Abstract
This work presents a new concept to implement the elliptic curve point multiplication (PM). This computation is based on a new modular arithmetic over Gaussian integer fields. Gaussian integers are a subset of the complex numbers such that the real and imaginary parts are integers. Since Gaussian integer fields are isomorphic to prime fields, this arithmetic is suitable for many elliptic curves. Representing the key by a Gaussian integer expansion is beneficial to reduce the computational complexity and the memory requirements of secure hardware implementations, which are robust against attacks. Furthermore, an area-efficient coprocessor design is proposed with an arithmetic unit that enables Montgomery modular arithmetic over Gaussian integers. The proposed architecture and the new arithmetic provide high flexibility, i.e., binary and non-binary key expansions as well as protected and unprotected PM calculations are supported. The proposed coprocessor is a competitive solution for a compact ECC processor suitable for applications in small embedded systems.
Highlights
Many resource-constrained systems, such as embedded systems, still rely on symmetric cryptography for message authentication
We consider the key expansion algorithm from [41] which improves the resistance against timing attacks (TA) and simple power analysis (SPA). We demonstrate that this key expansion employing Gaussian integers reduces the memory requirements and computational complexity compared with other τ-adic expansions
We compare the latency of a point multiplication (PM) with binary keys, according to Algorithm 1, with a PM using τ-adic key expansions, according to Algorithm 3, to demonstrate that a protected PM using Gaussian integers can be as fast as an unprotected binary
Summary
Many resource-constrained systems, such as embedded systems, still rely on symmetric cryptography for message authentication. For digital signatures and key exchange protocols, asymmetric cryptography is required. Software implementations of public-key algorithms such as the Rivest-Shamir-Adleman (RSA) algorithm or elliptic curve cryptography (ECC) may be too slow on small embedded systems. Such systems benefit from hardware assistance, i.e., coprocessors optimized for public-key operations. For applications in embedded systems, ECC algorithms dominate, because shorter key lengths are required [1,2,3,4,5,6]. We investigate the elliptic curve cryptography over Gaussian integers
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