Integer factorization and discrete logarithm problem are. This problem is the fundamental building block for elliptic curve cryptography and pairingbased cryptography, and has been a major area of research in computational number. Discrete logarithm problem on the other hand, given c and. As all of the npcomplete problems turned out to be impossible to solve in polynomial time by a classical computer, heuristic approaches or algorithms for restricted types of inputs need. What is the difference between discrete logarithm and. We outline some of the important cryptographic systems that use discrete logarithms. The main purpose of this paper is to examine the con ditions under which the dl problem with a composite. Suppose h gx for some g in the finite field and secret integer x.
The definition of a logarithm indicates that a logarithm is an exponent. Informally, the oracle complexity of a problem is the number of queries of such an oracle that are needed in order to solve the problem in polynomial time. 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. We show that for any sequences of prime powers q i i. Solving discrete logarithms with partial knowledge of the key. The discrete logarithm problem is the computational task of. Find an integer k such that where a and m are relatively prime. Solving a 112bit prime elliptic curve discrete logarithm problem on game consoles using sloppy reduction joppe w. Consider the discrete logarithm problem in the group of integers modulo p under addition. Discrete logarithms, diffiehellman, and reductions 3 oracle that gives correct answers to yesorno questions or, equivalently, to queries asking for one bit of data. Introduction let p be a prime number and n a positive integer, and let q d pn.
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. That is, no efficient classical algorithm is known for computing discrete logarithms in general. The discrete logarithm problem for g is to find, for given, a nonnegative integer x if it exists such that. The discrete logarithm problem is to find the exponent in the expression base exponent power mod modulus this applet works for both prime and composite moduli. Discrete logarithm cryptography, in its broadest sense, is concerned with cryptographic schemes whose security relies on the intractability of the discrete logarithm problem dlp, together with the underlying mathematical structures, implementation methods, performanceusability comparisons etc. 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. This is not the right place to completely describe the discrete logarithm problem. Pdf the discrete logarithm problem characteristics are not only used. The elliptic curve discrete logarithm problem ecdlp is the following computational problem. The pohlighellman algorithm is best optimized for solving gx b mod p when p1 has small prime factors. We shall see that discrete logarithm algorithms for finite fields are similar. The discrete logarithm dl problem with modulus n and base a is that of solving w ax mod n for the integer x when the integers a, n, w are given, and in general is a hard problem. The smallest such integer x is called the discrete logarithm of to the base, and is written.
Pairingfriendly curves with discrete logarithm trapdoor. Logarithms and their properties definition of a logarithm. Network vulnerability, system security, discrete logarithm, integer factorization, multilevel decomposition, complexity analysis. The problem of computing discrete logarithms is fundamental in computational alge. The discrete logarithm problem with auxiliary inputs. I do not understand the difference between these two concepts. Shors discrete logarithm quantum algorithm for elliptic. Used algorithms for prime generationcheck fermats test and miler rabins test to implement discrete logarithm bsgs, and inverse and exponentiation extended euclids algorithm. Interestingly, no constructions for elliptic curves in the. Let g be a cyclic group of finite order n with a generator g. Several important algorithms in publickey cryptography base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Math 5410 discrete logarithm problem let f gfq and take as a primitive element of f. Any mixer could be chosen by the voter as a transmitter to send the ballot and cut off the network address on the way.
Furthermore, there are no choices for s1 that more readily allow the veri. Assume t and u are elements in f q with the property that u is in. On discrete logarithm problem cryptography stack exchange. It is intrinsically related to the di ehellman problem dhp. At asiacrypt 2000, paillier proposed several encryption schemes invoking such curves 14, and more recently teske 19 suggested an elliptic curve cryptosystem with a trapdoor for the discrete logarithm problem. In our model, an adversary may use a very large amount of precomputationtoproduceanadvice stringaboutaspeci. But, since the discrete logarithm problem is hard, it is computationally infeasible to determine any signi. A secure anonymous evoting system based on discrete logarithm problem chinling chen1. The elliptic curve discrete logarithm problem and equivalent. If solving the discrete logarithm problem is easy, the elgamal and di ehellman systems can easily be broken. An oracle is a theoretical constanttime \black box function. Algorithms for discrete logarithms in finite fields and elliptic curves. A public key cryptosystem and a signature scheme based on discrete logarithms author. As far as we know, this problem is very hard to solve quickly.
The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. With the exception of dixons algorithm, these running times are all obtained using heuristic arguments. F we explain the basic algorithms based on combining congruences for solving the integer factorization and the discrete logarithm problems. The discretelogarithm problem with preprocessing henrycorrigangibbsanddmitrykogan stanforduniversity may12,2018 abstract. On the discrete logarithm problem in elliptic curves claus diem august 9, 2010 dedicated to gerhard frey abstract we study the elliptic curve discrete logarithm problem over. Elementary thoughts on discrete logarithms the library at msri. This video was made by 6 multimedia university students. The problem has survived scrutiny for a few decades and no easy solution has yet been publicized. The difficulty of this general discrete logarithm problem depends on the representation of the group. Di ehellman problem reduces to the discrete logarithm problem, imagine you have an algorithm to e ciently compute discrete logs and you are given the task of solving the di ehellman problem.
A public key cryptosystem and a signature scheme based on. The discrete logarithm problem is considered to be computationally intractable. Let p be a prime number and n a positive integer, and let q d p n. Dlp can be stated in various 1 this chapter is the 9th chapter of the book guide to pairingbased cryptography, edited by nadia. The discrete logarithm problem dlp is a classical hard problem in computational number theory, and forms the basis of many cryptographic schemes. On the discrete logarithm problem in elliptic curves. Say, given 12, find the exponent three needs to be raised to. A general algorithm for computing log b a in finite groups g is to raise b to larger and larger powers k until the desired a is found.
If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. Why is the discrete logarithm problem assumed to be hard. The focus in this book is on algebraic groups for which the dlp seems to be hard. Thispaperstudiesdiscretelogalgorithmsthatusepreprocessing.
This applet works for both prime and composite moduli. The discrete logarithm problem dlp is one of the most used mathematical problems. In the equation is referred to as the logarithm, is the base, and is the argument. The impact of the number field sieve on the discrete logarithm. The author deeply regrets that, due to space and time constraints, it is not exhaustive. Discrete logarithm find an integer k such that ak is congruent modulo b given three integers a, b and m.
Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. The discrete logarithm problem is to find a given only the integers c,e and m. Request pdf the discrete logarithm problem for large prime numbers p, computing discrete logarithms of elements of the multiplicative group z. Elliptic curve elliptic curf discrete logarithm discrete logarithm problem chinese remainder theorem. That formulation of the problem is incompatible with the complexity classes p, bpp, np, and so forth which people prefer to consider, which concern only decision yesno problems.
Put another way, compute, when as far as we know, this problem is very hard to solve quickly. The discrete logarithm problem is a critical problem in number theory, and is similar in many ways to the integer factorization problem. Q2efq to nd an integer a, if it exists, such that q ap. Any c in f has a unique representation as c m, for 0 discrete logarithm problem is not always hard. Discrete logarithms are quickly computable in a few special cases.
Notice that the discrete logarithm function is the inverse of the discrete exponentiation. If it were possible to compute discrete logs efficiently, it would be possible to break numerous thoughttobe unbreakable cryptographic schemes. Nevertheless, the most important examples of publickey cryptography using discrete logarithms, in terms of wide use and. Lenstra laboratory for cryptologic algorithms, ecole polytechnique f. Discrete logarithm records are the best results achieved to date in solving the discrete logarithm problem, which is the problem of finding solutions x to the equation g x h given elements g and h of a finite cyclic group g. Public key cryptography using discrete logarithms this is an introduction to a series of pages that look at public key cryptography using the properties of discrete logarithms. The discrete logarithm problem is to find the exponent in the expression base exponent power mod modulus. There are several ancillary problems to the dlp, the closest ones being the. As of 2006, the most efficient means known to solve the dhp is to solve the discrete logarithm problem dlp, which is to find x given g and g x.
If it is not possible for any k to satisfy this relation, print 1. On the discrete logarithm problem for primefield elliptic curves. Discrete logarithin hash function that is collision free. Pdf on the discrete logarithm problem researchgate. The discrete logarithm problem with auxiliary inputs yongsoo song. The hardness of finding discrete logarithms depends on the groups. Recent progress on the elliptic curve discrete logarithm problem. Discrete logarithms in finite fields and their cryptographic. Then you could easily compute afrom ga mod pand then compute gba mod p gab mod p. If taking a power is of ot time, then finding a logarithm is of o2t2 time. However, no efficient method is known for computing them in general. Given 2 g, the discrete logarithm problem is to determine such that g. A secure anonymous evoting system based on discrete.
To avoid confusion with ordinary logs, we sometimes call this the. For example, a popular choice of groups for discrete logarithm based cryptosystems is z p where p is a prime number. Integer factorization and discrete logarithm problem mod composite are shown to be neither in p nor npcomplete. Integer factorization and discrete logarithm problems pierrick gaudry october 2014 abstract these are notes for a lecture given at cirm in 2014, for the journees nationales du calcul ormel. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. The discrete logarithm problem is interesting because its used in public key cryptography rsa and the like. Public key cryptography using discrete logarithms in. We also relate the problem of eds association to the tate pairing and the mov, freyruc k and shipsey eds attacks on the elliptic curve discrete logarithm problem in the cases where these apply. As to cut the link between the voters and the ballots, the theorem of blind signature is used in many proposals 2,3,4,5,6,7,8,9 to solve such a problem. For example, consider g to be the cyclic group of order n. Here is a list of some factoring algorithms and their running times. Oct 20, 20 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 elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences kristin e.
Python implementation of the some tools for solving instances of the discrete log problem over galois fields finite fields. What is the difference between discrete logarithm and logarithm. This paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime. The shanks method and the kangaroo method of pollard can also be used to compute the discrete logarithm of in about j ehg6i steps when this discrete log is known to lie in an interval of. Babystepgianstep algorithm is better for p1 with larger prime factors. The discrete log problem is the analogue of this problem modulo. Fermats theorem and discrete logarithms css322, l11, y14 duration. These algorithms take a number of steps polynomial in the input size, e. The discrete logarithm problem with auxiliary inputs jung hee cheon, taechan kim and yongsoo song abstract. This video is about the brief explanation of discrete logarithm used in cryptography. Clearly, the discrete logarithm problem for a general group g is exactly the problem of inverting the exponentiation function defined by where n is the order of. The cryptoimmunity of numerous public key cryptogra phic protocols is based on the computational complexity of the discrete logarithm problems 1,2. The discrete logarithm problem is to find the e slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising.
And this can be made prohibitively large if t log 2 q is large. Discrete logarithm find an integer k such that ak is. The discrete logarithm problem dlp is one of the most used mathematical problems in asymmetric cryptography design, the other one being the integer factorization. That formulation of the problem is incompatible with the complexity classes p, bpp, np, and so forth which people prefer to consider, which concern only decision yesno. The discrete logarithm problem is most often formulated as a function problem, mapping tuples of integers to another integer. Show that the discrete logarithm problem in this case can be solved in polynomialtime. To show this claim, we rst introduce a way to model such a solution. Elliptic curve, discrete logarithm problem dlp, prime field.
It covers methods of domain parameter generation, domain parameter validation, key pair generation, public key validation, shared secret value calculation, key derivation, and test message authentication code computation for discrete logarithm problem based key agreement schemes. Integer factorization and discrete logarithm problems. Briefly, in elgammal cryptosystem with underlying group the group of units modulo a prime number p im told to find a subgroup of index 2 to solve discrete logarithm problem in order to break the system. Aug 19, 2016 this video is about the brief explanation of discrete logarithm used in cryptography. The only restriction is that the base and the modulus, and the power and the modulus must be relatively prime. Nobody has admitted publicly to having proved that the discrete log cant be solved quickly, but many very smart people have tried hard and.
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