Discrete logarithms modular exponentiation coursera. The discrete logarithm problem dlp refers to the problem of finding logarithms modulo some integer. Diffiehellman key exchange is a method of securely exchanging cryptographic keys over a public channel and was one of the first publickey protocols as conceived by ralph merkle and named after whitfield diffie and martin hellman. An oracle is a theoretical constanttime \black box function. A public key cryptosystem and a signature scheme based on. This in turn allows chinese remainder theorem based attacks on dlp. We outline some of the important cryptographic systems that use discrete logarithms.
If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. And elliptic elgamal has proved to be a strong cryptosystem using elliptic curves and discrete logarithms. In the mathematics of the real numbers, the logarithm log b a is a number x such that bx a, for given numbers a and b. Say, given 12, find the exponent three needs to be raised to. This shift in the way data and information is being transported then calls.
Discrete logarithms to cryptography or the invention of public key cryptography sam wagsta computer sciences and mathematics october 21, 2008 1. Proceedings of workshop on the theory and application of cryptographic techniques, 1984. Svore, and kristin lauter microsoft research, usa abstract. Applications of factoring and discrete logarithms to. 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. Dh is one of the earliest practical examples of public key exchange implemented within the field of cryptography. In this chapter, we will introduce and study another computationally difficult number theory problem, that of computing discrete logarithms, with an eventual goal of. The discrete logarithm of u is sometimes referred to as the index of u. In any of the cryptographic systems that are based on discrete logarithms, p must be chosen such that p 1 has at least one large prime factor. This recommendation specifies key establishment schemes using discrete logarithm cryptography, based on standards developed by the accredited standards committee asc x9, inc ans x9. Recallthe tonellishanksalgorithmfor computing squarerootsmodulo p from section 2. Discrete logarithm diffiehellman key exchange coursera.
Here is a list of some factoring algorithms and their running times. The exponent ais called the discrete logarithm of ain base. The functions are mainly based on the ieee p63a standard. Pdf on the discrete logarithm problem semantic scholar. Similarly, if g and h are elements of a finite cyclic group g then a solution x of the equation g h is called a discrete logarithm to. The latter will require us to introduce the weil pairing. The first one is a direct adaptation of the beckercoronjoux bcj algorithm for subset sum to the discrete logarithm setting. Then, with the development of cryptography, theirimportance raised considerably, especially after di. 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.
If p 1 has only small prime fac tors, then computing discrete logarithms is easy see a. Introductionbefore the middle of the last century, discrete logarithms were just common tools usedto perform calculations in. Apr 28, 2014 khan academy has been translated into dozens of languages, and 100 million people use our platform worldwide every year. Currently, the dlp based on the hyperelliptic curve of genus 2 hcdlp is widely used in industry and also a research field of hot interest. Fermats little theorem states that when p is a prime, then for any integer a that is coprime to p, the following. 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. The algorithm for computing discrete logarithms implied by the proof of the propo sition does more than just group operations, it also compares group elements. Aside from the intrinsic interest that the problem of computing discrete logarithms has, it is of considerable importance in cryptography. While pbc has attracted most of the attention during the. It is also relevant for applications in cryptography. Public key cryptography unlike symmetric key cryptography, we do not find historical use of publickey cryptography. The discrete log problem is dificult in some groups and is easy in other groups. Quantum algorithm for solving hyperelliptic curve discrete. What is the difference between discrete logarithm and logarithm.
Suppose h gx for some g in the finite field and secret integer x. Discrete logarithms to cryptography or the invention of public key cryptography sam wagsta computer sciences and mathematics 1. Computing elliptic curve discrete logarithms with improved. The discrete logarithm problem is to find the e slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Quantum resource estimates for computing elliptic curve. In mathematics, specifically in abstract algebra and its applications, discrete logarithms are grouptheoretic analogues of ordinary logarithms. Shors algorithm 29,30 solves the discrete logarithm problem for nite abelian groups with only polynomial cost. Hence one generally uses elements of prime order r for cryptography. Compressing elements in discrete logarithm cryptography philip nicholas james eagle, esq. In this video series different topics will be explained which will help you to understand blockchain. In this module, we will cover the squareandmultiply method, euliers totient theorem and function, and demonstrate the use of discrete logarithms.
Before we dive in, lets take a quick look at the underlying mathematics. Discrete logarithms are quickly computable in a few special cases. The dlp was rst proposed in the multiplicative groups of nite elds. The integer factorization problem ifp, the finite field discrete logarithm problem dlp and the elliptic curve discrete logarithm problem ecdlp are essentially the only three mathematical problems that the practical publickey cryptographic systems are based on. However, no efficient method is known for computing them in general. Discrete logarithms have a natural extension into the realm of elliptic curves and hyperelliptic curves. This paper refers to other papers by teske, who presented a improvement of pollards original function in on random walks for pollards rho method. Submitted in total ful lment of the requirements of the degree of philosophi. May 23, 2015 and they are logarithms because they are analogous to ordinary logarithms. The discrete logarithm problem dipartimento di matematica tor. In particular, an ordinary logarithm logab is a solution of the equation a b over the real or complex numbers. A more in depth understanding of modular exponentiation is crucial to understanding cryptographic mathematics. Discrete logarithms in cryptography esat ku leuven.
The discrete logarithm problem dlp in nite groups is an important computational problem in modern cryptography. At the same time, quantum computing, a new paradigm for computing based on quantum mechanics, provides the. This advantage arises from the fact that the currently known best algorithms to compute elliptic curve discrete logarithms are exponential in the size of the input parameters2, whereas there exist subexponential algorithms for factoring 30, 9 and nite eld discrete logarithms 18,24. Sp 80056a revised, recommendation for pairwise key. As the name suggests, we are concerned with discrete logarithms. Due to its speed, spread and ease of use, the internet has now become a popular means through which useful data and information are transported from one location to another. Symmetric cryptography was well suited for organizations such as governments, military, and big financial corporations were involved in. 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. Rather than rely only on big integers, dh exploits the difficulty of the discrete logarithm problem dlp. Designing good algorithms to compute discrete logarithms is a problem that is of interest in itself.
The second one significantly improves on this adaptation for all possible weights using a more involved application of the representation technique together with some new markov chain analysis. Ecc requires smaller keys compared to nonec cryptography based on plain galois fields to provide equivalent security. This is part 9 of the blockchain tutorial explaining what discrete logarithms are. Voiceover we need a numerical procedure, which is easy in one direction and hard in the other. Discrete logarithms an overview sciencedirect topics. Earlier, we proved a few basic properties about orders. Cryptography before the 1970s cryptography has been used to hide messages at least since the time of julius caesar more than 2000 years ago. Public key cryptography using discrete logarithms in. The discrete logarithm problem asks for a solution of something like this. The discrete logarithm problem is the computational task of finding a.
We shall see that discrete logarithm algorithms for finite fields are similar. Recommendations for discrete logarithmbased cryptography. When run on a largescale, faulttolerant quantum computer, its variant for ellipticcurve groups could e ciently break elliptic curve cryptography with parameters that are. It is well known that the multiplicative group of nonzero elements of, denoted by, is a cyclic group of order q1. 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. Applications of factoring and discrete logarithms to cryptography. Analogously, in any group g, powers bk can be defined for all integers k, and the discrete logarithm log b a is an integer k such that bk a. 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. Doctor june 2008 information security group royal holloway college, university of london. Public key cryptography using discrete logarithms in finite.
This recommendation specifies keyestablishment schemes based on the discrete logarithm problem over finite fields and elliptic curves, including several variations of diffiehellman and menezesquvanstone mqv key establishment schemes. Its presumed hardness provides the basis for security for a number of cryptographic systems. We say a call to an oracle is a use of the function on a speci ed input, giving us. The discrete logarithm problem journey into cryptography. Discrete logarithm cryptography, in its broadest sense, is concerned with cryptographic schemes whose security relies on the intractability of the discrete logarithm problem dlp, together with the underlying mathematical structures, implementation methods, performanceusability comparisons etc. Implementation of the digital signature operations is based on fips pub 1862. Diffiehellman key exchange and the discrete log problem by christof paar duration. Pdf comparative analysis of discrete logarithm and rsa. Ellipticcurve cryptography ecc is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields. Unless otherwise specified, all content on this website is licensed under a creative commons attributionnoncommercialsharealike 4. Yet another application of this theorem will be in the speedup of the modular exponentiation algorithm that is presented in section 12.
Choose random numbers r, each time compute g r mod p, and save any that. Applications of factoring and discrete logarithms to cryptography or the invention of public key cryptography sam wagsta computer sciences and mathematics. Recommendation for pairwise key establishment schemes. Quantum resource estimates for computing elliptic curve discrete logarithms martin roetteler, michael naehrig, krysta m. Discrete logarithms and elliptic curves in cryptography. This is because the best classical integer factoring. Block ciphers digital signatures discrete logarithms elliptic curves finite fields hash functions historical ciphers informationtheoretic security key exchange message authentication codes primality testing provable security publickey cryptography secure multiparty computation stream ciphers symmetrickey cryptography the enigma machine. We give precise quantum resource estimates for shors algorithm to compute discrete logarithms on elliptic curves over prime elds. A public key cryptosystem and a signature scheme based on discrete logarithms. Implementing elliptic curve cryptography leonidas deligiannidis wentworth institute of technology dept. A faster method to compute primitive elements and discrete. The discrete logarithm problem dlp plays an important role in modern cryptography since it cannot be efficiently solved on a classical computer. In the next part of the chapter, we will take a look at the discrete logarithm problem and discuss its application to cryptography. Cryptography based on the discrete logarithm chapter 4.
Cryptography is the process of writing using various methods ciphers to keep messages secret. Jan 17, 2017 the curious case of the discrete logarithm. We normally define a logarithm with base b such that. The most obvious approach to breaking modern cryptosystems is to attack the underlying mathematical problem. When enough of these linear combinations are found, the unknown, smaller logarithms can be solved for as if they were variables in a linear system. Cryptosystems based on discrete logarithms let be a finite field of q elements so that for some prime p and integer n. Due to this method, small primes give no added security in discrete logarithm systems. We will then discuss the discrete logarithm problem for elliptic curves. This brings us to modular arithmetic, also known as clock arithmetic. With the exception of dixons algorithm, these running times are all obtained using heuristic arguments.
The belief in the intractability of this computational problem in many groups is based on anecdotal evidence rather than on mathematical proof. Briefly, in elgammal cryptosystem with underlying group the group of units modulo a prime number p im told to find a subgroup of index 2 to. However, a single large prime factor will still spell trouble for the attacker. This course also describes some mathematical concepts, e. The difficulty or intractability of the discrete log problem may and will depend on the specfic group. Sage implementation of discrete logarithm in subgroup of group of units. Recall that when we mod out by an integer n, we are left with only finitely many integers a discrete set usually represented as 0, 1, 2, n1. For each a2gthere exists an unique 0 a n 1 such that a a. Discrete logarithms in finite fields and their cryptographic.