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: y 2 = x 3 + Ax in prime characteristic p ≡ 1 mod 4, and E: y 2 = x 3 + B in prime characteristic p ≡ 1 mod 3. 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 x 2 −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. 1

Cross-platform Identity-based Cryptography using WebAssembly

The explosive spread of the devices connected to the Internet has increased the need for efficient and portable cryptographic routines. Despite this fact, truly platformindependent implementations are still hard to find. In this paper, an Identitybased Cryptography library, called CryptID is introduced. The main goal of this library is to provide an efficient and opensource IBC implementation for the desktop, the mobile, and the IoT platforms. Powered by WebAssembly, which is a specification aiming to securely speed up code execution in various embedding environments, CryptID can be utilized on both the client and the server-side. The second novelty of CrpytID is the use of structured public keys, opening up a wide range of domain-specific use cases via arbitrary metadata embedded into the public key. Embedded metadata can include, for example, a geolocation value when working with geolocation-based Identitybased Cryptography, or a timestamp, enabling simple and efficient generation of singleuse keypairs. Thanks to these characteristics, we think, that CryptID could serve as a real alternative to the current Identitybased Cryptography implementations