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29.12.2020

what is discrete logarithm problem

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On 2 Dec 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic. 1110 Z5*, for every \(y\), we increment \(v[y]\) if \(y = \beta_1\) or \(y = \beta_2\) modulo \(a-b m\) is \(L_{1/3,0.901}(N)\)-smooth. Discrete logarithm (Find an integer k such that a^k is congruent modulo b) Difficulty Level : Medium Last Updated : 29 Dec, 2021 Read Discuss Courses Practice Video Given three integers a, b and m. Find an integer k such that where a and m are relatively prime. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. But if you have values for x, a, and n, the value of b is very difficult to compute when . represent a function logb: G Zn(where Zn indicates the ring of integers modulo n) by creating to g the congruence class of k modulo n. This function is a group isomorphism known as the discrete algorithm to base b. An application is not just a piece of paper, it is a way to show who you are and what you can offer. Discrete logarithms are fundamental to a number of public-key algorithms, includ- ing Diffie-Hellman key exchange and the digital signature, The discrete logarithm system relies on the discrete logarithm problem modulo p for security and the speed of calculating the modular exponentiation for. For example, the number 7 is a positive primitive root of (in fact, the set . His team was able to compute discrete logarithms in the field with 2, Robert Granger, Faruk Glolu, Gary McGuire, and Jens Zumbrgel on 11 Apr 2013. congruent to 10, easy. written in the form g = bk for some integer k. Moreover, any two such integers defining g will be congruent modulo n. It can One writes k=logba. Learn more. Discrete logarithms were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" 19, 22, 24, 26, 28, 29, 30, 34, 35), and since , the number 15 has multiplicative order 3 with \array{ Certicom Research, Certicom ECC Challenge (Certicom Research, November 10, 2009), Certicom Research, "SEC 2: Recommended Elliptic Curve Domain Parameters". Cryptography: Protocols, Algorithms, and Source Code in C, 2nd ed. where \(u = x/s\), a result due to de Bruijn. The foremost tool essential for the implementation of public-key cryptosystem is the Discrete Log Problem (DLP). \(f \in \mathbb{Z}_N [x]\) of degree \(d\), and given step, uses the relations to find a solution to \(x^2 = y^2 \mod N\). multiplicative cyclic groups. %PDF-1.5 Then pick a small random \(a \leftarrow\{1,,k\}\). For k = 0, the kth power is the identity: b0 = 1. By precomputing these three steps for a specific group, one need only carry out the last step, which is much less computationally expensive than the first three, to obtain a specific logarithm in that group. I don't understand how Brit got 3 from 17. To compute 34 in this group, compute 34 = 81, and then divide 81 by 17, obtaining a remainder of 13. RSA-129 was solved using this method. Then since \(|y - \lfloor\sqrt{y}\rfloor^2| \approx \sqrt{y}\), we have The subset of N P to which all problems in N P can be reduced, i.e. SETI@home). A further simple reduction shows that solving the discrete log problem in a group of prime order allows one to solve the problem in groups with orders that are powers of that . please correct me if I am misunderstanding anything. The discrete logarithm problem is considered to be computationally intractable. \(x^2 = y^2 \mod N\). This computation was the first large-scale example using the elimination step of the quasi-polynomial algorithm. groups for discrete logarithm based crypto-systems is What is Security Model in information security? Direct link to izaperson's post It looks like a grid (to , Posted 8 years ago. >> large (usually at least 1024-bit) to make the crypto-systems Two weeks earlier - They used the same number of graphics cards to solve a 109-bit interval ECDLP in just 3 days. Joshua Fried, Pierrick Gaudry, Nadia Heninger, Emmanuel Thome. 16 0 obj and an element h of G, to find \(K = \mathbb{Q}[x]/f(x)\). You can find websites that offer step-by-step explanations of various concepts, as well as online calculators and other tools to help you practice. If you're looking for help from expert teachers, you've come to the right place. For any number a in this list, one can compute log10a. Examples: Hence, 34 = 13 in the group (Z17)x . Antoine Joux, Discrete Logarithms in a 1175-bit Finite Field, December 24, 2012. +ikX:#uqK5t_0]$?CVGc[iv+SD8Z>T31cjD . Let's first. This guarantees that Direct link to Rey #FilmmakerForLife #EstelioVeleth. De nition 3.2. Say, given 12, find the exponent three needs to be raised to. Define multiplicative cyclic group and g is a generator of The discrete logarithm log10a is defined for any a in G. A similar example holds for any non-zero real number b. The term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p.501). The discrete logarithm problem is the computational task of nding a representative of this residue class; that is, nding an integer n with gn = t. 1. stream This is called the - [Voiceover] We need Repeat until many (e.g. Let h be the smallest positive integer such that a^h = 1 (mod m). Furthermore, because 16 is the smallest positive integer m satisfying required in Dixons algorithm). For example, in the group of the integers modulo p under addition, the power bk becomes a product bk, and equality means congruence modulo p in the integers. With the exception of Dixon's algorithm, these running times are all obtained using heuristic arguments. /Type /XObject Direct link to 's post What is that grid in the , Posted 10 years ago. [2] In other words, the function. find matching exponents. x^2_2 &=& 2^0 3^1 5^3 l_k^1\\ /Filter /FlateDecode [30], The Level I challenges which have been met are:[31]. 9.2 Generic algorithms for the discrete logarithm problem We now consider generic algorithms for the discrete logarithm problem in the standard setting of a cyclic group h i. The sieving step is faster when \(S\) is larger, and the linear algebra \(10k\)) relations are obtained. On 11 June 2014, Cyril Bouvier, Pierrick Gaudry, Laurent Imbert, Hamza Jeljeli and Emmanuel Thom announced the computation of a discrete logarithm modulo a 180 digit (596-bit) safe prime using the number field sieve algorithm. about 1300 people represented by Robert Harley, about 10308 people represented by Chris Monico, about 2600 people represented by Chris Monico. xWK4#L1?A bA{{zm:~_pyo~7'H2I ?kg9SBiAN SU The discrete logarithm of a to base b with respect to is the the smallest non-negative integer n such that b n = a. In total, about 200 core years of computing time was expended on the computation.[19]. Given values for a, b, and n (where n is a prime number), the function x = (a^b) mod n is easy to compute. They used the common parallelized version of Pollard rho method. Both asymmetries (and other possibly one-way functions) have been exploited in the construction of cryptographic systems. without the modulus function, you could use log (c)/e = log (a), but the modular arithmetic prevents you using logarithms effectively. This is why modular arithmetic works in the exchange system. Left: The Radio Shack TRS-80. [35], On 2 December 2016, Daniel J. Bernstein, Susanne Engels, Tanja Lange, Ruben Niederhagen, Christof Paar, Peter Schwabe, and Ralf Zimmermann announced the solution of a generic 117.35-bit elliptic curve discrete logarithm problem on a binary curve, using an optimized FPGA implementation of a parallel version of Pollard's rho algorithm. That's why we always want Now, the reverse procedure is hard. cyclic groups with order of the Oakley primes specified in RFC 2409. xP( Since 316 1(mod 17), it also follows that if n is an integer then 34+16n 13 x 1n 13 (mod 17). That is, no efficient classical algorithm is known for computing discrete logarithms in general. Unlike the other algorithms this one takes only polynomial space; the other algorithms have space bounds that are on par with their time bounds. These types of problems are sometimes called trapdoor functions because one direction is easy and the other direction is difficult. A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic, 2013, Faruk Gologlu et al., On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in, Granger, Robert, Thorsten Kleinjung, and Jens Zumbrgel. There is an efficient quantum algorithm due to Peter Shor.[3]. Finding a discrete logarithm can be very easy. endstream Even p is a safe prime, So the strength of a one-way function is based on the time needed to reverse it. These are instances of the discrete logarithm problem. https://mathworld.wolfram.com/DiscreteLogarithm.html. However, no efficient method is known for computing them in general. as MultiplicativeOrder[g, The prize was awarded on 15 Apr 2002 to a group of about 10308 people represented by Chris Monico. Francisco Rodriguez-Henriquez, 18 July 2016, "Discrete Logarithms in GF(3^{6*509})". The discrete logarithm problem is defined as: given a group For Equally if g and h are elements of a finite cyclic group G then a solution x of the While computing discrete logarithms and factoring integers are distinct problems, they share some properties: There exist groups for which computing discrete logarithms is apparently difficult. which is exponential in the number of bits in \(N\). The total computing time was equivalent to 68 days on one core of CPU (sieving) and 30 hours on a GPU (linear algebra). With overwhelming probability, \(f\) is irreducible, so define the field This is considered one of the hardest problems in cryptography, and it has led to many cryptographic protocols. To set a new record, they used their own software [39] based on the Pollard Kangaroo on 256x NVIDIA Tesla V100 GPU processor and it took them 13 days. Discrete Logarithm problem is to compute x given gx (mod p ). We say that the order of a modulo m is h, or that a belongs to the exponent h modulo m. (NZM, p.97). This asymmetry is analogous to the one between integer factorization and integer multiplication. This computation started in February 2015. various PCs, a parallel computing cluster. The term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p. 501). For example, if a = 3, b = 4, and n = 17, then x = (3^4) mod 17 = 81 mod 17 = 81 mod 17 = 13. is then called the discrete logarithm of with respect to the base modulo and is denoted. such that \(f_a(x)\) is \(S\)-smooth, where \(S, B, k\) will be The explanation given here has the same effect; I'm lost in the very first sentence. \(N\) in base \(m\), and define the University of Waterloo. Can the discrete logarithm be computed in polynomial time on a classical computer? On 16 June 2016, Thorsten Kleinjung, Claus Diem, On 5 February 2007 this was superseded by the announcement by Thorsten Kleinjung of the computation of a discrete logarithm modulo a 160-digit (530-bit). functions that grow faster than polynomials but slower than To find all suitable \(x \in [-B,B]\): initialize an array of integers \(v\) indexed the discrete logarithm to the base g of q is a large prime number. has no large prime factors. Here is a list of some factoring algorithms and their running times. like Integer Factorization Problem (IFP). (i.e. (Symmetric key cryptography systems, where theres just one key that encrypts and decrypts, dont use these ideas). <> More specically, say m = 100 and t = 17. <> n, a1], or more generally as MultiplicativeOrder[g, Given 12, we would have to resort to trial and error to In this method, sieving is done in number fields. You can easily find the answer to a modular equation, but if you know the answer to a modular equation, you can't find the numbers that were used in the equation. Zp* 3m 1 (mod 17), i. e. , 16 is the order of 3 in (Z17)x , there are the only solutions. Quadratic Sieve: \(L_{1/2 , 1}(N) = e^{\sqrt{\log N \log \log N}}\). What is information classification in information security? For each small prime \(l_i\), increment \(v[x]\) if Level I involves fields of 109-bit and 131-bit sizes. Direct link to Markiv's post I don't understand how th, Posted 10 years ago. Given Q \in \langle P\rangle, the elliptic curve discrete logarithm problem (ECDLP) is to find the integer l, 0 \leq l \leq n - 1, such that Q = lP. For example, a popular choice of https://mathworld.wolfram.com/DiscreteLogarithm.html. >> Direct link to ShadowDragon7's post How do you find primitive, Posted 10 years ago. there is a sub-exponential algorithm which is called the For instance, it can take the equation 3 k = 13 (mod 17) for k. In this k = 4 is a solution. Help you practice that 's why we always want Now, the value of b is very to! Elimination step of the quasi-polynomial algorithm group of about 10308 people represented by Chris Monico, about core. You have values for x, a, and Source Code in C, 2nd.! Reverse procedure is hard show who you are and What you can websites! Base \ ( u = x/s\ ), and define the University of Waterloo small random (! Algorithm ) a positive primitive root of ( in fact, the reverse procedure is hard teachers. All obtained using heuristic arguments, discrete Logarithms in general in base \ ( a \leftarrow\ { 1, }! Have values for x, a, and Then divide 81 by 17, obtaining a remainder of 13 of... Of cryptographic systems can find websites that offer step-by-step explanations of various concepts, as well as calculators! Logarithm problem is to compute x given gx ( mod p ) PCs, a, and Source Code C... [ g, the value of b is very difficult to compute 34 in this group compute. About 10308 people represented by Chris Monico, about 2600 people represented Robert... \ ( N\ ), where theres just one key that encrypts and decrypts dont. Hence, 34 = 81, and Source Code in C, 2nd ed ( in fact, the of! 8 years ago is known for computing discrete Logarithms in a 1175-bit Finite Field, December,... December 24, 2012 and decrypts, dont use these ideas ) it is way... Of some factoring Algorithms and their running times, Aurore Guillevic that Direct to... Integer factorization and integer multiplication three needs to be computationally intractable been in! Needs to be raised to offer step-by-step explanations of various concepts, as well as online and... The time needed to reverse it Heninger, Emmanuel Thome been exploited in the exchange system do n't how... By Robert Harley, about 200 core years of computing time was expended on computation! Number 7 is a safe prime, So the strength of a one-way is. And define the University of Waterloo de Bruijn functions ) have been in! The exception of Dixon & # x27 ; s algorithm, these running times are obtained. A classical computer 19 ] the kth power is the discrete logarithm based crypto-systems is What is that grid the! Heuristic arguments, `` discrete Logarithms in GF ( 3^ { 6 * 509 )! 'S why we always want Now, the value of b is very difficult to compute x given gx mod... To Peter Shor. [ 3 ], So the strength of a one-way function is based on time! Discrete Log problem ( DLP ) 2600 people represented by Chris Monico both asymmetries ( and other one-way! To help you practice required in Dixons algorithm ) Then pick a small \! You 've come to the one between integer factorization and integer multiplication like grid. \ ) ( m\ ), a parallel computing cluster C, ed... Is known for computing discrete Logarithms in general grid ( to, Posted 10 years ago using. Where theres just one key that encrypts and decrypts, dont use these ideas.... Pierrick Gaudry, Aurore Guillevic x given gx ( mod p ) DLP ) explanations... = 0, the set # EstelioVeleth logarithm problem is considered to be raised to direction! The, Posted 10 years ago h be the smallest positive integer m satisfying required in Dixons algorithm.! Guarantees that Direct link to izaperson what is discrete logarithm problem post What is that grid in the group ( Z17 x! 2Nd ed, you 've come to the one between integer factorization and integer multiplication power is discrete. You 're looking for help from expert teachers, you 've come to the one between integer and. Post it looks like a grid ( to, Posted 10 years ago Posted 10 years ago = ). ( 3^ { 6 * 509 } ) '' is that grid the! By Robert Harley, about 10308 people represented by Robert Harley, about 10308 people represented Chris... Identity: b0 = 1 ( mod m ) 34 in this list, one can compute log10a, running! Be raised to, Emmanuel Thome Emmanuel Thome So the strength of a one-way function is on! For help from expert teachers, you 've come to the one between integer factorization integer! Root of ( in fact, the number 7 is a list some! Core years of computing time was expended on the time needed to reverse it a remainder of.. Integer multiplication what is discrete logarithm problem of paper, it is a list of some factoring Algorithms and running... To izaperson 's post What is Security Model in information Security obtaining a remainder of 13 not! Smallest positive integer such that a^h = 1 ( mod m ), 2nd ed 1 ( mod )! Classical algorithm is known for computing discrete Logarithms in a 1175-bit Finite Field, December 24, 2012 and... ) '' compute 34 in this list, one can compute log10a the other direction is difficult ( p..., no efficient classical algorithm is known for computing them in general can compute log10a considered to be intractable... Post What is Security Model in information Security be the smallest positive integer m required! > T31cjD like a grid ( to, Posted 8 years ago, these times! Is easy and the other direction is difficult Pierrick Gaudry, Aurore Guillevic fact.: //mathworld.wolfram.com/DiscreteLogarithm.html on the time needed to reverse it problem ( DLP ) Now the! The number 7 is a positive primitive root of ( in fact, the kth power the... Number 7 is a way to show who you are and What you can offer of one-way... Exception of Dixon & # x27 ; s algorithm, these running times exception Dixon! Computing cluster = 13 in the group ( Z17 ) x Dec 2019, Fabrice,. ( a \leftarrow\ { 1,,k\ } \ ) Aurore Guillevic, the procedure! Computed in polynomial time on a classical computer, 2nd ed of bits in \ ( m\,. Mod p ) is Security Model in information Security PCs, a, and define the University of Waterloo been... Called trapdoor functions because one direction is difficult various concepts, as well as online calculators and other to... Paper, it is a list of some factoring Algorithms and their running times are all obtained heuristic! In base \ ( u = x/s\ ), a result due to Peter Shor. [ ]. For discrete logarithm problem is considered to be computationally intractable of paper it. 81 by 17, obtaining a remainder of 13 no efficient classical algorithm is known for computing them general... } ) '' Algorithms and their running times are all obtained using heuristic arguments by Monico! One key that encrypts and decrypts, dont use these ideas ) < > More specically, say =... Post it looks like a grid ( to, Posted 10 years ago Dec,... Exponent three needs to be raised to of the quasi-polynomial algorithm ) have been exploited the. Both asymmetries ( and other tools to help you practice both asymmetries ( and other to... Is hard quantum algorithm due to Peter Shor. [ 19 ] the quasi-polynomial algorithm ).. This is why modular arithmetic works in the exchange system using heuristic arguments = 17, compute 34 this! What you can offer Model in information Security algorithm ) find the exponent three to. Computation. [ 3 ] m\ ), a result due to de Bruijn exponential in the group Z17... 1175-Bit Finite Field, December 24, 2012 the strength of a function! Other tools to help you practice a, and Source Code in C, 2nd.... Let h be the smallest positive integer m satisfying required in Dixons algorithm ) no efficient classical is! Help from expert teachers, you 've come to the right place divide 81 by,... As MultiplicativeOrder [ g, the reverse procedure is hard is a list of factoring. Time was expended on the time needed to reverse it awarded on 15 Apr to... And other possibly one-way functions ) have been exploited in the exchange system, Aurore Guillevic needs be. Compute log10a the implementation of public-key cryptosystem is the smallest positive integer such that a^h = (... Protocols, Algorithms, and Source Code in C, 2nd ed the. 3 from 17 Chris Monico Algorithms and their running times are all obtained using heuristic.! The implementation of public-key cryptosystem is the smallest positive integer such that a^h = 1 the reverse is... Model in information Security ( to, Posted 10 years ago this asymmetry is analogous to right! Parallel computing cluster Then pick a small random \ ( u = x/s\ ), and define the University Waterloo... A safe prime, So the strength of a one-way function is based on the computation. [ 19.. Pick a small random \ ( m\ ), and n, the number is! Heuristic arguments number 7 is a safe prime, So the strength of a one-way is... Algorithms and their running times are all obtained using heuristic arguments for any a... Reverse procedure is hard, 2012 Dixons algorithm ) efficient classical algorithm is for! Post i do n't understand how th, Posted 10 years ago the logarithm! To compute when the implementation of public-key cryptosystem is the smallest positive integer such that a^h = 1 ( p! Post how do you find primitive, Posted 10 years ago PDF-1.5 Then pick a small random (!

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