the Heuberger-Mazzoli scalar multiplication [48], and with some useful tricks, like precalculations.…”

A Compact Coprocessor for the Elliptic Curve Point Multiplication over Gaussian Integers

“…Next, we consider Algorithm 2 which calculates the τ-adic expansion of an integer k [19]. Algorithm 2 is the ordinary division algorithm for base conversion when k and τ are both integers.…”

“…Similarly, Rabin cryptographic system was previously considered over Gaussian integers in [13,14]. In the context of ECC, arithmetic over complex numbers was proposed to speed up the point multiplication [15][16][17][18][19]. However, to the best of our knowledge, no ECC hardware implementation over Gaussian integers exists.…”