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.
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
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