TY - JOUR

T1 - Faster maptopoint on supersingular elliptic curves in characteristic 3

AU - Kawahara, Yuto

AU - Kobayashi, Tetsutaro

AU - Takahashi, Gen

AU - Takagi, Tsuyoshi

N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2011/1

Y1 - 2011/1

N2 - Pairing-based cryptosystems are generally constructed us ing many functions such as pairing computation, arithmetic in finite fields, and arithmetic on elliptic curves. MapToPoint, which is a hashing algo rithm onto an elliptic curve point, is one of the functions for constructing pairing-based cryptosystems. There are two MapToPoint algorithms on supersingular elliptic curves in characteristic three, which is used by qy pairing. The first is computed by using a square root computation in IF and the computational cost of this algorithm is 0(log m) multiplications in F The second is computed by using an (m-1) × (m - 1) matrix over F It can be computed by O(1) multiplications in F However, this algorithm needs the off-line memory to store about m F3m -elements. In this paper, we propose an efficient MapToPoint algorithm on the supersingular ellip tic curves in characteristic three by using 1/3-trace over F3m We propose 1/3-trace over F3m which can compute solution x of x - x = c by using no multiplication in F3m The proposed algorithm is computed by O(1) multi plications in F 3m and it requires less than rn F3m -elements to be stored in the off-line memory to efficiently compute trace over F3m Moreover, in our software implementation of F 3m the proposed MapToPoint algorithm is approximately 35% faster than the conventional MapToPoint algorithm us ing the square root computation on an AMD Opteron processor (2.2 GHz).

AB - Pairing-based cryptosystems are generally constructed us ing many functions such as pairing computation, arithmetic in finite fields, and arithmetic on elliptic curves. MapToPoint, which is a hashing algo rithm onto an elliptic curve point, is one of the functions for constructing pairing-based cryptosystems. There are two MapToPoint algorithms on supersingular elliptic curves in characteristic three, which is used by qy pairing. The first is computed by using a square root computation in IF and the computational cost of this algorithm is 0(log m) multiplications in F The second is computed by using an (m-1) × (m - 1) matrix over F It can be computed by O(1) multiplications in F However, this algorithm needs the off-line memory to store about m F3m -elements. In this paper, we propose an efficient MapToPoint algorithm on the supersingular ellip tic curves in characteristic three by using 1/3-trace over F3m We propose 1/3-trace over F3m which can compute solution x of x - x = c by using no multiplication in F3m The proposed algorithm is computed by O(1) multi plications in F 3m and it requires less than rn F3m -elements to be stored in the off-line memory to efficiently compute trace over F3m Moreover, in our software implementation of F 3m the proposed MapToPoint algorithm is approximately 35% faster than the conventional MapToPoint algorithm us ing the square root computation on an AMD Opteron processor (2.2 GHz).

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U2 - 10.1587/transfun.E94.A.150

DO - 10.1587/transfun.E94.A.150

M3 - Article

AN - SCOPUS:78650945831

SN - 0916-8508

VL - E94-A

SP - 150

EP - 155

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

IS - 1

ER -