Sample exponential and logarithm problems 1 exponential problems. The rules of exponents apply to these and make simplifying logarithms easier. Discrete log problem dlp let g be a cyclic group of prime order p and let g be a generator of g. We outline some of the important cryptographic systems that use discrete logarithms. Logarithms which are not whole numbers logarithms do not have to be whole numbers.
Discrete logarithm problem using index calculus method. 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. Various so called squareroot attacks are discussed for the discrete logarithm problem in an arbitrary cyclic group. But then computing logg t is really solving the congruence ng. Last monday, the 1st ndpopulation numbered 2000 and the 2 numbered 2 048 000. Nobody has admitted publicly to having proved that the discrete log cant be solved quickly, but many very smart people have tried hard and. Solving logarithmic equations word problems example 1 investment mr.
He noted that the 1st ndpopulation doubled every day while the 2 population was reduced by half each day. The generalized discrete log problem and the security of diffiehellman by christof paar duration. Show that the discrete logarithm problem in this case can be solved in polynomialtime. Some logarithmic problems are solved by simply dropping the logarithms while others are solved by rewriting the logarithmic problem in exponential form. In the equation is referred to as the logarithm, is the base, and is the argument.
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. 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. 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. The discrete logarithm problem in a cyclic group g is to find the discrete logarithm of x to the base g, when x has been chosen uniformly at random from the group. Practice problems contributed by sarah leyden, typed solutions by scott.
You might skip it now, but should return to it when needed. The security of certain cryptosystems is based on the difficulty of this computation. The presumed computational difculty of solving the dlp in appropriate groups is the basis of many cryptosystems and protocols. Due to w ork of menezes, ok amoto and v anstone, 2, it. Solving discrete logarithms with partial knowledge of the key. 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.
Note that we are multiplying and dividing a logarithm by a plain number, not by another logarithm. Steps for solving logarithmic equations containing only logarithms step 1. The problem of nding this xis known as the discrete logarithm problem, and it is the basis of our trapdoor functions. 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. This brings us to modular arithmetic, also known as clock arithmetic. Logarithms appear in all sorts of calculations in engineering and science. Properties of exponents and logarithms exponents let a and b be real numbers and m and n be integers. To avoid confusion with ordinary logs, we sometimes call this the. The focus in this book is on algebraic groups for which the dlp seems to be hard. The definition of a logarithm indicates that a logarithm is an exponent. Isolate the logarithmic term on one side of the equation.
Given 2 g, the discrete logarithm problem is to determine such that g. 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. And this can be made prohibitively large if t log 2 q is large. Mitchell are saving for their daughters college education. Aug 19, 2016 this video is about the brief explanation of discrete logarithm used in cryptography. Since x log3 27 then, by the definition of a logarithm, we have. Thanks for contributing an answer to mathematics stack exchange. But avoid asking for help, clarification, or responding to other answers. The discrete logarithm problem is the computational task of. Any c in f has a unique representation as c m, for 0 discrete logarithm problem is to find a given only the integers c,e and m. Sample exponential and logarithm problems 1 exponential problems example 1. In this paper, we study the computational power of generic algorithms.
Nov 24, 2012 discrete logarithm problem modular arithmetic. The discrete logarithm problem on elliptic curves of trace one. The discrete log problem is the analogue of this problem modulo. If not, stop and use the steps for solving logarithmic equations containing terms without logarithms. The elliptic curve discrete logarithm problem and equivalent.
Remember that a logarithm without an indicated base is assumed to be base 10, the common logarithm. Interesting problem in logarithms mathematics stack exchange. Notice that by using the powers, we have changed a multiplication problem into one. Discrete logarithm problem on the other hand, given c and. The natural log and exponential this chapter treats the basic theory of logs and exponentials. Q2efq to nd an integer a, if it exists, such that q ap. Consider the discrete logarithm problem in the group of integers modulo p under addition. Math 5410 discrete logarithm problem let f gfq and take as a primitive element of f. This approach enables one to give a quick definition ofifand to overcome a number of technical difficulties, but it is an unnatural way to defme exponentiation.
The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences kristin e. The difficulty in solving the discrete logarithm problem is of extreme cryptographic importance since it is widely used in signature schemes, message encryption, key exchange, authentication and. The discrete logarithm problem plays an important role in cryptography. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete logarithms. Exponential and logarithmic word problems a microbiologist is studying two bacteria populations. Voiceover we need a numerical procedure, which is easy in one direction and hard in the other. This video was made by 6 multimedia university students. Recent progress on the elliptic curve discrete logarithm problem. Lower bounds for discrete logarithms and related problems. Pdf on the discrete logarithm problem researchgate.
The discrete logarithm problem is little more than an integer analog to a rotorcode cipher problem. The discrete logarithm problem is considered to be computationally intractable. Well email you at these times to remind you to study. That is, no efficient classical algorithm is known for computing discrete logarithms in general. Complex logarithm problem mathematics stack exchange. The later is a problem in finding concurrent zeros for two periodic functions where the periodicity of one tracks the value of x and the other the value of y. If taking a power is of ot time, then finding a logarithm is of o2t2 time. As far as we know, this problem is very hard to solve quickly.
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