密码学报（中英文）2024，Vol.11Issue(5)：1152-1159,8.DOI:10.13868/j.cnki.jcr.000735
关于素域上的Koblitz曲线
On Koblitz Curves over Prime Fields
摘要
Abstract
The well-known class of Koblitz curves Ea over binary fields F2m is among the earliest curves in cryptography that are of both theoretical and practical significance.The Frobenius map τ:Ea(F2m)→ Ea(F2m),which is critical in the fast arithmetics for this class of Koblitz curves,is also connecting the cardinality of the underlying field and the number of rational points of the curve in the following manner:2m=N(τm),≠Ea(F2m)=N(τm-1),where N is the norm over Z[τ],the point counting formula is obtained through zeta function.Recently the cryptographic choice by some platforms of block-chain makes the Koblitz curves Eb:y2=x3+b/Fp over a prime field attracting attention,where the prime p=1(mod 3).There is a classical result of Raj wade for the point counting of Eb/Fp,with a different approach from that using zeta function.Based on Rajwade's formula,this paper derives a concise expression for the number of points of Eb.Our representation involves only complex arithmetics without quadratic or cubic residues,nor six-piece formula.The new result is in terms of the ring Z[w]of Eisenstein integers,together with a prime decomposition of p,we prove that there is a primary prime π ∈ Z[ω]and a unit u ∈ Z[ω]such that p=N(π),#Eb(Fp)=N(π-u).This is interesting as it is so similar to the case for binary Koblitz curves:there are two elements of Z[ω]whose difference is just a unit and their norms are the cardinality of the underlying field and the number of rational points of the curve respectively.To this end,we also develop some computational tools for cubic residue,including a whole spectrum for cubic residue character of 2.关键词
Koblitz曲线/Eisenstein整数/有理点数Key words
Koblitz curves/Eisenstein integers/number of rational points分类
信息技术与安全科学引用本文复制引用
伍涵,许光午..关于素域上的Koblitz曲线[J].密码学报（中英文）,2024,11(5):1152-1159,8.基金项目
国家重点研发计划(2022YFB2701700,2018YFA0704702) （2022YFB2701700,2018YFA0704702）
国家自然科学基金(12271306)National Key Research and Development Program of China(2022YFB2701700,2018YFA0704702) （12271306）
National Natural Science Foundation of China(12271306) （12271306）
