On the discrete logarithm problem in elliptic curves. The discrete logarithm to the base g of h in the group g is defined to be x. We often use the idea that we have an oracle to show rough computational. The difficulty of this general discrete logarithm problem depends on the representation of the group. Elliptic curve elliptic curf discrete logarithm discrete logarithm problem chinese remainder theorem. Gf2discretelog class to facilitate this calculation. We present an argument for the fact that discrete logarithms of the numbers in any. If so, stop and use steps for solving logarithmic equations containing only logarithms. The motivation for this problem is that many security systems use oneway functions. Given 2 g, the discrete logarithm problem is to determine such that g. The function problem version of discrete logarithm is a problem to. It is a natural extension of the discrete logarithm problem.
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. Designing good algorithms to compute discrete logarithms is a problem that is. The definition of a logarithm indicates that a logarithm is an exponent. If and, then, so is a solution to the discrete logarithm problem if has order or or is a product of reasonably small primes, then there are some methods for attacking the discrete log problem on, which are beyond the scope of this book.
If taking a power is of ot time, then finding a logarithm is of o2t2 time. Computing discrete logarithms is believed to be difficult. The discrete log problem is the analogue of this problem modulo. For large prime numbers p, computing discrete logarithms of elements of the multiplicative group z. Put another way, compute, when as far as we know, this problem is very hard to solve quickly. Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. The rest of the silverpohlighellman algorithm raising to a power of each of the prime cofactors, and using the chinese remainder theorem to combine the discrete logarithm within each of the subgroups is very straightforward. Discrete logarithm problem is difficult under classical computing model. The discrete logarithm problem is most often formulated as a function problem, mapping tuples of integers to another integer. Discrete logarithin hash function that is collision free. We say a call to an oracle is a use of the function on a speci ed input, giving us our desired output. Adding is easy on elliptic curves, but undoing addition seems hard. 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. Recommendation for pairwise key establishment schemes.
An oracle is a theoretical constanttime \black box function. Why is the discrete logarithm problem hard to solve. Steps for solving logarithmic equations containing terms without logarithms step 1. Solving the discrete log with a composite modulus is exactly as hard as factoring 3. This paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime. 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.
This brings us to modular arithmetic, also known as clock arithmetic. We shall see that discrete logarithm algorithms for finite fields are similar. The discrete logarithm problem journey into cryptography. 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. Why is the discrete logarithm problem assumed to be hard.
The discrete logarithm problem is the computational task of. What i mean by this is usually called the discrete logarithm problem. Solving discrete logarithms with partial knowledge of the key. A kilobit hidden snfs discrete logarithm computation. No efficient general method for computing discrete logarithms on conventional computers is known. Modular exponentiation generates a permutation of all the numbers in the sequence less than the modulus. Cryptographic applications of the discrete logarithm problem rely on the fact that it is easy to compute.
Logarithms and their properties definition of a logarithm. Discrete logarithms are logarithms defined with regard to multiplicative cyclic groups. Solving a 676bit discrete logarithm problem in gf36n. Discrete logarithm problem alison free online courses. 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. 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. 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. We offer free personalized sat test prep in partnership with the test developer, the college board. The discrete logarithm problem dipartimento di matematica tor. The only restriction is that the base and the modulus, and the power and the modulus must be relatively prime. 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. This recommendation specifies keyestablishment schemes based on the discrete logarithm problem over finite fields and elliptic.
The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation ga mod n. Discrete logarithm problem mathematical and statistical. We show that for any sequences of prime powers q i i. 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.
There are many algorithms that will solve the discrete log problem much faster than this method, brute force search runs at a worst case of on, or in other words o2n. Integer factorization and discrete logarithm problem are. The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. Integer factorization and discrete logarithm problems. Time complexity exploration eulers totient function. 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. I will add here a simple bruteforce algorithm which tries every possible value from 1 to m and outputs a solution if it was found. Various so called squareroot attacks are discussed for the discrete logarithm problem in. Revision 3 recommendation for pairwise keyestablishment schemes using discrete logarithm cryptography elaine barker lily chen allen roginsky apostol vassilev.
For example, they enable encrypting a message, but reversing the encryption is. In the equation is referred to as the logarithm, is the base, and is the argument. 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. The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. This paper focuses on a prime modulus, p, for which it.
Using shors algorithm to solve the discrete logarithm problem. Sage implementation of discrete logarithm in subgroup of. In an elliptic curve dl problem, youll note the base field i. Linear feedback shift registers for the uninitiated, part. Solving discrete logarithms in smoothorder groups with cuda. The discrete logarithm problem 3 2 exponential algorithms following ideas in 11, section 7, we consider the following general problem. We begin with a formal statement of the discrete logarithm problem. Log in to save your progress and obtain a certificate in alisons free understanding cryptography and its role in digital communications online course. Pdf on the discrete logarithm problem researchgate. Discrete logarithm problem on the other hand, given c and. The security of certain cryptosystems is based on the difficulty of this computation. It is thus important to be able to compute efficiently, in order to verify that the elliptic curve one wishes to use for a cryptosystem doesnt have any.
Well email you at these times to remind you to study. The diffiehellman problem dhp is a mathematical problem first proposed by whitfield diffie and martin hellman in the context of cryptography. However, it is not known if solving discrete log with a prime modulus leads to efficient factoring 4. Voiceover we need a numerical procedure, which is easy in one direction and hard in the other. The above paper solves the discrete logarithm in time on3 not on3, two very different things. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete.
Koblitz and miller had insights aplenty, but the central observation in all of this is the following. Can shors algorithm, though, be used to solve this problem. The discretelogarithm problem with preprocessing cryptology. 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 dlp is one of the most used. The discrete logarithm problem is considered to be computationally intractable.
The main purpose of this paper is to examine the con ditions under which the dl problem with a composite. Introduction discrete logarithm problem motivations discrete logarithm problem dlp given g group and g. The difficulty of taking logarithms makes exponentiation in. For example, consider g to be the cyclic group of order n. Shuffling as in cards, if done precisely generates a permutation in the order of the cards.
That is, no efficient classical algorithm is known for computing discrete logarithms in general. Discrete logarithms are thus the finitegrouptheoretic analogue of ordinary logarithms, which solve the same equation for real numbers b and g, where b is the base of the logarithm and g is the. The most obvious approach to breaking modern cryptosystems is to attack the underlying mathematical problem. Computing computer science journey into cryptography modern cryptography. To avoid confusion with ordinary logs, we sometimes call this the. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g.
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