Elliptic curve scalar point multiplication using radix-4 Booth's algorithm [cryptosystems]

Abstract

The main back-bone operation in elliptic curve cryptosystems is scalar point multiplication. The most frequently used method implementing the scalar point multiplication, which is performed in the top level of GF (Galois field) multiplication and GF division, has been the double-and-add algorithm, which is being recently challenged by the NAF (non-adjacent format) algorithm. In this paper, we propose a more efficient and novel approach of a scalar multiplication method than existing double-and-add by applying redundant recoding which originates from radix-4 Booth's algorithm. We call the novel algorithm quad-and-add. After deriving the algorithm, we created a new GF operation, named point quadruple, and verified it with calculations of a real-world application. The derived numerical expressions were verified using both C programs and HDL (hardware description language). The proposed method can be utilized in many elliptic curve security applications for handling efficient and fast calculations.