Analysis on the Elliptic Scalar Multiplication Using Integer Sub-Decomposition Method

Abstract

This study proposes a new approach called, integer sub-decomposition (ISD), to compute any multiple kP of a point P of order n lying on an elliptic curve. Our method depends, in computations, on fast endomorphisms ψ 1 and ψ 2 of elliptic curve over prime fields. The integer sub-decomposition to multiple kP , when the value of k is decomposed into two values k 1 and k 2 , where both values or one of them is not bounded by ±C √ n, is illustrated in the following formula:where −C √ n < k 11 , k 12 , k 21 , k 22 < C √ n. The integers k 11 , k 12 , k 21 and k 22 are computed by solving a closest vector problem in lattice. Consequently, as for this sub-decomposition, we have managed to increase the percentage of a successful computation of kP . Moreover, the gap in the proof of the bound of kernel K vectors of the reduction map T : (a, b) → a + λb(mod n) on ISD method will be filled through the analysis of the multiplier k, using two fast endomorphisms with minimal polynomials X 2 + rX i + s i for i = 1, 2, 3. In particular, we prove an integer sub-decomposition (ISD) with explicit constant kP = k 11 P + k 12 ψ 1 (P ) + k 21 P + k 22 ψ 2 (P ), with

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.…”

“…(1.4) where −C √ n < k 11 , k 12 , k 21 , k 22 < C √ n, where C > 1 as proven in [9]. The mathematical proofs of the computational complexity to compute a scalar multiplication kP on elliptic curve over a finite field F p are determined in this paper.…”