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On Deterministic Polynomial-time Equivalence of Computing the CRT-RSA Secret Keys and Factoring

DEFENCE SCIENCE JOURNAL 2012年第2期62卷 122-126页

作者： Maitra, Subhamoy Sarkar, Santanu Indian Stat Inst Kolkata 700108 India

Let N = pq be the product of two large primes. Consider Chinese remainder theorem-Rivest, Shamir, Adleman (CRT-RSA) with the public encryption exponent e and private decryption exponents d(p), d(q). It is well known that given any one of d(p) or d(q) (or both) one can factorise N in probabilistic poly(log N) time with success probability almost equal to 1. Though this serves all the practical purposes, from theoretical point of view, this is not a deterministic polynomial time algorithm. In this paper, we present a lattice-based deterministic poly(log N) time algorithm that uses both d(p), d(q) (in addition to the public information e, N) to factorise N for certain ranges of d(p), d(q). We like to stress that proving the equivalence for all the values of d(p), d(q) may be a nontrivial task.

关键词： CRT-RSA cryptanalysis factorisation lll algorithm cryptosystems

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Some bounds and limits in the theory of Riemann's zeta function

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2012年第1期396卷 199-214页

作者： Arias de Reyna, Juan van de Lune, Jan Univ Seville Fac Matemat E-41080 Seville Spain CWI NL-1009 AB Amsterdam Netherlands

For any real a > 0 we determine the supremum of the real sigma such that zeta(sigma + it) = a for some real t. For 0 1 the results turn out to be quite different. We also determine the supremum E of the real parts... 详细信息

For any real a > 0 we determine the supremum of the real sigma such that zeta(sigma + it) = a for some real t. For 0 < a < 1, a = 1, and a > 1 the results turn out to be quite different. We also determine the supremum E of the real parts of the 'turning points', that is points sigma + it where a curve lm zeta(sigma + it) = 0 has a vertical tangent. This supremum E (also considered by Titchmarsh) coincides with the supremum of the real sigma such that zeta'(sigma + it) = 0 for some real t. We find a surprising connection between the three indicated problems: zeta(s) = 1, zeta'(s) = 0 and turning points of zeta(s). The almost extremal values for these three problems appear to be located at approximately the same height. (C) 2012 Elsevier Inc. All right reserved.

关键词： Zeta function lll algorithm Extreme values

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LOW-COMPLEXITY LATTICE REDUCTION ARCHITECTURE USING INTERPOLATION-BASED QR DECOMPOSITION FOR MIMO-OFDM SYSTEMS

LOW-COMPLEXITY LATTICE REDUCTION ARCHITECTURE USING INTERPOL...

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IEEE Asia Pacific Conference on Circuits and Systems (APCCAS)

作者： Liu, I-Wen Liao, Chun-Fu Lan, Fang-Chun Huang, Yuan-Hao Natl Tsing Hua Univ Inst Commun Engn Hsinchu Taiwan Natl Tsing Hua Univ Dept Elect Engn Hsinchu Taiwan

ISBN: (纸本)9781457717284;9781457717291

Lattice reduction (LR) technique is well known for its capability of improving MIMO detection performance. Nevertheless, the complexity increases significantly when LR is applied to all sub-carriers in the orthogonal-frequency division-multiplexing (OFDM) system. To address this issue, this research proposes an interpolation-based preprocessing architecture for LR-aided MIMO-OFDM system. The proposed architecture combines the interpolation-based QR decomposition and grouping architecture to decrease the iteration loops of LR by adopting lattice matrix of adjacent sub-carrier. The experiment compares the BER performance of the proposed method with those of other works in the channel of the 3GPP-LTE system. The experimental results show that the proposed LR architecture not only shortens latency but also reduce computational complexity and hardware cost for MIMO-OFDM systems.

关键词： Lattice Reduction QR decomposition lll algorithm MIMO OFDM

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An improved lll algorithm

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LINEAR ALGEBRA AND ITS APPLICATIONS 2008年第2-3期428卷 441-452页

作者： Luk, Franklin T. Tracy, Daniel A. Rensselaer Polytech Inst Dept Comp Sci Troy NY 12180 USA

The lll algorithm has received a lot of attention as an effective numerical tool for preconditioning an integer least squares problem. However, the workings of the algorithm are not well understood. In this paper, we present a new way to look at the lll reduction, which leads to a new implementation method that performs better than the original lll scheme. (c) 2007 Elsevier Inc. All rights reserved.

关键词： lll algorithm integer least squares unimodular transformation reduced basis Gauss transformation QR decomposition plane reflection condition number

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Inferring Sequences Produced by Nonlinear Pseudorandom Number Generators Using Coppersmith's Methods

Inferring Sequences Produced by Nonlinear Pseudorandom Numbe...

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15th International Conference on Practice and Theory in Public Key Cryptography (PKC)

作者： Bauer, Aurelie Vergnaud, Damien Zapalowicz, Jean-Christophe Agence Natl Securite Syst Informat 51 Blvd Tour Maubourg F-75700 Paris 07 SP France CNRS Ecole Normale Super INRIA F-75230 Paris 05 France INRIA Rennes Bretagne Atlantique F-35042 Rennes France

ISBN: (纸本)9783642300578;9783642300561

Number-theoretic pseudorandom generators work by iterating an algebraic map F (public or private) over a residue ring Z(N) on a secret random initial seed value v 0 is an element of Z(N) to compute values v(n+1) - F(v(n)) mod N for n is an element of N. They output some consecutive bits of the state value v(n) at each iteration and their efficiency and security are thus strongly related to the number of output bits. In 2005, Blackburn, Gomez-Perez, Gutierrez and Shparlinski proposed a deep analysis on the security of such generators. In this paper, we revisit the security of number-theoretic generators by proposing better attacks based on Coppersmith's techniques for finding small roots on polynomial equations. Using intricate constructions, we are able to significantly improve the security bounds obtained by Blackburn et al..

关键词： Nonlinear Pseudorandom number generators Euclidean lattice lll algorithm Coppersmith's techniques Unravelled linearization

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Cryptanalysis of RSA with Constrained Low Public Exponent

Cryptanalysis of RSA with Constrained Low Public Exponent

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中国密码学会2012年会

作者： Luo Ping Zhou Haijian Key Laboratory for Information System Security Ministry of Education Tsinghua National Laboratory for Information Science and Technology(TNlist)School of Software Tsinghua University Beijing 100084 China

In this paper,we study the RSA cryptosystem with low public key e =Nα and constrained private key d Let μ be a small integer and d ≡ d'modeμ When d' =eγ is sufficiently small and 0 ≤ γ ≤μ-1/8a(1+√9 +... 详细信息

关键词： RSA low public key cryptanalysis lattice basis reduction lll algorithm

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Solving Generalized Small Inverse Problems

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2011年第6期E94A卷 1274-1284页

作者： Kunihiro, Noboru Univ Tokyo Kashiwa Chiba 2778561 Japan

We introduce a "generalized small inverse problem (GSIP)" and present an algorithm for solving this problem. GSIP is formulated as finding small solutions of f(x(0), x(1),...x(n)) = x(0)h(x(1),...,x(n))+C = 0(mod M) for an n-variate polynomial h, non-zero integers C and M. Our algorithm is based on lattice-based Coppersmith technique. We provide a strategy for construction of a lattice basis for solving f = 0, which is systematically transformed from a lattice basis for solving h = 0. Then, we derive an upper bound such that the target problem can be solved in polynomial time in log M in an explicit form. Since GSIPs include some RSA-related problems, our algorithm is applicable to them. For example, the small key attacks by Boneh and Durfee are re-found automatically.

关键词： lll algorithm small inverse problem RSA lattice-based cryptanalysis

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THE TRACE PROBLEM FOR TOTALLY POSITIVE ALGEBRAIC INTEGERS

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JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2011年第3期90卷 341-354页

作者： Liang, Yanhua Wu, Qiang SW Univ China Dept Math Chongqing 400715 Peoples R China

Let a be a totally positive algebraic integer of degree d >= 2 and alpha(1) = alpha, alpha(2), ... alpha(d), be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of S-k/d, where S-k = Sigma(d)(i=1) alpha(k)(i) and k = 1, 2, 3. These improvements have consequences for the search of Salem numbers with negative traces.

关键词： absolute trace totally positive algebraic integer Salem number explicit auxiliary function semi-infinite linear programming lll algorithm

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Decoding by Embedding: Correct Decoding Radius and DMT Optimality

Decoding by Embedding: Correct Decoding Radius and DMT Optim...

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IEEE International Symposium on Information Theory (ISIT)

作者： Ling, Cong Liu, Shuiyin Luzzi, Laura Stehle, Damien Univ London Imperial Coll Sci Technol & Med Dept Elect & Elect Engn London SW7 2AZ England

ISBN: (纸本)9781457705953

In lattice-coded multiple-input multiple-output (MIMO) systems, optimal decoding amounts to solving the closest vector problem (CVP). Embedding is a powerful technique for the approximate CVP, yet its remarkable performance is not well understood. In this paper, we analyze the embedding technique from a bounded distance decoding (BDD) viewpoint. 1/(2 gamma)-BDD is referred to as a decoder that finds the closest vector when the noise norm is smaller than lambda(1)/(2 gamma), where lambda(1) is the minimum distance of the lattice. We prove that the Lenstra, Lenstra and Lovasz (lll) algorithm can achieve 1/(2 gamma)-BDD for gamma approximate to O(2(n/4)). This substantially improves the existing result gamma = O(2(n)) for embedding decoding. We also prove that BDD of the regularized lattice is optimal in terms of the diversity-multiplexing gain tradeoff (DMT).

关键词： BDD viewpoint Boolean functions CVP Complexity theory DMT optimality Data structures lll algorithm Lattices MIMO communication MIMO system Maximum likelihood decoding Signal to noise ratio bounded distance decoding viewpoint closest vector problem decoding diversity reception diversity-multiplexing gain tradeoff embedding decoding radius lattice-coded multiple-input multiple-output systems

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LATENCY-CONSTRAINED LOW-COMPLEXITY LATTICE REDUCTION FOR MIMO-OFDM SYSTEMS

LATENCY-CONSTRAINED LOW-COMPLEXITY LATTICE REDUCTION FOR MIM...

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IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Liao, Chun-Fu Lan, Fang-Chun Huang, Yuan-Hao Chiu, Po-Lin Natl Tsing Hua Univ Inst Commun Engn Hsinchu Taiwan

ISBN: (纸本)9781457705397

Recent studies have investigated lattice-reduction (LR) preprocessing technique for multiple-input multiple-output (MIMO) detection. However, if LR is applied to the orthogonal-frequency-division-multiplexing (OFDM) system, its complexity and latency increase greatly because of the large number of sub-carriers. This paper proposes a new processing architecture for LR-aided MIMO-OFDM system. This LR processing architecture reduces the number of iteration loops by using preprocessing matrix of adjacent sub-carrier. Beside, the grouping of sub-carriers can break the long critical computational path so as to comprise the computational complexity and latency. We simulate the proposed LR-aided MIMO-OFDM processing in the 3GPP-LTE system. The proposed method not only reduces the computational complexity but also shortens the latency for the lattice reduction.

关键词： MIMO OFDM Lattice Reduction lll algorithm

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