Point Multiplication using Integer Sub-Decomposition for Elliptic Curve Cryptography

Abstract

Abstract:In this work, we proposed a new approach called integer sub-decomposition (ISD) based on the GLV idea to compute any multiple kP of a point P of order n lying on an elliptic curve E. This approach uses two fast endomorphisms ψ 1 and ψ 2 of E over prime field F p to calculate kP. The basic idea of ISD method is to sub-decompose the returned values k 1 and k 2 lying outside the range √ n from the GLV decomposition of a multiplier k into integers k 11 , k 12 , k 21 and k 22 with − √ n < k 11 , k 12 , k 21 , k 22 < √ n. These integers are computed by solving a closest vector problem in lattice. The new proposed algorithms and implementation results are shown and discussed in this study.

Upper Bound of Scalars in the Integer Sub-decomposition Method: The Theoretical Aspects

“…indicates the sub-decomposition of k using algorithm (2) in Ref. [6,7] of the ISD sub-decomposition for a scalar, ), (mod (that have been proved in Ref. [6]).…”

Mathematical analysis of the computational complexity of integer sub-decomposition algorithm

“…Run ISD generators Algorithm (1) given in [9], [10] with input (n, λ 1 , λ 2 ) to find {v 3 , v 4 } and 12 and k 21 , k 22 . Use generalized computing wNAF Algorithm (1) or (2) in [11] to compute w j N AF expansions for j from 1 to 4 of integers k 11 , k 12 , k 21 and k 22 .…”

“…For that we have generated the EEA and develop a modified algorithm to perform the sub-decomposition process. On the other hand, cost of the necessary condition part (NCP) of ISD generators algorithm (1) given in [9], [10] can be used to determine the cost of the ISD generators.…”

“…These computations include identifying linearly independent vectors v 3 , v 4 and v 5 , v 6 in the kerT , which considers the integer lattice points in two dimensions. The components in each vector must satisfy the conditions represented by the components of each vector being less than √ n and the relation between the components in each vector is a relative prime to form ISD (1) given in [9], [10], is…”

“…Computing the ISD generator algorithms (1) given in [9], [10] can be implemented with the necessary condition defined in the second part of this algorithm. The computational cost of ISD generators can be determined as follows.…”