Doing More with Fewer Bits

Abstract. Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden" element α of a finite field IFp of p elements from rather short strings of the most significant bits of the remainder modulo p of αt for several values of t selected uniformly at random from IF * p . We use some recent bounds of exponential sums to generalize this algorithm to the case when t is selected from a quite small subgroup of IF * p . Namely, our results apply to subgroups of size at least p 1/3+ε for all primes p and to subgroups of size at least p ε for almost all primes p, for any fixed ε > 0. We also use this generalization to improve (and correct) one of the statements of the aforementioned work about the computational security of the most significant bits of the Diffie-Hellman key.

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“…The results of that paper and some of their improvements in [15] have applications to the security of the new cryptosystem designed in [4,10].…”

Section: Remarksmentioning

confidence: 96%

On the Security of Diffie-Hellman Bits

Vasco

Shparlinski

2001

Cryptography and Computational Number Theory

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“…Section 4) based on their representations c and c k , respectively, so that the value of c k+1 follows as the trace over GF(p 2 ) of g * g k ∈ GF(p 6 ). More precisely, the owner of the private key k computes c k = T r(g k ) given c = T r(g) and k. ) is the 'smallest' (or 'largest') 1 . It follows that c k+1 is the 'smallest' possibility given the pair (c, c k ).…”

Section: Ifmentioning

confidence: 99%

“…In [1] it was shown that conjugates of elements of a subgroup of GF(p 6 ) * of order dividing φ 6 (p) = p 2 − p + 1 can be represented using 2 log 2 (p) bits, as opposed to the 6 log 2 (p) bits that would be required for their traditional representation. In [4] an improved version of the method from [1] was introduced that achieves the same communication advantage at a much lower computational cost.…”

Section: Introductionmentioning

confidence: 99%

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Key Improvements to XTR

Lenstra

Verheul

2000

Advances in Cryptology — ASIACRYPT 2000

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Abstract. This paper describes improved methods for XTR key representation and parameter generation (cf. [4]). If the field characteristic is properly chosen, the size of the XTR public key for signature applications can be reduced by a factor of three at the cost of a small one time computation for the recipient of the key. Furthermore, the parameter set-up for an XTR system can be simplified because the trace of a proper subgroup generator can, with very high probability, be computed directly, thus avoiding the probabilistic approach from [4]. These non-trivial extensions further enhance the practical potential of XTR.

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“…The algebraic torus T d (F q ) is a (cyclic) subgroup of F × q n of order Φ d (q) for d | n, where Φ d (X) is the d-th cyclotomic polynomial. Such subgroups are recently used in several cryptographic schemes [3], [11], [16], [17], [21], which are called torus-based cryptosystems, in order to reduce the length of ciphertexts and public keys without violating the specified Manuscript received March 26, 2013. Manuscript revised July 19, 2013.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Computing Discrete Logarithms over Algebraic Tori

Isobe

Koizumi

Nishigaki

et al. 2014

IEICE Trans. Inf. & Syst.

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SUMMARYThis paper studies the complexity of computing discrete logarithms over algebraic tori. We show that the order certified version of the discrete logarithm problem over general finite fields (OCDL, in symbols) reduces to the discrete logarithm problem over algebraic tori (TDL, in symbols) with respect to the polynomial-time Turing reducibility. This reduction means that if the prime factorization can be computed in polynomial time, then TDL is equivalent to the discrete logarithm problem over general finite fields with respect to the Turing reducibility. key words: algebraic tori, order certified discrete logarithm, Turing reduction

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Doing More with Fewer Bits

Cited by 48 publications

References 13 publications

On the Security of Diffie-Hellman Bits

On the Security of Diffie-Hellman Bits

Key Improvements to XTR

On the Complexity of Computing Discrete Logarithms over Algebraic Tori

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