Low-Cost, Low-Power FPGA Implementation of ED25519 and CURVE25519 Point Multiplication

Mohamad Ali Mehrabi (1) , Christophe Doche (1, 2)

1 Macquarie University
2 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

Twisted Edwards curves have been at the center of attention since their introduction by Bernstein et al. in 2007. The curve ED25519, used for Edwards-curve Digital Signature Algorithm (EdDSA), provides faster digital signatures than existing schemes without sacrificing security. The CURVE25519 is a Montgomery curve that is closely related to ED25519. It provides a simple, constant time, and fast point multiplication, which is used by the key exchange protocol X25519. Software implementations of EdDSA and X25519 are used in many web-based PC and Mobile applications. In this paper, we introduce a low-power, low-area FPGA implementation of the ED25519 and CURVE25519 scalar multiplication that is particularly relevant for Internet of Things (IoT) applications. The efficiency of the arithmetic modulo the prime number 2 255 - 19 , in particular the modular reduction and modular multiplication, are key to the efficiency of both EdDSA and X25519. To reduce the complexity of the hardware implementation, we propose a high-radix interleaved modular multiplication algorithm. One benefit of this architecture is to avoid the use of large-integer multipliers relying on FPGA DSP modules.

Domaines: Géométrie algébrique [math.AG]

Mohamad Ali Mehrabi, Christophe Doche. Low-Cost, Low-Power FPGA Implementation of ED25519 and CURVE25519 Point Multiplication. Information, 2019, 10 (9), pp.285. ⟨10.3390/info10090285⟩. ⟨hal-03602232⟩