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Using LLLReduction for Solving RSA and Factorization Problems: A Survey
, 2007
"... 25 years ago, Lenstra, Lenstra and Lovasz presented their celebrated LLL lattice reduction algorithm. Among the various applications of the LLL algorithm is a method due to Coppersmith for finding small roots of polynomial equations. We give a survey of the applications of this root finding method ..."
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Cited by 16 (0 self)
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25 years ago, Lenstra, Lenstra and Lovasz presented their celebrated LLL lattice reduction algorithm. Among the various applications of the LLL algorithm is a method due to Coppersmith for finding small roots of polynomial equations. We give a survey of the applications of this root finding method to the problem of inverting the RSA function and the factorization problem. As we will see, most of the results are of a dual nature: They can either be interpreted as cryptanalytic results or as hardness/security results.
A unified approach to deterministic encryption: New constructions and a connection to computational entropy
 TCC 2012, volume 7194 of LNCS
, 2012
"... We propose a general construction of deterministic encryption schemes that unifies prior work and gives novel schemes. Specifically, its instantiations provide: • A construction from any trapdoor function that has sufficiently many hardcore bits. • A construction that provides “bounded ” multimessa ..."
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Cited by 11 (1 self)
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We propose a general construction of deterministic encryption schemes that unifies prior work and gives novel schemes. Specifically, its instantiations provide: • A construction from any trapdoor function that has sufficiently many hardcore bits. • A construction that provides “bounded ” multimessage security from lossy trapdoor functions. The security proofs for these schemes are enabled by three tools that are of broader interest: • A weaker and more precise sufficient condition for semantic security on a highentropy message distribution. Namely, we show that to establish semantic security on a distribution M of messages, it suffices to establish indistinguishability for all conditional distribution ME, where E is an event of probability at least 1/4. (Prior work required indistinguishability on all distributions of a given entropy.) • A result about computational entropy of conditional distributions. Namely, we show that conditioning on an event E of probability p reduces the quality of computational entropy by a factor of p and its quantity by log 2 1/p. • A generalization of leftover hash lemma to correlated distributions. We also extend our result about computational entropy to the average case, which is useful in reasoning about leakageresilient cryptography: leaking λ bits of information reduces the quality of computational entropy by a factor of 2 λ and its quantity by λ.
Efficient pseudorandom generators based on the ddh assumption, ePrint 2006/321
 In PKC 2007, volume ???? of LNCS
, 2007
"... Abstract. A family of pseudorandom generators based on the decisional DiffieHellman assumption is proposed. The new construction is a modified and generalized version of the Dual Elliptic Curve generator proposed by Barker and Kelsey. Although the original Dual Elliptic Curve generator is shown to ..."
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Abstract. A family of pseudorandom generators based on the decisional DiffieHellman assumption is proposed. The new construction is a modified and generalized version of the Dual Elliptic Curve generator proposed by Barker and Kelsey. Although the original Dual Elliptic Curve generator is shown to be insecure, the modified version is provably secure and very efficient in comparison with the other pseudorandom generators based on discrete log assumptions. Our generator can be based on any group of prime order provided that an additional requirement is met (i.e., there exists an efficiently computable function that in some sense enumerates the elements of the group). Two specific instances are presented. The techniques used to design the instances, for example, the new probabilistic randomness extractor are of independent interest for other applications. 1
Secure PRNGs from Specialized Polynomial Maps over Any Fq
"... Abstract. Berbain, Gilbert, and Patarin presented QUAD, a pseudo random number generator (PRNG) at Eurocrypt 2006. QUAD (as PRNG and stream cipher) may be proved secure based on an interesting hardness assumption about the onewayness of multivariate quadratic polynomial systems over F2. The origina ..."
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Abstract. Berbain, Gilbert, and Patarin presented QUAD, a pseudo random number generator (PRNG) at Eurocrypt 2006. QUAD (as PRNG and stream cipher) may be proved secure based on an interesting hardness assumption about the onewayness of multivariate quadratic polynomial systems over F2. The original BGP proof only worked for F2 and left a gap to general Fq. We show that the result can be generalized to any arbitrary finite field Fq, and thus produces a stream cipher with alphabets in Fq. Further, we generalize the underlying hardness assumption to specialized systems in Fq (including F2) that can be evaluated more efficiently. Barring breakthroughs in the current stateoftheart for systemsolving, a rough implementation of a provably secure instance of our new PRNG is twice as fast and takes 1/10 the storage of an instance of QUAD with the same level of provable security. Recent results on specialization on security are also examined. And we conclude that our ideas are consistent with these new developments and complement them. This gives a clue that we may build secure primitives based on specialized polynomial maps which are more efficient.