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

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.

Montgomery Reduction for Gaussian Integers

“…Public-key cryptography based on the Rivest-Shamir-Adleman (RSA) system benefits from the efficient Montgomery modulo reduction [2,3]. Similarly, elliptic curve cryptography (ECC) systems over prime fields apply Montgomery reduction to support arbitrary prime curves [4][5][6][7][8][9][10][11][12][13][14].…”

“…Due to the isomorphism between Gaussian integer rings and ordinary integer rings, this arithmetic is suitable for many cryptography systems. The Rabin cryptography system and elliptic curve cryptography over Gaussian integers were considered in [19][20][21][22]. Moreover, coding applications over Gaussian integers use Gaussian integer arithmetic [23][24][25][26][27][28].…”

“…The proposed concept of Montgomery reduction was applied in [22] for the efficient calculation of the elliptic curve point multiplication. In particular, an area-efficient coprocessor was designed in [22]. This processor uses the proposed Montgomery modular arithmetic over Gaussian integers.…”

“…This processor uses the proposed Montgomery modular arithmetic over Gaussian integers. It is shown in [21,22] that a key representation with a Gaussian integer expansion is beneficial for the calculation of the point multiplication. This key representation reduces the computational complexity and the memory requirements of a secure hardware implementation, which is protected against side-channel attacks [30][31][32][33][34][35][36][37].…”

“…This key representation reduces the computational complexity and the memory requirements of a secure hardware implementation, which is protected against side-channel attacks [30][31][32][33][34][35][36][37]. In this work, we provide the theoretical justification for the reduction algorithms applied in [22]. Furthermore, we present new results for protected implementations of the point multiplication.…”