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Practical Cryptanalysis of a Public Key Cryptosystem Based on the Morphism of Polynomials Problem

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Tsinghua Science and Technology 2018年第6期23卷 671-679页

作者： Jaihui Chen Chik How Tan Xiaoyu Li School of Computer Guangdong University of Technology Guangzhou 510006 China Temasek Laboratories NationalUniversity of Singapore Singapore 117411 Singapore the School of Computer Zhengzhou University of Aeronautics Zhengzhou 450046 China

Multivariate Public Key Cryptography （MPKC） has intensively and rapidly developed during the past three decades. MPKC is a promising candidate for post-quantum cryptography. However, designing it is universally regarded as a difficult task to design a secure MPKC foundation scheme, such as an encryption scheme and key exchange scheme. In this work, we investigate the security of a new public key cryptosystem that is based on the Morphism of Polynomials （MP）. The public key cryptosystem proposed by Wang et al. （Wuhan University, China） comprises a key exchange scheme and encryption scheme. Its security can be provably reduced to the hardness of solving a new difficult problem, namely, the Decisional Multivariate Diffie Hellman （DMDH） problem. This problem Js a variant of the MP problem, which is difficult to solve by random systems. We present a proposition that reduces the DMDH problem to an easy example of the MP problem. Then, we propose an efficient algorithm for the Key Recover Attack （KRA） on the schemes of the public key cryptosystem. In practice, we are able to entirely break the cryptosystem＇s claimed parameter of 96 security levels in less than 17.252 s. Furthermore, we show that finding parameters that yield a secure and practical scheme is impossible.

关键词： Cryptanalysis post-quantum cryptography multivariate public key cryptosystems morphism ofpolynomials problem

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Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem

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Science China(Information Sciences) 2016年第5期59卷 133-143页

作者： Jinhui LIU Huanguo ZHANG Jianwei JIA Houzhen WANG Shaowu MAO Wanqing WU Computer School of Wuhan University Key Laboratory of Aerospace Information security and trusted computing Ministry of Education

Advances in quantum computation threaten to break public key cryptosystems such as RSA, ECC,and El Gamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now,no quantum algorithms have been found to be able to solve certain mathematical problems on non-commutative algebraic structures. Against this background, Raulynaitis et al. have proposed a novel asymmetric cipher protocol using a matrix decomposition problem. Their proposed scheme is vulnerable to a linear algebra attack based on the probable occurrence of weak keys in the generation process. In this paper, we show that the asymmetric cipher of the non-commutative cryptography scheme is vulnerable to a linear algebra attack and that it only requires polynomial time to obtain the equivalent keys for some given public keys. We also propose an improvement to enhance the scheme of Raulynaitis et al.

关键词： cryptography post-quantum computational cryptography asymmetric cipher protocol Cryptanalysis matrix decomposition

Cryptanalysis of Schemes Based on Polynomial Symmetrical Decomposition

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Chinese Journal of Electronics 2017年第6期26卷 1139-1146页

作者： LIU Jinhui ZHANG Huanguo JIA Jianwei Computer School of Wuhan University Key Laboratory of Aerospace Information Security and Trusted Computing Ministry of Education

Advances in quantum computation threaten to break public key cryptosystems such as RSA,ECC, and El Gamal that are based on the difficulty of factorization or taking a discrete logarithm, although up to now, no quantum algorithms have been found that are able to solve certain mathematical problems on non-commutative algebraic structures. Against this background, some novel public key cryptography based on Polynomial symmetrical decomposition(PSD) problem have been proposed. We find that these schemes are not secure. We present that they are vulnerable to structural attack, linearization equations attack, overdefined systems of multivariate polynomial equations attack and that, they only require polynomial time complexity to retrieve the same secret key for some given public keys respectively.We also propose an improvement to enhance public key cryptography based on PSD problem. In addition, we discuss possible lines of future work.

关键词： Cryptography Post-quantum computational cryptography Cryptanalysis Polynomial symmetrical decomposition(PSD) problem Computational complexity

Cryptanalysis and improvement of quantum broadcast communication and authentication protocol with a quantum one-time pad

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Chinese Physics B 2016年第8期25卷 59-64页

作者：刘志昊陈汉武 School of Computer Science and Engineering Southeast University Key Laboratory of Computer Network and Information Integration (Southeast University) Ministry of Education Center for Quantum Computation and Intelligent Systems Faculty of Engineering and Information TechnologyUniversity of Technology Sydney

The security of quantum broadcast communication（QBC） and authentication protocol based on Greenberger–Horne–Zeilinger（GHZ） state and quantum one-time pad is analyzed. It is shown that there are some security issues in this protocol.Firstly, an external eavesdropper can take the intercept–measure–resend attack strategy to eavesdrop on 0.369 bit of every bit of the identity string of each receiver without being detected. Meanwhile, 0.524 bit of every bit of the secret message can be eavesdropped on without being detected. Secondly, an inner receiver can take the intercept–measure–resend attack strategy to eavesdrop on half of the identity string of the other＇s definitely without being checked. In addition, an alternative attack called the CNOT-operation attack is discussed. As for the multi-party QBC protocol, the attack efficiency increases with the increase of the number of users. Finally, the QBC protocol is improved to a secure one.

关键词： Cryptanalysis quantum broadcast communication information leakage intercept-measure-resendattack CNOT-operation attack

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Cryptanalysis of full PRIDE block cipher

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Science China(Information Sciences) 2017年第5期60卷 169-180页

作者： Yibin DAI Shaozhen CHEN State Key Laboratory of Mathematical Engineering and Advanced Computing Zhengzhou Information Science and Technology Institute

PRIDE is a lightweight block cipher proposed at CRYPTO 2014 by Albrecht et al., who claimed that the construction of linear layers is efficient and secure. In this paper, we investigate the key schedule and find eight 2-round iterative related-key differential characteristics, which can be used to construct 18-round related-key differentials. A study of the first subkey derivation function reveals that there exist three weak-key classes, as a result of which all the differences of subkeys for each round are identical. For the weak-key classes,we also find eight 2-round iterative related-key differential characteristics. Based on one of the related-key differentials, we launch an attack on the full PRIDE block cipher. The data and time complexity are 2;chosen plaintexts and 2;encryptions, respectively. Moreover, by using multiple related-key differentials, we improve the Cryptanalysis, which then requires 2;chosen plaintexts and 2;encryptions, respectively. Finally, we use two 17-round related-key differentials to analyze full PRIDE, which requires 2;plaintexts and 2;encryptions.These are the first results on full PRIDE, and show that the PRIDE block cipher is not secure against related-key differential attack.

关键词： Cryptanalysis block cipher PRIDE iterative characteristics related-key differential

Cryptanalysis of a Key Exchange Protocol Based on Commuting Matrices

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Chinese Journal of Electronics 2017年第5期26卷 947-951页

作者： JIA Jianwei LIU Jinhui ZHANG Huanguo Computer School of Wuhan University Key Laboratory of Aerospace Information Security and Trusted Computing Ministry of Education

A key exchange protocol is considered unsafe.The scheme is based on a set of m commuting square singular matrices of dimension n×n over a finite field,and its security is claimed to rely on the hardness of the matrix version discrete logarithm problem.However,the proposal’s design allows for a clean attack strategy.We show that the key exchange protocol is vulnerable to a linear algebra attack which only requires polynomial time to obtain the equivalent keys for all given public keys.We conduct a detailed analysis on the attack method and provide some improved suggestions on the key exchange protocol based on commuting matrices.

关键词： Cryptography Key exchange protocol Cryptanalysis Discrete logarithm problem Linear equations

Cryptanalysis of an MOR cryptosystem based on a finite associative algebra

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Science China(Information Sciences) 2016年第3期59卷 137-145页

作者： Wanqing WU Huanguo ZHANG Houzhen WANG Shaowu MAO Shuomei WU Haiqing HAN Computer School of Wuhan University Computer Department Shijiazhuang University School of Mathematics and Physics Hubei Polytechnic University

The Shor algorithm is effective for public-key cryptosystems based on an abelian group. At CRYPTO 2001, Paeng（2001） presented a MOR cryptosystem using a non-abelian group, which can be considered as a candidate scheme for post-quantum attack. This paper analyses the security of a MOR cryptosystem based on a finite associative algebra using a quantum algorithm. Specifically, let L be a finite associative algebra over a finite field F. Consider a homomorphism φ : Aut（L） → Aut（H） × Aut（I）, where I is an ideal of L and H ■ L/I.We compute dim Im（φ） and dim Ker（φ）, and combine them by dim Aut（L） = dim Im（φ） + dim Ker（φ）. We prove that Im（φ） = Stab Comp（H,I）(μ + B;（H, I）) and Ker（φ）■ Z;（H, I）. Thus, we can obtain dim Im（φ）, since the algorithm for the stabilizer is a standard algorithm among abelian hidden subgroup algorithms. In addition,Z;（H, I） is equivalent to the solution space of the linear equation group over the Galois fields GF（p）, and it is possible to obtain dim Ker（φ） by the enumeration theorem. Furthermore, we can obtain the dimension of the automorphism group Aut（L）. When the map ? ∈ Aut（L）, it is possible to effectively compute the cyclic group ? and recover the private key a. Therefore, the MOR scheme is insecure when based on a finite associative algebra in quantum computation.

关键词： MOR cryptosystem Cryptanalysis quantum algorithm finite associative algebra hidden subgroup problem stabilizer

Generalized Cryptanalysis of RSA with small public exponent

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Science China(Information Sciences) 2016年第3期59卷 97-106页

作者： Mengce ZHENG Honggang HU Zilong WANG Key Laboratory of Electromagnetic Space Information Chinese Academy of Sciences (CAS) School of Information Science and Technology University of Science and Technology of China

In this paper, we demonstrate that there exist weak keys in the RSA public-key cryptosystem with the public exponent e = N;N;. In 1999, Boneh and Durfee showed that when α≈ 1 and the private exponent d = N;< N;, the system is insecure. Moreover, their attack is still effective for 0.5 < α < 1.875.We propose a generalized cryptanalytic method to attack the RSA cryptosystem with α≤0.5. For c = [（1-α）/α] and e;≡ d（mod e;）, when γ, β satisfy γ < 1+1/c-1/（2αc） and β < αc +7/6- αγc-1/3（6α + 6αc + 1- 6αγc）;, we can perform cryptanalytic attacks based on the LLL algorithm. The basic idea is an application of Coppersmith’s techniques and we further adapt the technique of unravelled linearization, which leads to an optimized lattice.Our advantage is that we achieve new attacks on RSA with α 0.5 and consequently, there exist weak keys in RSA for most α.

关键词： Cryptanalysis RSA LLL algorithm Coppersmith’s techniques unravelled linearization

Cryptanalysis of Prime Power RSA with two private exponents

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Science China(Information Sciences) 2015年第11期58卷 59-66页

作者： ZHENG MengCe HU HongGang Key Laboratory of Electromagnetic Space Information Chinese Academy of Sciences School of Information Science and Technology University of Science and Technology of China

In this paper, we consider a variant of RSA schemes called Prime Power RSA with modulus N= prq for r 2, where p, q are of the same bit-size. May showed that when private exponent d 详细信息

In this paper, we consider a variant of RSA schemes called Prime Power RSA with modulus N= prq for r 2, where p, q are of the same bit-size. May showed that when private exponent d

关键词： Cryptanalysis Prime Power RSA two private exponents LLL algorithm Coppersmith’s techniques