The difficulty of this problem is the basis for the security of several cryptographic systems, including diffiehellman key agreement, elgamal encryption, the elgamal. Nobody has admitted publicly to having proved that the discrete log cant be solved quickly, but many very smart people have tried hard and. Lower bounds for discrete logarithms and related problems victor shoup ibm researchzurich, saumerstr. For example, to find 46 mod 12, we could take a rope of length 46 units and rap it around a clock of 12 units, which is called the modulus, and where the rope ends is the. 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 discrete logarithm problem, circulant matrices. The elliptic curve discrete logarithm problem ecdlp is the following computational problem. These publickey schemes are used in todays security infrastructure to provide publickey encryption and authenticated key exchange. Consider the discrete logarithm problem in the group of integers modulo p under addition. On the discrete logarithm problem in elliptic curves.
For any element a of the group, one can compute log 10 a. Say, given 12, find the exponent three needs to be raised to. It can be denoted by log e but is usually denoted by ln. 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. As far as we know, this problem is very hard to solve quickly. Sample exponential and logarithm problems 1 exponential problems example 1. The only restriction is that the base and the modulus, and the power and the modulus must be relatively prime. In particular, we are interested in how their properties di.
The discrete logarithm problem is the computational task of. This brings us to modular arithmetic, also known as clock arithmetic. Pdf the discrete logarithm problem on elliptic curves of. Discrete logarithm problem imperial college london. 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. The discrete logarithm problem plays a central role in cryptographic protocols. If taking a power is of ot time, then finding a logarithm is of o2t2 time. Om log m bit operations even with an optimal sorting algorithm such as merge. Jan 22, 2014 consider the problem of computing the discrete logarithm in a generic group of a known prime order.
Exponential and logarithmic word problems solutions population 1. Steps for solving logarithmic equations containing only logarithms step 1. The focus in this book is on algebraic groups for which the dlp seems to be hard. The security of certain cryptosystems is based on the difficulty of this computation. Sometimes, we know that the discrete log lies in a subinterval. 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. 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. Generalized jacobian and discrete logarithm problem on elliptic.
To establish the exact complexity, not only of the discrete logarithm problem but also of its relatives, the diffiehellman dh problem and the decision dh problem, is of some importance. In this paper we study the discrete logarithm problem in the group of circulant matrices over a. The discrete logarithm problem is considered to be computationally intractable. Discrete logarithm problem or just integer exponentiation problem. The past, evolving present and future of discrete logarithm. The discrete logarithm problem in the group of nonsingular. In this paper, we study the computational power of generic algorithms. From discrete logarithm problem to menelaus theorem windows. This security is based on the difficulty of solving elliptic curve discrete logarithm problem. Discrete logarithm problem how is discrete logarithm.
Show that the discrete logarithm problem in this case can be solved in polynomialtime. But then computing logg t is really solving the congruence ng. Even though quantum computers are not commercial yet, they are speculated. Given 2 g, the discrete logarithm problem is to determine such that g.
The discrete logarithm problem in the group of non. Integer factorization and discrete logarithm problem mod composite are shown to be neither in p nor npcomplete. This paper presents a new methodology for the precomputation phase of the index calculus method icm, which is a popular attack on solving the discrete logarithm problem dlp. The discrete log problem is the analogue of this problem modulo. The second law of logarithms log a xm mlog a x 5 7.
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. If it is not possible for any k to satisfy this relation, print 1. 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. Logarithms which are not whole numbers logarithms do not have to be whole numbers.
This is usually done by calculating the logarithm of x to base 10, and dividing that by the logarithm of b to base 10 we can do this in modular arithmetic too. Here is a list of some factoring algorithms and their running times. Well email you at these times to remind you to study. If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. 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. Integer factorization and discrete logarithm problems. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete logarithms. The second scheme, in its attempt to improve mobile agents computation efficiency, applied the cryptosystem based on the difficulty of discrete logarithm problem to reduce the.
The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. Pdf for elliptic curve based cryptosystems, the discrete loga rithm problem must. Recent progress on the elliptic curve discrete logarithm problem. Due to w ork of menezes, ok amoto and v anstone, 2, it. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. The problem of nding this xis known as the discrete logarithm problem, and it is the basis of our trapdoor functions. However, although most mathematicians and computer scientists. We say that we solve the discrete logarithm problem dlp in g if, given any. Now we use that exponential base 3 and logarithm base 3 are inverse functions to see that log3 344. With the exception of dixons algorithm, these running times are all obtained using heuristic arguments. Pdf we survey recent work on the elliptic curve discrete logarithm problem. Put another way, compute, when as far as we know, this problem is very hard to solve quickly. Note that we are multiplying and dividing a logarithm by a plain number, not by another logarithm.
The complex logarithm, exponential and power functions in these notes, we examine the logarithm, exponential and power functions, where the arguments. 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. Common logarithm natural logarithm log 10 x log x log e x ln x evaluating common and natural logarithms evaluate a log 8 and b ln 0. The hardness of finding discrete logarithms depends on the groups. 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. But, the shors algorithm, created by peter shor, in the year 1994, can break these algorithms, if run on a quantum computer. Q2efq to nd an integer a, if it exists, such that q ap.
Solving discrete logarithms with partial knowledge of the key. If not, stop and use the steps for solving logarithmic equations containing terms without logarithms. The complex logarithm, exponential and power functions. Vanessa vitse uvsq elliptic curve discrete logarithm problem october 19, 2009 9 30. The discrete logarithm problem is a critical problem in number theory, and is similar in many ways to the integer factorization problem. Find an integer k such that where a and m are relatively prime. Recent progress on the elliptic curve discrete logarithm. Discrete logarithm problem mathematical and statistical. The elliptic curve discrete logarithm problem and equivalent. Lower bounds for discrete logarithms and related problems. This is a logarithm of base 4, so we write 16 as an exponential of base 4. Complex logarithm problem mathematics stack exchange.
The discrete logarithm problem plays an important role in cryptography. To avoid confusion with ordinary logs, we sometimes call this the. On discrete logarithm problem cryptography stack exchange. This paper considers the computational complexity of the discrete logarithm and related problems in the context of generic algorithmsthat is, algorithms which do not exploit any special prop. Instead of having access to the group itself, we may only manipulate encodings of its elements basically, a random mapping of the group to a sufficiently large alphabet via a group oracle. On the complexity of the discrete logarithm and diffie. This is usually done by calculating the logarithm of x to base 10, and dividing that by the logarithm of b to base 10. Sample exponential and logarithm problems 1 exponential. Solving discrete logarithms in smoothorder groups with cuda. Discrete logarithm find an integer k such that ak is. This set g is a cyclic group under multiplication, and 10 is a generator. If it were possible to compute discrete logs efficiently, it would be possible to break numerous thoughttobe unbreakable cryptographic schemes.
The discrete logarithm problem on elliptic curves of trace one. On improving integer factorization and discrete logarithm. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete. Latticebased cryptography for iot in a quantum world. Discrete logarithm problem on the other hand, given c and. These are instances of the discrete logarithm problem. The discrete logarithm problem plays a central role in cryptographic protocols and computational number theory. Consider the problem of computing the discrete logarithm in a generic group of a known prime order. Discrete logarithm problem using index calculus method. This paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime. Integer factorization and discrete logarithm problem are.
From discrete logarithm problem to menelaus theorem. Voiceover we need a numerical procedure, which is easy in one direction and hard in the other. Integer factorization and discrete logarithm problems halinria. G is called the elliptic curve discrete logarithm problem ecdlp. We shall see that discrete logarithm algorithms for finite fields are similar. We outline some of the important cryptographic systems that use discrete logarithms. That is, no efficient classical algorithm is known for computing discrete logarithms in general. This is an introduction to a series of pages that look at public key cryptography using the properties of discrete logarithms.
To summarize this process in one line, log3 81 log3 3 44 problem. Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. 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. 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. The discrete logarithm problem on elliptic curves of trace one article pdf available in journal of cryptology 123. Index calculus basic outlines remarks on the index calculus tradeo for the smoothness bound b if b too small, very few elements are decomposable if b too large, many relations needed and expensive linear algebra step.
Pdf recent progress on the elliptic curve discrete logarithm problem. We provide algorithms to solve the discrete logarithm problem for generic. 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. And this can be made prohibitively large if t log 2 q is large. We show that for any sequences of prime powers q i i. The number field sieve and the discrete logarithm problem 401 step 2. The discrete logarithm problem is one of the backbones in public key cryptography. Pdf solving discrete logarithms from partial knowledge of the key. We begin with a formal statement of the discrete logarithm problem. Sample exponential and logarithm problems 1 exponential problems. For large prime numbers p, computing discrete logarithms of elements of the multiplicative group z.