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�������պ����ѧѧ�� 2011, Vol. 37 Issue (11) :1451-1455    DOI:
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����ǿ1, �����2, ����ƻ2, ������2*
1. �������պ����ѧ ����������������ص�ʵ����, ���� 100191;
2. ����ˮ��ˮ��ѧԺ ��Ϣ����ѧԺ, ֣�� 450011
Scalar multiplication algorithm of ECC based on precomputation and periodicity
Zhang Xiaoqiang1, Zhu Guiliang2, Wang Weiping2, Wang Mengmeng2*
1. State Key Lab of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China;
2. Department of Information Engineering, North China University of Water Conservancy and Electric Power, Zhengzhou 450011, China

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ժҪ ���о������ơ������ŵĶ�����(NAF,Non-Adjacent Form)�ȳ��������˷��㷨�Ļ�����,�����Բ���߻�����������Ժ�Ԥ���㱶�����з�ʽ,�����һ���µı����˷��㷨,�����������㷨����ϸ����.��������Ժ�ϵ��������ֱ�ӽ��б����˷����㻹��ת��Ϊ������Ԫ,Ԥ���㱶�����з�ʽ��������Բ������������(ECC,Elliptic Curve Cryptosystem)�ӽ��ܹ����д������ظ�����.Ϊ��֤�㷨����ȷ��,������Կ����Ϊ192 bit��Բ����,������һ������ʵ��.ʵ��������㷨��������:������ƺ�NAF�㷨���,���㷨��ռ����һЩ�洢�ռ�,��ʡȥ�˱��������ʱ�俪��,ͬʱ�����˵�ӵ��������,���������˱����˷������Ч��.���㷨�����������ECC���ۺͼӿ�ECC��ʵ���е�Ӧ�þ�����Ҫ����.
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Abstract�� Based on the binary method, non-adjacent form (NAF) method, etc., a new scalar multiplication algorithm was proposed, which uses the periodicity of based point and the precomputation mean. Meanwhile, the steps of new algorithm were given. The periodicity of based point and the coefficient of scalar multiplication determine performing the operation of scalar multiplication directly or computing its inverse element. The precomputation mean can void a quantity of repeat computation during the encryption and decryption processes of elliptic curve cryptosystem (ECC). To verify the correctness of new algorithm, a concrete experiment was offered with an elliptic curve, whose key length is 192 bit. The experimental results and algorithm analyses show that comparing with binary and NAF methods, although the new algorithm requires a little extra space to store precomputed points, it does not need the operation of point doublings and reduces the operation times of point addition. Therefore, the new algorithm can improve the efficiency of scalar multiplication sharply. The research achievement is significant for completing the theory of ECC and accelerating its application in practice.
Keywords�� elliptic curve cryptosystem(ECC)   binary method   non-adjacent form(NAF) method   scalar multiplication   precomputation   periodicity     
Received 2010-07-02;
About author: ����ǿ(1983-),��,�����ڻ���,��ʿ��,grayqiang@163.com.
����ǿ, �����, ����ƻ, ������.����Ԥ����������Ե�ECC�����˷��㷨[J]  �������պ����ѧѧ��, 2011,V37(11): 1451-1455
Zhang Xiaoqiang, Zhu Guiliang, Wang Weiping, Wang Mengmeng.Scalar multiplication algorithm of ECC based on precomputation and periodicity[J]  JOURNAL OF BEIJING UNIVERSITY OF AERONAUTICS AND A, 2011,V37(11): 1451-1455
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