A new approach to the discrete logarithm problem with auxiliary inputs

Cheon, Jung Hee; Kim, Tae-Chan

doi:10.1112/s1461157015000303

Cited by 5 publications

(3 citation statements)

References 20 publications

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“…The approach typically uses private and public keys to offer a safe conduit for information privacy and confidentiality. The complexity of most cryptosystems is rooted in the problems of Discrete Logarithm Problem [ 31 , 32 ] or the Integer Factorization Problem [ 33 ].…”

Section: Proposed Solutionmentioning

confidence: 99%

Method to Increase Dependability in a Cloud-Fog-Edge Environment

Stan

Enyedi

Corches

et al. 2021

Sensors

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Robots can be very different, from humanoids to intelligent self-driving cars or just IoT systems that collect and process local sensors’ information. This paper presents a way to increase dependability for information exchange and processing in systems with Cloud-Fog-Edge architectures. In an ideal interconnected world, the recognized and registered robots must be able to communicate with each other if they are close enough, or through the Fog access points without overloading the Cloud. In essence, the presented work addresses the Edge area and how the devices can communicate in a safe and secure environment using cryptographic methods for structured systems. The presented work emphasizes the importance of security in a system’s dependability and offers a communication mechanism for several robots without overburdening the Cloud. This solution is ideal to be used where various monitoring and control aspects demand extra degrees of safety. The extra private keys employed by this procedure further enhance algorithm complexity, limiting the probability that the method may be broken by brute force or systemic attacks.

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Section: Proposed Solutionmentioning

confidence: 99%

Method to Increase Dependability in a Cloud-Fog-Edge Environment

Stan

Enyedi

Corches

et al. 2021

Sensors

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show abstract

“…Given d dividing p − 1, either a number of queries to a Diffie-Hellman oracle or appropriate auxiliary input can be used to "lift" the problem to a order (p − 1)/d subgroup, where the discrete logarithm can be computed more easily. Kushwaha and Mahalanobis [24] observed that when α already lies in a sufficiently small subgroup of F * p , the modified baby-step giant-step algorithm of [9] and [11] can be used to find α without any calls to a Diffie-Hellman oracle [13] or auxiliary input [14]. Our main observation in this paper is that, although the approach of [24] does not appear to result in a faster method for computing discrete logarithms in general, it does reveal a new type of weak key for discrete logarithm based cryptosystems.…”

Section: Introductionmentioning

confidence: 99%

Removable weak keys for discrete logarithm-based cryptography

Jacobson

Kushwaha

2020

J Cryptogr Eng

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We describe a novel type of weak cryptographic private key that can exist in any discrete logarithm based public-key cryptosystem set in a group of prime order p where p − 1 has small divisors. Unlike the weak private keys based on numerical size (such as smaller private keys, or private keys lying in an interval) that will always exist in any DLP cryptosystems, our type of weak private keys occurs purely due to parameter choice of p, and hence, can be removed with appropriate value of p. Using the theory of implicit group representations, we present algorithms that can determine whether a key is weak, and if so, recover the private key from the corresponding public key. We analyze several elliptic curves proposed in the literature and in various standards, giving counts of the number of keys that can be broken with relatively small amounts of computation. Our results show that many of these curves, including some from standards, have a considerable number of such weak private keys. We also use our methods to show that none of the 14 outstanding Certicom Challenge problem instances are weak in our sense, up to a certain weakness bound.

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“…A second motivation comes from one of the classical questions of the theory of finite fields: constructing rational functions with a small image set or, more generally, with many repeated values. It has been shown that results of this type are of interest for certain cryptographic attacks; see [9,10,11,24,25], for example. More precisely, for algorithms of [9,10,11,24,25] it is important to have a polynomial or a rational function f ∈ F q (X) of prescribed degree (or with the degree in a prescribed dyadic interval) such that the map f : F q → F q has many "collisions", or, more formally, the equation…”

mentioning

confidence: 99%

Finding elliptic curves with a subgroup of prescribed size

Shparlinski

Sutherland

2016

Int. J. Number Theory

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Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m = o(p 1/2 (log p) −4 ), outputs an elliptic curve E over the finite field Fp for which the cardinality of E(Fp) is divisible by m. The running time of the algorithm is mp 1/2+o(1) , and this leads to more efficient constructions of rational functions over Fp whose image is small relative to p. We also give an unconditional version of the algorithm that works for almost all primes p, and give a probabilistic algorithm with subexponential time complexity.

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A new approach to the discrete logarithm problem with auxiliary inputs

Cited by 5 publications

References 20 publications

Method to Increase Dependability in a Cloud-Fog-Edge Environment

Method to Increase Dependability in a Cloud-Fog-Edge Environment

Removable weak keys for discrete logarithm-based cryptography

Finding elliptic curves with a subgroup of prescribed size

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