Robert granger, thorsten kleinjung, jens zumbragel download. For large prime numbers p, computing discrete logarithms of elements of the multiplicative group z. Find an integer k such that where a and m are relatively prime. As many cryptography techniques are based on integer factorization or discrete logarithm problem, the computational complexity of these problems are crucially important to ensure the computer security514. The discrete logarithm problem is most often formulated as a function problem, mapping tuples of integers to another integer. To avoid confusion with ordinary logs, we sometimes call this the. The discrete logarithm problem on elliptic curves of trace.
Solving discrete logarithms in smoothorder groups with cuda. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. Nobody has admitted publicly to having proved that the discrete log cant be solved quickly, but many very smart people have tried hard and not succeeded. An integer is a primitive root modulo p if for every relatively prime to p there is an integer x such that x mod p. Pdf on the discrete logarithm problem semantic scholar. Solving discrete logarithm problems faster with the aid of. The problem of finding x is called the discrete logarithm problem. Pollard, kangaroo method, solving discrete logarithm problem dlp using pseudorandom walks. A subexponential algorithm for the discrete logarithm. Consider the discrete logarithm problem in the group of integers modulo p under addition. We begin with a formal statement of the discrete logarithm problem. The rest of the thesis is devoted to a study of the discrete logarithm problem over. In contrast, it is not known if the discrete logarithm.
Example code for the algorithm is also provided by the author of that paper. Asian journal of research 12 12, 2017 issn 2433 202x. I have read about shors algorithm and my understanding is that it can be used to factor large numbers efficiently. In the general discrete logarithm problem 0 d discrete logarithm problem. The function problem version of discrete logarithm is a problem to. On the discrete logarithm problem in finite fields of fixed characteristic. The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. If we formulate an appropriate decision problem version of the discrete logarithm problem, we can show that it belongs to the intersection of the complexity classes np, conp, and bqp a decision problem version of discrete log. The discrete logarithm problem is the computational task of.
Nist recommended elliptic curves, previously specified in fips 1864 appendix d, are now included in draft special publication sp 800186, recommendations for discrete logarithmbased cryptography. Can shors algorithm, though, be used to solve this problem. Nist requests comments on the set of recommended and allowed elliptic curves included in draft nist sp 800186. The security of certain cryptosystems is based on the difficulty of this computation. Ecc requires smaller keys compared to nonec cryptography based on plain galois fields to provide equivalent security elliptic curves are applicable for key agreement, digital signatures, pseudorandom generators and other tasks. Clearly, as the group of units modulo a prime number is cyclic, if x is a generator then x2 generates a subgroup of index 2. If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. For the elliptic curve discrete logarithm problem, there are some particular cases where a. Discrete logarithm find an integer k such that ak is. The discrete logarithm problem is solvable by a deterministic polynomial time algorithm in on3.
Using shors algorithm to solve the discrete logarithm problem. Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. When n is a prime p, the complexity is then op p groupoperations. Computing prime factorization and discrete logarithms. Solution of the dlp modulo p given a factorization of p 1 adelmans subexponential algorithm for the discrete logarithm problem. Suppose i tell you that i have a secret number a that satisfies mathae \mod m cmath the discrete logarithm problem is to find a given only the integers c,e and m. The past, evolving present and future of discrete logarithm. The integer factorization problem ifp, the finite field discrete logarithm problem dlp and the elliptic curve discrete logarithm problem ecdlp are essentially the only three mathematical problems that the practical publickey cryptographic systems are based on. Well email you at these times to remind you to study. On improving integer factorization and discrete logarithm.
Given 2 g, the discrete logarithm problem is to determine such that g. Logarithms and discrete logarithm in block chain new. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms. Discrete logarithms are perhaps simplest to understand in the group z p, where p is the prime number. There are also algorithms which can solve a dlp with online complexity smaller than. Integer factorization and discrete logarithm problem are. This paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves whose group orders are equal to the order of the. Algorithm 14, are based on the discrete logarithm problem, and these primitives are used.
For the free group on n generators we prove that the discrete logarithm is distributed according to the standard gaussian when the logarithm is renormalized appropriately. Wiener, parallel collision search with cryptanalytic applications, j. Discrete logarithms an overview sciencedirect topics. If taking a power is of ot time, then finding a logarithm is of o2t2 time. In some sense, the discrete logarithm has a long history in number theory. Summation polynomials and the discrete logarithm problem on elliptic curves igor semaev department of mathematics university of leuven,celestijnenlaan 200b 3001 heverlee,belgium igor. An algorithm for solving the discrete log problem on. Used algorithms for prime generationcheck fermats test and miler rabins test to implement discrete logarithm bsgs, and inverse and exponentiation extended euclids algorithm. If it is not possible for any k to satisfy this relation, print 1. This video is about the brief explanation of discrete logarithm used in cryptography.
Google a paper titled computing a discrete logarithm in on3, which can be found at cornells arxiv website. Put another way, compute, when as far as we know, this problem is very hard to solve quickly. The discrete log problem is the analogue of this problem modulo. In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem on elliptic curves of trace one. The discrete logarithm problem is to compute d logg xgiven the group elements gand x. Given p, g and ga mod p, determine a othis would break diffiehellman and elgamal discrete log algorithms analogous to factoring, except no sieving othis makes discrete log harder to solve oimplies smaller numbers can be used for equivalent security, compared to factoring. Sage implementation of discrete logarithm in subgroup of. Discrete logarithm problem on the other hand, given c and.
Show that the discrete logarithm problem in this case can be solved in polynomialtime. Ellipticcurve cryptography ecc is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields. Solving discrete logarithms with partial knowledge of the key. Before we can describe the diffiehellman protocol, we must establish the. Discrete logarithm problem mathematical and statistical.
Pdf on the discrete logarithm problem researchgate. A subexponential algorithm for the discrete logarithm problem with applications to cryptography published in. It is called generator because applying the multiplication operation on one single element. Pramod pandya, in cyber security and it infrastructure protection, 2014. The discrete logarithm problem computer security and.
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