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Topic: Discrete logarithm

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	Discrete logarithm - Wikipedia, the free encyclopedia
		In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms.
		The problem of computing discrete logarithms is a sort of sibling to the problem of integer factorization, in that both problems are difficult (no efficient algorithms are known), algorithms from one problem are often adapted to the other, and the difficulty of both problems has been exploited to construct various cryptographic (code) systems.
		Computing discrete logarithms is apparently difficult (no efficient algorithm is known), while the inverse problem of discrete exponentiation is not (it can be computed efficiently using exponentiation by squaring, for example).
	en.wikipedia.org /wiki/Discrete_logarithm (521 words)

	Logarithm - Wikipedia, the free encyclopedia
		Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7.
		Logarithms were originally invented to make lengthy numerical operations easier to perform and, before the advent of electronic computers, they were widely used for this purpose in fields such as astronomy, engineering and celestial navigation.
		The discrete logarithm is a related notion in the theory of finite groups.
	en.wikipedia.org /wiki/Logarithm (2471 words)

	ipedia.com: Logarithm Article (Site not responding. Last check: 2007-11-07)
		Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are represented by simpler operations performed on the former.
		Logarithms are useful in order to solve equations in which the unknown appears in the exponent, and they often occur as the solution of differential equations because of their simple derivatives.
		The base of the logarithm is hardly ever mentioned explicitly when analyzing the asymptotic complexity of algorithms in terms of Big O notation, since for all valid bases b and c.
	www.ipedia.com /logarithm.html (1749 words)

	PlanetMath: elliptic curve discrete logarithm problem (Site not responding. Last check: 2007-11-07)
		The elliptic curve discrete logarithm problem is the cornerstone of much of present-day elliptic curve cryptography.
		The problem is computationally difficult unless the curve has a “bad” number of points over the given field, where the term “bad” encompasses various collections of numbers of points which make the elliptic curve discrete logarithm problem breakable.
		This is version 4 of elliptic curve discrete logarithm problem, born on 2003-07-17, modified 2005-03-18.
	planetmath.org /encyclopedia/EllipticCurveDiscreteLogarithmProblem.html (216 words)

	Lexias Glossary 3
		discrete logarithm - Given two elements d, g, in a group such that there is an integer r satisfying g r = d, r is called the discrete logarithm**.
		discrete logarithm problem - The problem of given d and g in a group, to find r such that g ** r = d.
		For some groups, the discrete log problem is a hard problem that can be used in public-key cryptography.
	www.lexias.com /2.0/glossary3.html (532 words)

	Discrete Logarithm Problem -- Chris Studholme (Site not responding. Last check: 2007-11-07)
		This paper discusses the discrete logarithm problem both in general and specifically in the multiplicative group of integers modulo a prime.
		Code for computing discrete logarithms in the multiplicative group of GF(p), p prime, using the index calculus method of Coppersmith, Odlzyko, and Schroeppel is available in dlog-1.0.tar.gz (41.0KiB).
		If your interest is in the computation of discrete logarithms for some other project, and your modulus, p, is no larger than about 120 bits, then this index calculus code is as fast and more robust than my number field sieve code.
	www.cs.toronto.edu /~cvs/dlog (506 words)

	ECDL FAQ (Site not responding. Last check: 2007-11-07)
		In the case of elliptic curve discrete logarithms, no such extra structure is known except in a few special cases.
		The discrete logarithm is just division, although we know in advance that the quotient is an integer (there is no remainder).
		The discrete logarithm is an ordinary logarithm, except that we know that the result is an integer.
	pauillac.inria.fr /~harley/ecdl6/FAQ.html (1825 words)

	PlanetMath: discrete logarithm (Site not responding. Last check: 2007-11-07)
		is called the discrete logarithm or index of
		It is a difficult problem to compute the discrete logarithm, while powering is very easy.
		This is version 3 of discrete logarithm, born on 2004-12-21, modified 2004-12-25.
	planetmath.org /encyclopedia/DiscreteLogarithm.html (72 words)

	Discrete Logarithms and Factoring (Site not responding. Last check: 2007-11-07)
		We present probabilistic polynomial-time reduction that show: 1) To factor n, it suffices to be able to compute discrete logarithms modulo n.
		3) To compute a discrete logarithm modulo any n, it suffices to be able to factor and compute discrete logarithms modulo primes.
		To summarize: solving the discrete logarithm problem for a composite modulus is exactly as hard as factoring and solving it modulo primes.
	techreports.lib.berkeley.edu /accessPages/CSD-84-186 (97 words)

	MUG: discrete logarithm of a value (9.5.02)
		Strictly speaking, all I need to know is whether the logarithm is odd or even, but a general answer would also be useful.
		Deciding whether a particular number divides the logarithm is much easier than computing the logarithm itself.
		Discrete Logarithms Author: Carl Devore 13 May 2002 Input: x, b elements of the finite field F p a prime F a finite field returned by the GF command.
	www.math.rwth-aachen.de /mapleAnswers/html/1543.html (687 words)

	Q31: What are Elliptic Curve Cryptosystems? (Site not responding. Last check: 2007-11-07)
		The curves used in elliptic curve analogs of discrete logarithm cryptosystems are normally of the form
		The lack of specialized attacks means that shorter key sizes for elliptic cryptosystems give the same security as larger keys in cryptosystems that are based on discrete logarithm problem.
		It is possible that algorithm development in this area will change the security of elliptic curve discrete logarithm cryptosystems to be equivalent to that of general discrete logarithm cryptosystems; this is an open research problem.
	www.x5.net /faqs/crypto/q31.html (236 words)

	Discrete Logarithm
		The Discrete Logarithm (DL) of a vector of n bits is the ranking number of this vector in a maximum sequence generated by a Linear Feedback Shift Register (LFSR) defined by a logarithm base.
		By convention, the vector having all bits equal to 1, has a discrete logarithm equal to 0.
		In a Linear Feedback Shift Register, the transformation of one vector to the next one is made by the shift of one position by each bit, the first bit being determined by an OR-exclusive function operated on 2 bits of this vector.
	www.ulb.ac.be /di/scsi/classicsys/dl.htm (293 words)

	Q52: What is the Discrete Logarithm Problem? (Site not responding. Last check: 2007-11-07)
		Like the factoring problem, the discrete logarithm problem is believed to be difficult and also to be the hard direction of a one-way function.
		The discrete logarithm problem bears the same relation to these systems as factoring does to RSA: the security of these systems rests on the assumption that discrete logarithms are difficult to compute.
		The best discrete logarithm problems have expected running times similar to those of the best factoring algorithms.
	www.x5.net /faqs/crypto/q52.html (242 words)

	Kryptographie FAQ: Frage 52: What is the Discrete Logarithm Problem? (Site not responding. Last check: 2007-11-07)
		The discrete logarithm problem has received much attention in recent years; descriptions of some of the most efficient algorithms for discrete logarithms over finite fields can be found in [Odl84] [LL90] [COS86] [Gor93] [GM93].
		Rivest [Riv92a] has analyzed the expected time to solve the discrete logarithm problem both in terms of computing power and cost.
		The discrete logarithm problem appears to be much harder over arbitrary groups than over finite fields; this is the motivation for cryptosystems based on elliptic curves (see Question 31).
	www.iks-jena.de /mitarb/lutz/security/cryptfaq/q52.html (306 words)

	Cryptosystems Based on Discrete Logarithms
		The difficulty of this general discrete logarithm problem depends on the representation of the group.
		(As we have indicated, no one has been able to prove that these discrete logarithm problems are really hard, but they have been studied by number theorists for considerable time with only limited success.) For recent surveys and a more detailed study of the discrete logarithm problem, we refer the reader to [15, 18, 19].
		If one can solve the discrete logarithm problem, then it is clear that one can solve the Diffie-Hellman problem; hence the latter problem is no harder than the former.
	www.math.clemson.edu /faculty/Gao/crypto_mod/node4.html (1067 words)

	Number Theory Seminar (Site not responding. Last check: 2007-11-07)
		An important question in cryptography is how to generate elliptic curves over finite fields whose discrete logarithm problem is "hard" -- at least as hard as it is on other comparable curves.
		Low degree isogenies provide efficient reductions of the discrete logarithm problem on curves they connect, and expander graphs have the property that random walks mix rapidly.
		This provides a means to spread the difficulty of the discrete logarithm problem fairly uniformly over the family of curves in question.
	www.eecs.harvard.edu /theory/05-06/millabs.html (189 words)

	No Title (Site not responding. Last check: 2007-11-07)
		While the discrete logarithm problem is widely believed to be computationally intractable, a proof to this effect is neither known nor thought to be forthcoming.
		Up until the mid 1970's, the best algorithm known for the discrete logarithm problem was the giant-step baby-step method, which is exponential in both time and space.
		The discrete logarithm problem, apart from having applications to cryptography, is of fundamental significance in mathematics and algorithmic number theory.
	www.dms.auburn.edu /~rodgec1/cadcom/applications/menesnap/menesnap.html (514 words)

	ECDL FAQ (Site not responding. Last check: 2007-11-07)
		It can be shown that computing discrete logarithms takes exponential time in general, although it can take less in some cases.
		This is a bad choice for cryptography since it makes the discrete logarithm easier.
		However discrete logarithms (and big factorisations) are different.
	cristal.inria.fr /~harley/ecdl7/FAQ.html (1775 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The case of d=2 is of special interest since it corresponds to the representation of the rightmost bit of the discrete logarithm and defines whether the argument is a quadratic residue.
		These results are used to obtain lower bounds on the parallel complexity of computing the discrete logarithm.
		For example, we prove that any unbounded fan-in Boolean circuit of sublogarithmic depth computing the the discrete logarithm modulo p must be of superpolynomial size.
	www.cs.newcastle.edu.au /Seminars/97/Mar14.html (229 words)

	LASEC (Site not responding. Last check: 2007-11-07)
		A number of signature schemes and standards have been recently designed, based on the discrete logarithm problem.
		In this paper we try to minimize the use of ideal hash functions for several Discrete Logarithm (DSS-like) signatures (abstracted as generic schemes).
		In addition, schemes with formal validation which is made public, may ease global standardization since they neutralize much of the suspicions regarding potential knowledge gaps and unfair advantages gained by the scheme designer's country (e.g.
	lasecwww.epfl.ch /php_code/publications/search.php?ref=BPVY00 (217 words)

	VCSG-705/VCSS-482 homeworks (Site not responding. Last check: 2007-11-07)
		discrete logarithm of 2 base 3 mod 7
		discrete logarithm of 2 base 3 mod 25
		discrete logarithm of 3 base 3 mod 101
	www.cs.rit.edu /~spr/COURSES/CRYPTO/hmwk705.html (649 words)

	Math 5410 Discrete Logarithm Problem
		On the other hand, given c and µ, finding m is a more difficult proposition and is called the discrete logarithm problem.
		The difficulty of taking logarithms makes exponentiation in a finite field a one-way function (not a trapdoor function however).
		To decrypt, Bob raises the first component to his secret exponent a, finds the inverse mod p of this number, and multiplies the second component by this inverse to get the message back.
	www-math.cudenver.edu /~wcherowi/courses/m5410/ctcdlp.html (670 words)

	A chosen text attack on the RSA cryptosystem and some discrete logarithm schemes - Desmedt, Odlyzko (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Desmedt and A. Odlyzko, A chosen text attack on the RSA cryptosystem and some discrete logarithms schemes, Advances in Cryptology -- Crypto '85 (H. Williams, ed.), Lecture Notes in Computer Science, vol.
		46 Discrete logarithms in finite fields and their cryptographic..
		34 Discrete logarithms in GF (context) - Coppersmith, Odlyzko et al.
	sherry.ifi.unizh.ch /desmedt86chosen.html (472 words)

	ECC Cryptography Tutorial - 5.2 The Elliptic Curve Discrete Logarithm Problem (Site not responding. Last check: 2007-11-07)
		In the multiplicative group Zp, the discrete* logarithm problem is: given elements r and q of the group, and a prime p, find a number k such that r = qk mod p.
		What is the discrete logarithm k of Q = (4,5) to the base P = (16,5)?
		Since 9P = (4,5) = Q, the discrete logarithm of Q to the base P is k = 9.
	www.certicom.com /index.php?action=ecc,ecc_tut_5_2 (168 words)

	Operations on Elements
		Two operations are provided: computing the order of an element, and computing the discrete logarithm of an element to a given base.
		Assume that g and d are elements of the generic abelian group A. This function returns the discrete logarithm of d to the base g.
		Here follow some statements illustrating the computation of the order of an element and of the discrete logarithm relative to a given base.
	www.umich.edu /~gpcc/scs/magma/text266.htm (669 words)

	Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic
		Estimates for Discrete Logarithm Computations in Finite Fields of Small Characteristic.
		We give estimates for the running-time of the function field sieve (FFS) to compute discrete logarithms in $\F_{p^n}^{\times}$ for small~$p$.
		We also give evidence that for any fixed field size some may be weaker than others of a different characteristic or field representation, and compare the relative difficulty of computing discrete logarithms via the FFS in such cases.
	www.cs.bris.ac.uk /Publications/pub_info.jsp?id=2000029 (160 words)

	Andreas Enge - Publikationen
		Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time.
		A general framework for subexponential discrete logarithm algorithms.
		A general framework for subexponential discrete logarithm algorithms in groups of unknown order.
	www.lix.polytechnique.fr /Labo/Andreas.Enge/Publikationen.html (348 words)

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