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Solving the Multi-discrete logarithm Problems over a Group of Elliptic Curves with Prime Order

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Acta Mathematica Sinica,English Series 2005年第6期21卷 1443-1450页

作者： Jun Quan LI Mu Lan LIU Liang Liang XIAO Academy of Mathematics and Systems Science Key Laboratory of Mathematics Mechanization Chinese Academy of Sciences Beijing 100080 P. R. China

In this paper, we discuss the expected number of steps in solving multi-discrete logarithm problems over a group of elliptic curves with prime order by using Pollard＇s rho method and parallel collision search algorithm. We prove that when using these algorithms to compute discrete logarithms, the knowledge gained through computing many logarithms does not make it easier for finding other logarithms. Hence in an elliptic cryptosystem, it is safe for many users to share the same curve, with different private keys.

关键词： Pollard's rho method Parallel collision search algorithm Elliptic curve discrete logarithm Distinguished point

Threshold Signature Scheme Based on discrete logarithm and Quadratic Residue

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Wuhan University Journal of Natural Sciences 2004年第5期9卷 770-774页

作者： FEI Ru-chun 1,2 , WANG Li-na 1 1.School of Computer, Wuhan University, Wuhan 430072, Hubei, China 2.Department of Information Engineering, Benxi College of Metallurgy, Benxi 117022,Liaoning,China School of Computer Wuhan University Wuhan Hubei China Department of Information Engineering Benxi College of Metallurgy Benxi Liaoning China

Digital signature scheme is a very important research field in computer security and modern cryptography. A (k, n) threshold digital signature scheme is proposed by integrating digital signature scheme with Shamir secret sharing scheme. It can realize group-oriented digital signature, and its security is based on the difficulty in computing discrete logarithm and quadratic residue on some special conditions. In this scheme, effective digital signature can not be generated by anyk?1 or fewer legal users, or only by signature executive. In addition, this scheme can identify any legal user who presents incorrect partial digital signature to disrupt correct signature, or any illegal user who forges digital signature. A method of extending this scheme to an Abelian group such as elliptical curve group is also discussed. The extended scheme can provide rapider computing speed and stronger security in the case of using shorter key. Key words threshold scheme - digital signature - discrete logarithm - quadratic residuc - threshold digital signature CLC number TP 309. 7 Foundation item: Supported the National Nature Science Foundation of China, Hubei Province (90104005, 2002 AB0039)Biography: FEI Ru-chun (1964-), male, Ph. D candidate, Associated professor, research direction: information security and cryptography.

关键词： threshold scheme digital signature discrete logarithm quadratic residuc threshold digital signature

The discrete logarithm problem from a local duality perspective

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Science China Mathematics 2013年第7期56卷 1421-1427页

作者： HUANG MingDeh Department of Computer Science University of Southern CaliforniaLos AngelesCA 90089-0781 USA

The discrete logarithm problem is analyzed from the perspective of Tate local duality. Local duality in the multiplicative case and the case of Jacobians of curves over p-adic local fields are considered. When the local field contains the necessary roots of unity, the case of curves over local fields is polynomial time reducible to the multiplicative case, and the multiplicative case is polynomial time equivalent to computing discrete logarithm in finite fields. When the local field does not contains the necessary roots of unity, similar results can be obtained at the cost of going to an extension that contains these roots of unity. There was evidence in the analysis that suggests that the minimal extension where the local duality can be rationally and algorithmically defined must contain the roots of unity. Therefore, the discrete logarithm problem appears to be well protected against an attack using local duality. These results are also of independent interest for algorithmic study of arithmetic duality as they explicitly relate local duality in the case of curves over local fields to the multiplicative case and Tate-Lichtenbaum pairing (over finite fields).

关键词： discrete logarithm local duality

Security Analysis of discrete logarithm Based Cryptosystems

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Wuhan University Journal of Natural Sciences 2006年第6期11卷 1715-1717页

作者： WANG Yuzhu LIAO Xiaofeng School of Computer Chongqing University Chongqing 400030 China

discrete logarithm based cryptosysterns have subtle problems that make the schemes vulnerable. This paper gives a comprehensive listing of security issues in the systems and analyzes three classes of attacks which are based on mathematical structure of the group which is used in the schemes, the disclosed information of the subgroup and implementation details respectively. The analysis will, in turn, allow us to motivate protocol design and implementation decisions.

关键词： discrete logarithm security analysis prime order subgroup secure implementation

Fault Attacks on Hyperelliptic Curve discrete logarithm Problem over Finite Fields

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China Communications 2012年第11期9卷 150-161页

作者： Wang Mingqiang Xue Haiyang Zhan Tao School of Mathematics Shandong University Jinan 250100 P. R. China State Key Laboratory of Information Security Institute of Information Engineering Chinese Academy of Sciences Beijing 100864. P. R. China

In this paper, we present two explicit invalid-curve attacks on the genus 2 hyperelliptic curve over a finite field. First, we propose two explicit attack models by injecting a one-bit fault in a given divisor. Then, we discuss the construction of an invalid curve based on the faulted divisor. Our attacks are based on the fact that the Hyperelliptic Curve Scalar Multiplication (HECSM) algorithm does not utilize the curve parameters and We consider three hyperelliptic curves as the attack targets. For curve with security level 186 (in bits), our attack method can get the weakest invalid curve with security level 42 (in bits); there are 93 invalid curves with security level less than 50. We also estimate the theoretical probability of getting a weak hyperelliptic curve whose cardinality is a smooth integer. Finally, we show that the complexity of the fault attack is subexponential if the attacker can freely inject a fault in the input divisor. Cryptosystems based on the genus 2 hyperelliptic curves cannot work against our attack algorithm in practice.

关键词： hyperelliptic curve discrete logarithm finite field genus cryptosystem

Forgery Attacks on Wang＇s Signature Schemes Based on Factoring and discrete logarithm

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Journal of Systems Science and Information 2009年第1期7卷 43-48页

作者： Dezhi Gao Xiangqian Liang Information Science and Engineering College Shandong University of Science and TechnologyQingdao 266510 Shandong China

To enhance the security of signature schemes, Wang proposed two signature schemes based on the difficulties of simultaneously solving the factoring and discrete logarithm＇ problems with almost the same sizes of arithmetic modulus. This paper, firstly, shows that Wang＇s signature scheme 1 do not satisfy the claimed properties. Moreover, we will point out that the all variants of Wang＇s scheme 2 are not secure if attackers can solve discrete logarithm problems.

关键词： cryptography security digital signature factoring discrete logarithm

Threshold Signature Scheme Based on discrete logarithm and Quadratic Residue

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Threshold Signature Scheme Based on Discrete Logarithm and Q...

Reducing discrete logarithm Problem of Conic Curve Over GF(2n) to discrete logarithm Problem of Finite Field

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作者： FEI Ru-chun, WANG Li-na School of Computer, Wuhan University, Wuhan 430072, Hubei, China Department of Information Engineering, Benxi College of Metallurgy, Benxi 117022,Liaoning,China

＜正＞ Digital signature scheme is a very important research field in computer security and modern cryptography. A （k,n） threshold digital signature scheme is proposed by integrating digital signature scheme with Shamir secret sharing scheme. It can realize group-oriented digital signature, and its security is based on the difficulty in computing discrete logarithm and quadratic residue on some special conditions. In this scheme, effective digital signature can not be generated by any k- 1 or fewer legal users, or only by signature executive. In addition, this scheme can identify any legal user who presents incorrect partial digital signature to disrupt correct signature, or any illegal user who forges digital signature. A method of extending this scheme to an Abelian group such as elliptical curve group is also discussed. The extended scheme can provide rapider computing speed and stronger security in the case of using shorter key.

关键词： threshold scheme digital signature discrete logarithm quadratic residue threshold digital signature

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Reducing Discrete Logarithm Problem of Conic Curve Over GF(2...

Weak-Keys in Public Key Cryptosystems Based on discrete logarithms

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作者： Liu Duo Tan Zhiyong Dai Yiqi Department of Computer Science and Technology,Tsinghua University,Beijing,100084,Peoples Republic of China

The isomorphism from conic curve C over GF(2)to a subgroup of GF(2) is presented. Therefore it conclude that the security of cryptosystems base on the C over GF(2)is no stronger than those based on the multiplicative ... 详细信息

关键词： Cryptology Conic Curve Finite Field discrete logarithm

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Tsinghua Science and Technology 2005年第5期10卷 579-581页

作者：戴瑞恩罗平彭小宁王道顺 Department of Computer Science and Technology Tsinghua University Beijing 100084 China Department of Computer Science and Technology Huaihua University Huaihua 418008 China

The discrete logarithm method is the foundation of many public key algorithms. However, one type of key, defined as a weak-key, reduces the security of public key cryptosystems based on the discrete logarithm method. The weak-key occurs if the public key is a factor or multiple of the primitive element, in which case the user＇s private key is not needed but can be obtained based on the character of the public key. An algorithm is presented that can easily test whether there is a weak-key in the cryptosystem. An example is given to show that an attack can be completed for the Elgamal digital signature if a weak-key exists, therefore validating the danger of weak-keys. Methods are given to prevent the generation of these weak-keys.

关键词： discrete logarithm public key cryptography weak-key Elgamal digital signature