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 elliptic curve discrete logarithm problem and equivalent. Pdf discrete logarithms in gfp using the number field sieve. With the exception of dixons algorithm, these running times are all obtained using heuristic arguments. Obviously it is easy if the target element t is the group identity, or in general, some small power of g. The secure identi cation option of the sun network file system, for example, uses discrete logarithms in a eld gfp with p a prime of 192 bits. 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.
Algorithms for discrete logarithms in finite fields and elliptic curves. The elliptic curve discrete logarithm problem ecdlp is the following computational problem. Discrete logarithms are quickly computable in a few special cases. Nobody has admitted publicly to having proved that the discrete log cant be solved quickly, but many very smart people have tried hard and. But then computing logg t is really solving the congruence ng. Faster individual discrete logarithms cryptology eprint archive iacr. We outline some of the important cryptographic systems that use discrete logarithms.
Computing discrete logarithms in finite fields is a main concern in cryptography. Logarithms transform multiplication and division processes to addition and subtraction processes which are much simpler. For example, a popular choice of groups for discrete logarithm based cryptosystems is z p where p is a prime number. All uploads and downloads are deemed secure and files are permanently deleted from the smallpdf servers within an hour. Typical examples are e cient proofs of knowledge of a discrete logarithm which are based on schnorrs digital signature scheme 18 and systems for proving the equality of two discrete logarithms, as used in 8. A public key cryptosystem and a signature scheme based on discrete logarithms author. Introduction to logarithms dear reader logarithms are a tool originally designed to simplify complicated arithmetic calculations. Discrete logarithm problem on the other hand, given c and. While there have been substantial advances in discrete log algorithms in the last two decades. Elementary thoughts on discrete logarithms the library at msri. 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. They were extensively used before the advent of calculators. Sep 16, 2017 last time we talked about the multiplicative inverse in finite fields, which is rather boring and mundane, and has an easy solution with blankinships algorithm discrete logarithms, on the other hand, are much more interesting, and this article covers only the tip of the iceberg. Pdf solving discrete logarithms from partial knowledge of the key.
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. The first practical public key cryptosystem to be published, the diffiehellman key exchange algorithm, was based on the assumption that discrete logarithms are hard to compute. The algorithm succeeds when one of the following sets contains the discrete logarithm we are looking for. Solving discrete logarithms with partial knowledge of the key. What is the difference between discrete logarithm and logarithm. On the discrete logarithm problem in elliptic curves.
Here is a list of some factoring algorithms and their running times. 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. Computation of discrete logarithms in a multiplicative. We shall see that discrete logarithm algorithms for finite fields are similar. First, make a table that translates your list of numbers into logarithmic form by taking the log base 10 or common logarithm of each value.
Proof systems for general statements about discrete logarithms. To avoid confusion with ordinary logs, we sometimes call this the. As far as we know, this problem is very hard to solve quickly. Discrete logarithm in gf2 809 with ffs razvan barbulescu cnrs. Jan 12, 2012 lesson 4a introduction to logarithms mat12x 6 lets use logarithms and create a logarithmic scale and see how that works.
The hardness of finding discrete logarithms depends on the groups. The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences kristin e. The argument does not use the primality of p, but it may. A public key cryptosystem and a signature scheme based on.
Q2efq to nd an integer a, if it exists, such that q ap. Pdf discrete logarithms, diffiehellman, and reductions. Discrete logarithm computation in finite fields fpn with nfs. Discrete logarithms in finite fields and their cryptographic. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The discrete log problem is the analogue of this problem modulo.
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. Before you combine the files into one pdf file, use merge pdf to draganddrop pages to reorder or to delete them as you like. If taking a power is of ot time, then finding a logarithm is of o2t2 time. Nov 24, 2015 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. In the equation is referred to as the logarithm, is the base, and is the argument.
Discrete logarithms via the index calculus method included in this package is an implementation of the index calculus method for solving discrete logarithms in the multiplicative group of gfp, for p prime. We give an introduction to the discrete logarithm problem in cyclic groups and treat the most important methods for solving them. Solving discrete logarithms with partial knowledge of the key k. If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. However, no efficient method is known for computing them in general. Assume t and u are elements in f q with the property that u is in. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. More specifically, we consider the following abelian groups. We show that for any sequences of prime powers q i i. Given a primitive element g of a finite field gfq, the discrete logarithm of a. Linear feedback shift registers for the uninitiated, part iv. Elementary thoughts on discrete logarithms 387 the next reduction considers the case when the order of g is a prime power, say pa, where a1. Explaining logarithms a progression of ideas illuminating an important mathematical concept by dan umbarger.
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. Before the days of calculators they were used to assist in the process of multiplication by replacing. Applications of factoring and discrete logarithms to. 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. Say, given 12, find the exponent three needs to be raised to. Computation of discrete logarithms in prime fields citeseerx. What happens if a logarithm to a di erent base, for example 2, is required. Logarithms and their properties definition of a logarithm. This intractability hypothesis is also the foundation for the presumed security of a variety of other public key schemes. In this version of the discrete logarithm calculator only the pohlighellman algorithm is implemented, so the execution time is proportional to the square root of the largest prime factor of the modulus minus 1. Discrete logarithms in gfp using the number field sieve article pdf available in siam journal on discrete mathematics 61. Public key cryptography using discrete logarithms in finite. Then the index calculus method and the number field sieve method for solving discrete logarithms modulo a prime are introduced and their runtime is analyzed.
Recent progress on the elliptic curve discrete logarithm problem. 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 length at most j. The applet works in a reasonable amount of time if this factor is less than 10 17. Pdf we consider the oneprimenotp and allprimesbutp variants of the discrete logarithm dl problem in a group of prime order p. And this can be made prohibitively large if t log 2 q is large. We provide algorithms to solve the discrete logarithm problem for generic. Cryptosystems based on discrete logarithms let be a finite field of q elements so that for some prime p and integer n.
Pdf discrete logarithms in gfp using the number field. Pdf for elliptic curve based cryptosystems, the discrete loga rithm problem must. Hence cryptosystem designers have to be careful not to limit the range in which discrete logs lie. We summarize recent developments on the computation of discrete logarithms in general groups as well as in some specialized settings. The discrete logarithm problem is the computational task of. Rearrange individual pages or entire files in the desired order. Applications of factoring and discrete logarithms to cryptography or the invention of public key cryptography sam wagsta computer sciences and mathematics. It is well known that the multiplicative group of nonzero elements of, denoted by, is a cyclic group of order q1. Logarithms mcty logarithms 20091 logarithms appear in all sorts of calculations in engineering and science, business and economics.
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