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On the coefficients of the polynomial in the number field sieve

Science China(Information Sciences) 2015年第11期58卷 182-190页

作者： YANG Min MENG QingShu WANG ZhangYi LI Li ZHANG HuanGuo International School of Software Wuhan University Computer School Wuhan University

Polynomial selection is very important in the number field sieve. If the number of relations a pair of polynomials can generate is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first, which can speed up the selection of good polynomials. In this paper, we aim to study the correlation between polynomial coefficients and the number of relations the polynomials can generate. By analyzing the zero roots, it is found that a polynomial with the ending coefficient containing more small primes usually can generate more relations than the one whose ending coefficient contains less. As a polynomial with more real roots usually can generate more relations, using the complete discrimination system,the requirements on the coefficients of a polynomial to obtain more real roots are analyzed. For instance, a necessary condition for a polynomial of degree d to have d distinct real roots is that the coefficient of degree d- 2should be negative or small enough. The result in the case d = 3 can be used directly in selecting polynomials generated by the nonlinear method, where d = 3 is already enough for practical purpose.

关键词： cryptography integer factorization number field sieve polynomial selection coefficients zero roots complete discrimination system

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New Weak Keys in RSA

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Wuhan University Journal of Natural Sciences 2006年第6期11卷 1529-1532页

作者： WANG Baocang LIU Shuanggen HU Yupu Key Laboratory of Computer Network and Information Security Ministry of Edueation Xidian University Xi＇an 710071 Shaanxi China College of Computer Information Engineering Jiangxi Normal University Nanchang 330022 Jiangxi China

The security of the RSA system with the prime pairs of some special form is investigated. A new special-purpose algorithm for factoring RSA numbers is proposed. The basic idea of the method is to factor RSA numbers by factoring a well-chosen quadratic polynomial with integral coefficients. When viewed as a general-purpose algorithm, the new algorithm has a high computational complexity. It is shown thai the RSA number n = pq can be easily factored if p and q have the special form of p = as＋b, q=cs＋d, where a, b, c, d are relatively small numbers. Such prime pairs （p, q） are the weak keys of RSA, so when we generate RSA modulus, we should avoid using such prime pairs （p, q）.

关键词： integer factorization RSA number public key cryptosystem special-purpose algorithm

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Further research on public-key cryptosystems based on third-order recurrence sequence

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Frontiers of Electrical and Electronic Engineering in China 2006年第3期1卷 287-290页

作者： JIANG Zheng-tao SUN Xi TIAN Lei WANG Yu-min National Key laboratory of Integrated service Networks Xidian UniversityXi’an 710071China Present address:School of Computer Science Beihang UniversityBeijing 100083China National Key Laboratory of Integrated Service Networks Xidian UniversityXi’an 710071China Vocational College of Qingdao University Qingdao 266101China

Properties of third-order recurrence sequences were investigated and a new variant of the GH public-key cryptosystem,which was further improved to be a probabil-istic public-key cryptosystem,was *** security analysis of the proposed scheme was provided and it was proved that the one-wayness of the proposed scheme is equivalent to partial discrete logarithm and its semantic se-curity is equivalent to decisional Diffie-Hellman problem in ring ***,efficiency analysis of the proposed scheme was provided,and that these two encryption schemes need to transfer 2log N and 4log N bits data re-spectively.

关键词： public-key cryptosystem third-order linear recurrence sequence (trapdoor)discrete logarithm integer factorization

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A NEW PUBLIC-KEY ENCRYPTION SCHEME BASED ON LUCAS SEQUENCE

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Journal of Electronics(China) 2005年第5期22卷 490-497页

作者： Jiang Zhengtao Hao Yanhua Wang Yumin National Key Lab of Integrated Service Networks Xidian Univ. Xi＇an 710071 China

Investigated the properties of LUCas sequence(LUC), the paper proposed a new variant of (probabilistic) public-key encryption scheme. Security analysis of the proposed encryption schemes shows that its one-wayness is equivalent to partial LUC discrete logarithm problem in ZN, and for the proposed probabilistic encryption scheme, its semantic security is equivalent to decisional LUC Diffie-Hellman problem in ZN. At last, the efficiency of the proposed schemes is briefly analyzed.

关键词： Probabilistic public-key encryption scheme LUCas sequence(LUC) Discrete logarithm integer factorization

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