On the discrete logarithm problem for primefield elliptic curves article pdf available in finite fields and their applications 51. To avoid confusion with ordinary logs, we sometimes call this the. On the discrete logarithm problem in elliptic curves. Discrete logarithms, diffiehellman, and reductions 3 oracle that gives correct answers to yesorno questions or, equivalently, to queries asking for one bit of data. Solving discrete logarithms with partial knowledge of the key. The discrete logarithm problem on elliptic curves of trace one. Discrete logarithin hash function that is collision free.
The discrete logarithm problem is the computational task of finding a representative of this residue class. The discrete logarithm to the base g of h in the group g is defined to be x. In the following two theorems, qis always a prime power and na natural number. Discrete logarithm problem on the other hand, given c and. Discrete logarithm find an integer k such that ak is. Later we discuss the result for more general adversaries. Thispaperstudiesdiscretelogalgorithmsthatusepreprocessing. We try to establish that initial minors are a viable way to solve this problem. Find an integer k such that where a and m are relatively prime.
Similarly, if g and h are elements of a finite cyclic group g then a solution x of the equation g h is called a discrete logarithm to. Pdf blind signatures based on the discrete logarithm. 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. Original research paper new collisions to improve pollards. Its related to the usual logarithm, by the fact that if isnt an integer power of then is a lower bound on. An l algorithm for discrete logarithm computation and. The diffiehellman problem dhp is a mathematical problem first proposed by whitfield diffie and martin hellman in the context of cryptography. The discrete logarithm problem is the computational task of. If taking a power is of ot time, then finding a logarithm is of o2t2 time. 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 considered to be computationally intractable. Discrete logarithm problem or just integer exponentiation problem. Since it is not easy to compare all these assumptions, in an attempt to simplify the. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms. This video was made by 6 multimedia university students. For example, they enable encrypting a message, but reversing the encryption is. That is, no efficient classical algorithm is known for computing discrete logarithms in general.
Put another way, compute, when as far as we know, this problem is very hard to solve quickly. The discretelogarithm problem with preprocessing cryptology. Even if d is too large to be recovered by discrete logarithm methods, however, it may still be. If it were possible to compute discrete logs efficiently, it would be possible to break numerous thoughttobe unbreakable cryptographic schemes.
Consider the discrete logarithm problem in the group of integers modulo p under addition. Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. Given 2 g, the discrete logarithm problem is to determine such that g. Two new blind signature schemes based on the discrete logarithm problem are presented. Elgamal proposed a publickey cryptosystem and a signature scheme, in which the difficulty of breaking the system is based on the difficulty of computing a discrete logarithm in a finite group. An efficient approach is to use babystep, giantstep algorithm by using meet in the middle trick babystep giantstep algorithm. 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 difficulty of this general discrete logarithm problem depends on the representation of the group. The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. Discrete logarithm problem shanks, pollard rho, pohlighellman, index calculus discrete logarithm notation the computation of x in y gx mod p is called the dlp here x is equal to the discrete analogue of the logarithm x log g y mod p 1 the modulus is p 1 since the powers are added and multiplied mod p 1 according to fermats theorem. The discrete logarithm problem department of computer science. 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.
For large prime numbers p, computing discrete logarithms of elements of the multiplicative group. In contrast, it is not known if the discrete logarithm problem in the groups of rational points of elliptic. Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. The problem of nding this xis known as the discrete logarithm problem, and it is the basis of our trapdoor functions. To show this claim, we rst introduce a way to model such a solution. Its the lowest value such that, for given being integers as well as the unknowns being integer. The main purpose of this paper is to examine the con ditions under which the dl problem with a composite. And this can be made prohibitively large if t log 2 q is large. What is the difference between discrete logarithm and. The motivation for this problem is that many security systems use oneway functions. Discrete logarithin hash function that is collision free and.
The discrete logarithm problem is most often formulated as a function problem, mapping tuples of integers to another integer. A only partially related value is the discrete logarithm, used in cryptography via modular arithmetic. The discrete log problem is the analogue of this problem modulo. The problem has survived scrutiny for a few decades and no easy solution has yet been publicized. Pdf on the discrete logarithm problem researchgate. Pramod pandya, in cyber security and it infrastructure protection, 2014. Public key cryptography suffers from the defect that the systems that seem safe. Discrete logarithms in finite fields and their cryptographic.
Summation polynomials and the discrete logarithm problem on. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. On the discrete logarithm problem in elliptic curves ii. When n is a prime p, the complexity is then op p groupoperations.
In mathematics, specifically in abstract algebra and its applications, discrete logarithms are grouptheoretic analogues of ordinary logarithms. 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. For example, consider g to be the cyclic group of order n. If g is a multiplicative cyclic group and g is a generator of g, then from the definition of cyclic groups, we know every element h in g can be written as g x for some x. Given a cyclic group g of order m, a generator a of the group and a group element b, the problem is to find an integer k such that. The past, evolving present and future of discrete logarithm. Discrete logarithms are perhaps simplest to understand in the group z p, where p is the prime number. Blind signature schemes, an important cryptographic primitive, are useful in protocols that guarantee the anonymity of the participants. The discrete logarithm problem dlp is one of the most used mathematical.
However, although most mathematicians and computer scientists. Pdf on the discrete logarithm problem semantic scholar. If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. Before we can describe the diffiehellman protocol, we must establish the. If it is not possible for any k to satisfy this relation, print 1. Show that the discrete logarithm problem in this case can be solved in polynomialtime. The only restriction is that the base and the modulus, and the power and the modulus must be relatively prime. Elementary thoughts on discrete logarithms the library at msri. The discrete logarithm problem journey into cryptography. If d is too small say, less than 160 bits, then an adversary might be able to recover it by the baby stepgiant step method. Integer factorization and discrete logarithm problems. Integer factorization and discrete logarithm problem are. The hardness of finding discrete logarithms depends on the groups.
If solving the discrete logarithm problem is easy, the elgamal and di ehellman systems can easily be broken. The problem of computing discrete logarithms w as just a mathematical curiosity until, in 1 9 76, diffie and h ellman discribed a method of e x. Factoring and discrete logarithms the most obvious approach to breaking modern cryptosystems is to attack the underlying mathematical 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. For large prime numbers p, computing discrete logarithms of elements of the multiplicative group z. The problem of computing discrete logarithms is fundamental in computational alge.
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. Publickey cryptosystem based on the discrete logarithm. 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. 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. Summation polynomials and the discrete logarithm problem on elliptic curves igor semaev department of mathematics university of leuven,celestijnenlaan 200b 3001 heverlee,belgium igor. The discretelogarithm problem with preprocessing henrycorrigangibbsanddmitrykogan stanforduniversity may12,2018 abstract. 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. The security of certain cryptosystems is based on the difficulty of this computation. Then the discrete logarithm problem in the degree 0 class groups of curves cfq of genus. Well email you at these times to remind you to study.
Introductionbefore the middle of the last century, discrete logarithms were just common tools usedto perform calculations in. 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. Pdf on the discrete logarithm problem for primefield.
Then, with the development of cryptography, theirimportance raised considerably, especially after di. The focus in this book is on algebraic groups for which the dlp seems to be hard. In some sense, the discrete logarithm has a long history in number theory. Discrete logarithms an overview sciencedirect topics. We achieve the subexponential complexity in ol,o1 when both the discriminant and the degree of the extension tend to infinity by using techniques due to enge, gaudry and thome in the context of algebraic curves over finite fields. Due to w ork of menezes, ok amoto and v anstone, 2, it. On arithmetic and the discrete logarithm problem in class. We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. Discrete logarithm problem mathematical and statistical. 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. In contrast, it is not known if the discrete logarithm.
An oracle is a theoretical constanttime \black box function. Original research paper new collisions to improve pollard. The function problem version of discrete logarithm is a problem to. In particular, an ordinary logarithm logab is a solution of the equation a b over the real or complex numbers.
In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete logarithms. On discrete logarithm problem cryptography stack exchange. 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. The discrete logarithm problem this paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime.
19 422 904 190 1154 1010 79 684 406 1213 935 568 998 1502 1096 370 352 1060 1011 65 1001 974 57 656 83 1625 1644 373 627 633 729 1018 73 1157 940 436 855 1238 1145 885 1356 1072