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Cryptography on twisted edwards curves over local fields

Science China(Information Sciences) 2015年第1期58卷 150-164页

作者： TANG ChunMing XU MaoZhi QI YanFeng School of Mathematics and Information China West Normal University Laboratory of Mathematics and Applied Mathematics School of Mathematical SciencesPeking University School of Science Hangzhou Dianzi University

twisted edwards curves over finite fields have attracted great interest for their efficient and unified addition formula. In this paper, we consider twisted edwards curves over local fields and introduce a cryptosystem based on quotient groups of twisted edwards curves over local fields. From the study of formal groups of twisted edwards curves and twisted edwards curves over local fields, we give the choice of cryptographic groups.An element in these groups can be uniformly represented by two n digit p-adic numbers, whereas an element in the elliptic curves in Weierstrass form over local fields is represented by a 3n-2 digit p-adic number and a4n-3 digit p-adic number. In the cryptography on elliptic curves in Weierstrass form over local fields, five cases for different input point pairs in computing points addition have to be considered and sometimes points have to be lifted. In the cryptography on twisted edwards curves over local fields, the addition formula is simple,unified, and complete, which is efficient, does not need lifting points, and is against the side channel analysis.Finally, a speedy point multiplication algorithm and some concrete instances are given.

关键词： local fields formal groups elliptic curve cryptography twisted edwards curves addition law

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Efficient Pairing Computation on twisted Weierstrass curves

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Chinese Journal of Electronics 2018年第4期27卷 739-745页

作者： WANG Bei OUYANG Yi HU Honggang School of Mathematical Sciences University of Science and Technology of China School of Information Science and Technology University of Science and Technology of China

In this paper, we construct the twists of twisted edwards curves in Weierstrass form. Then we define a new twisted Ate pairing on twisted Weierstrass curves named the Tx-Ate pairing. Following Miller’s algorithm, we give a computation of the Tx-Ate pairing on high degree twisted Weierstrass curves, where the point operations are over edwards form, and the computation of Miller function is over Weierstrass form. Although, in one doubling loop, our method to compute the Tx-Ate pairing is a litter slower than the previously fastest method.By twists, the Tx-Ate pairing can be calculated on more twisted Weierstrass curves with short loop length. The TxAte pairing is even competitive with optimal Ate pairing when they have the same short loop length.

关键词： Weierstrass curves twisted Ate pairing twisted edwards curves Miller function

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