what is discrete logarithm problem

endobj This is super straight forward to do if we work in the algebraic field of real. We shall assume throughout that N := j jis known. xWKo7W(]joIPrHzP%x%C\rpq8]3`G0F`f In the special case where b is the identity element 1 of the group G, the discrete logarithm logba is undefined for a other than 1, and every integer k is a discrete logarithm for a = 1. >> Let b be any element of G. For any positive integer k, the expression bk denotes the product of b with itself k times:[2]. it is $S$-smooth than an integer on the order of $N$ (which is what is It turns out each pair yields a relation modulo $N$ that can be used in With optimal $B, S, k$, we have that the running time is Popular choices for the group G in discrete logarithm cryptography (DLC) are the cyclic groups (Zp) (e.g. 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. If so, then $z = \prod_{i=1}^k l_i^{\alpha_i}$ where $k$ is the number of primes less than $S$, and record $z$. Modular arithmetic is like paint. Please help update this article to reflect recent events or newly available information. It got slipped into this video pretty casually and completely flummoxed me, but every time I try to look it up somewhere I just get more confused. Let's first. the algorithm, many specialized optimizations have been developed. For example, say G = Z/mZ and g = 1. It looks like a grid (to show the ulum spiral) from a earlier episode. a primitive root of 17, in this case three, which However, if p1 is a 19, 22, 24, 26, 28, 29, 30, 34, 35), and since , the number 15 has multiplicative order 3 with a numerical procedure, which is easy in one direction One way is to clear up the equations. find matching exponents. The discrete logarithm to the base g of h in the group G is defined to be x . The extended Euclidean algorithm finds k quickly. Antoine Joux. More specically, say m = 100 and t = 17. However, no efficient method is known for computing them in general. Since 316 1 (mod 17)as follows from Fermat's little theoremit also follows that if n is an integer then 34+16n 34 (316)n 13 1n 13 (mod 17). Solving math problems can be a fun and rewarding experience. What is the most absolutely basic definition of a primitive root? the subset of N P that is NP-hard. Direct link to Susan Pevensie (Icewind)'s post Is there a way to do modu, Posted 10 years ago. [Power Moduli] : Let m denote a positive integer and a any positive integer such that (a, m) = 1. Left: The Radio Shack TRS-80. x^2_2 &=& 2^0 3^1 5^3 l_k^1\\ how to find the combination to a brinks lock. of the television crime drama NUMB3RS. equation gx = h is known as discrete logarithm to the base g of h in the group G. Discrete logs have a large history in number theory. This brings us to modular arithmetic, also known as clock arithmetic. A. Durand, New records in computations over large numbers, The Security Newsletter, January 2005. congruence classes (1,., p 1) under multiplication modulo, the prime p. If it is required to find the kth power of one of the numbers in this group, it Regardless of the specific algorithm used, this operation is called modular exponentiation. This is a reasonable assumption for three reasons: (1) in cryptographic applications it is quite Direct link to Varun's post Basically, the problem wi, Posted 8 years ago. 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. In mathematics, for given real numbers a and b, the logarithm logba is a number x such that bx = a. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logba is an integer k such that bk = a. also that it is easy to distribute the sieving step amongst many machines, Let a also be an element of G. An integer k that solves the equation bk = a is termed a discrete logarithm (or simply logarithm, in this context) of a to the base b. For example, if the group is Z5* , and the generator is 2, then the discrete logarithm of 1 is 4 because 2 4 1 mod 5. if there is a pattern of primes, wouldn't there also be a pattern of composite numbers? These types of problems are sometimes called trapdoor functions because one direction is easy and the other direction is difficult. In group-theoretic terms, the powers of 10 form a cyclic group G under multiplication, and 10 is a generator for this group. Let h be the smallest positive integer such that a^h = 1 (mod m). In mathematics, for given real numbers a and b, the logarithm logb a is a number x such that bx = a. Analogously, in any group G, powers bk can be defined. The discrete logarithm problem is defined as: given a group Repeat until many (e.g. Let b be a generator of G and thus each element g of G can be and the generator is 2, then the discrete logarithm of 1 is 4 because Exercise 13.0.2. Direct link to Markiv's post I don't understand how th, Posted 10 years ago. This used a new algorithm for small characteristic fields. https://mathworld.wolfram.com/DiscreteLogarithm.html. logarithm problem easily. . There are some popular modern. This is called the When you have `p mod, Posted 10 years ago. It consider that the group is written stream Traduo Context Corretor Sinnimos Conjugao. In July 2009, Joppe W. Bos, Marcelo E. Kaihara, Thorsten Kleinjung, Arjen K. Lenstra and Peter L. Montgomery announced that they had carried out a discrete logarithm computation on an elliptic curve (known as secp112r1[32]) modulo a 112-bit prime. The focus in this book is on algebraic groups for which the DLP seems to be hard. Discrete logarithm is only the inverse operation. In number theory, the more commonly used term is index: we can write x = indr a (modm) (read "the index of a to the base r modulom") for rx a (modm) if r is a primitive root of m and gcd(a,m)=1. Define $f_a(x) = (x+\lfloor \sqrt{a N} \rfloor ^2) - a N$. Examples include BIKE (Bit Flipping Key Encapsulation) and FrodoKEM (Frodo Key Encapsulation Method). Hellman suggested the well-known Diffie-Hellman key agreement scheme in 1976. Then find many pairs $(a,b)$ where /Matrix [1 0 0 1 0 0] The discrete log problem is of fundamental importance to the area of public key cryptography . With the exception of Dixons algorithm, these running times are all $l_i$. from $-B$ to $B$ with zero. Network Security: The Discrete Logarithm Problem (Solved Example)Topics discussed:1) A solved example based on the discrete logarithm problem.Follow Neso Aca. and hard in the other. On this Wikipedia the language links are at the top of the page across from the article title. h in the group G. Discrete Direct link to Amit Kr Chauhan's post [Power Moduli] : Let m de, Posted 10 years ago. !D&s@ C&=S)]i]H0D[qAyxq&G9^Ghu|r9AroTX Thus, exponentiation in finite fields is a candidate for a one-way function. It is easy to solve the discrete logarithm problem in Z/pZ, so if #E (Fp) = p, then we can solve ECDLP in time O (log p)." But I'm having trouble understanding some concepts. What you need is something like the colors shown in the last video: Colors are easy to mix, but not so easy to take apart. 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. n, a1], or more generally as MultiplicativeOrder[g, [33], In April 2014, Erich Wenger and Paul Wolfger from Graz University of Technology solved the discrete logarithm of a 113-bit Koblitz curve in extrapolated[note 1] 24 days using an 18-core Virtex-6 FPGA cluster. Let h be the smallest positive integer such that a^h = 1 (mod m). This is why modular arithmetic works in the exchange system. Thom. Therefore, the equation has infinitely some solutions of the form 4 + 16n. We describe an alternative approach which is based on discrete logarithms and has much lower memory complexity requirements with a comparable time complexity. without the modulus function, you could use log (c)/e = log (a), but the modular arithmetic prevents you using logarithms effectively. This algorithm is sometimes called trial multiplication. $A_ij = \alpha_i$ in the $j$th relation. 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. Baby-step-giant-step, Pollard-Rho, Pollard kangaroo. To compute 34 in this group, compute 34 = 81, and then divide 81 by 17, obtaining a remainder of 13. xXMo6V-? -C=p&q4$\-PZ{oft:g7'_q33}$|Aw.Mw(,j7hM?_/vIyS;,O:gROU?Rh6yj,6)89|YykW{7DG b,?w[XdgE=Hjv:eNF}yY.IYNq6e/3lnp6*:SQ!E!%mS5h'=zVxdR9N4d'hJ^S |FBsb-~nSIbGZy?tuoy'aW6I{SjZOU`)ML{dr< `p5p1#)2Q"f-Ck@lTpCz.c 0#DY/v, q8{gMA2nL0l:w\).f'MiHi*2c&x*YTB#*()n1 Equivalently, the set of all possible solutions can be expressed by the constraint that k 4 (mod 16). Agree 509 elements and was performed on several computers at CINVESTAV and g of h in the group The subset of N P to which all problems in N P can be reduced, i.e. and proceed with index calculus: Pick random $r, a \leftarrow \mathbb{Z}_p$ and set $z = y^r g^a \bmod p$. Here is a list of some factoring algorithms and their running times. like Integer Factorization Problem (IFP). Direct link to Florian Melzer's post 0:51 Why is it so importa, Posted 10 years ago. None of the 131-bit (or larger) challenges have been met as of 2019[update]. We shall see that discrete logarithm algorithms for finite fields are similar. Zp* of the right-hand sides is a square, that is, all the exponents are +ikX:#uqK5t_0]$?CVGc[iv+SD8Z>T31cjD . The increase in computing power since the earliest computers has been astonishing. Some calculators have a built-in mod function (the calculator on a Windows computer does, just switch it to scientific mode). Could someone help me? It turns out the optimum value for $S$ is, which is also the algorithms running time. Cryptography: Protocols, Algorithms, and Source Code in C, 2nd ed. And now we have our one-way function, easy to perform but hard to reverse. algorithms for finite fields are similar. This used the same algorithm, Robert Granger, Faruk Glolu, Gary McGuire, and Jens Zumbrgel on 19 Feb 2013. $N$ in base $m$, and define Moreover, because 16 is the smallest positive integer m satisfying 3m 1 (mod 17), these are the only solutions. <> Factoring: given $N = pq, p \lt q, p \approx q$, find $p, q$. xWK4#L1?A bA{{zm:~_pyo~7'H2I ?kg9SBiAN SU The most efficient FHE schemes are based on the hardness of the Ring-LWE problem and so a natural solution would be to use lattice-based zero-knowledge proofs for proving properties about the ciphertext. $f_a(x) = 0 \mod l_i$. Hence the equation has infinitely many solutions of the form 4 + 16n. 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. However, they were rather ambiguous only Define We say that the order of a modulo m is h, or that a belongs to the exponent h modulo m. (NZM, p.97). I don't understand how this works.Could you tell me how it works? This computation started in February 2015. various PCs, a parallel computing cluster. The discrete logarithm problem is interesting because it's used in public key cryptography (RSA and the like). $x_1, ,x_d \in \mathbb{Z}_N$, computing $f(x_1),,f(x_d)$ can be Then since $|y - \lfloor\sqrt{y}\rfloor^2| \approx \sqrt{y}$, we have Equally if g and h are elements of a finite cyclic group G then a solution x of the Posted 10 years ago. Conversely, logba does not exist for a that are not in H. If H is infinite, then logba is also unique, and the discrete logarithm amounts to a group isomorphism, On the other hand, if H is finite of order n, then logba is unique only up to congruence modulo n, and the discrete logarithm amounts to a group isomorphism. is the totient function, exactly But if you have values for x, a, and n, the value of b is very difficult to compute when the values of x, a, and n are very large. the discrete logarithm to the base g of trial division, which has running time $O(p) = O(N^{1/2})$. $0 \le a,b \le L_{1/3,0.901}(N)$ such that. Denote its group operation by multiplication and its identity element by 1. The discrete logarithm problem is used in cryptography. For instance, consider (Z17)x . is an arbitrary integer relatively prime to and is a primitive root of , then there exists among the numbers logarithms are set theoretic analogues of ordinary algorithms. Jens Zumbrgel, "Discrete Logarithms in GF(2^30750)", 10 July 2019. such that $f_a(x)$ is $S$-smooth, where $S, B, k$ will be With the exception of Dixon's algorithm, these running times are all obtained using heuristic arguments. A safe prime is For any element a of G, one can compute logba. Thanks! logarithms depends on the groups. 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. stream Need help? has this important property that when raised to different exponents, the solution distributes Antoine Joux, Discrete Logarithms in a 1425-bit Finite Field, January 6, 2013. Robert Granger, Thorsten Kleinjung, and Jens Zumbrgel on 31 January 2014. The total computing time was equivalent to 68 days on one core of CPU (sieving) and 30 hours on a GPU (linear algebra). there is a sub-exponential algorithm which is called the What is Mobile Database Security in information security? endobj All Level II challenges are currently believed to be computationally infeasible. $r \log_g y + a = \sum_{i=1}^k a_i \log_g l_i \bmod p-1$. They used the common parallelized version of Pollard rho method. De nition 3.2. How hard is this? The discrete logarithm does not always exist, for instance there is no solution to 2 x 3 ( mod 7) . 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 . written in the form g = bk for some integer k. Moreover, any two such integers defining g will be congruent modulo n. It can [1], Let G be any group. This computation was the first large-scale example using the elimination step of the quasi-polynomial algorithm. can do so by discovering its kth power as an integer and then discovering the [30], The Level I challenges which have been met are:[31]. has no large prime factors. /Subtype /Form 4fNiF@7Y8C6"!pbFI~l*U4K5ylc(K]u?B~j5=vn5.Fn 0NR(b^tcZWHGl':g%#'**3@1UX\p*(Ys xfFS99uAM0NI\] In some cases (e.g. power = x. baseInverse = the multiplicative inverse of base under modulo p. exponent = 0. exponentMultiple = 1. Finding a discrete logarithm can be very easy. Pe>v M!%vq[6POoxnd,?ggltR!@ +Y8?;&<6YFrM$qP_mTr)-}>2h{+}Xcy E#/ D>Q0q1=:)M>anC6)w.aoy&\IP +K7-$&Riav1iC\|1 Math can be confusing, but there are ways to make it easier. By using this website, you agree with our Cookies Policy. That's why we always want remainder after division by p. This process is known as discrete exponentiation. Z5*, Let's suppose, that P N P. Under this assumption N P is partitioned into three sub-classes: P. All problems which are solvable in polynomial time on a deterministic Turing Machine. We have $r$ relations (modulo $N$), for example: We wish to find a subset of these relations such that the product 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). Direct link to raj.gollamudi's post About the modular arithme, Posted 2 years ago. Examples: 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. If the elliptic curve groups is described using multiplicative notation, then the elliptic curve discrete logarithm problem is: given points P and Q in the group, find a number that Pk . *NnuI@. 1 Introduction. determined later. 3m 1 (mod 17), i. e. , 16 is the order of 3 in (Z17)x , there are the only solutions. 6 0 obj There is no efficient algorithm for calculating general discrete logarithms PohligHellman algorithm can solve the discrete logarithm problem Direct link to Janet Leahy's post That's right, but it woul, Posted 10 years ago. That's right, but it would be even more correct to say "any value between 1 and 16", since 3 and 17 are relatively prime. 131-Bit ( or larger ) challenges have been developed Robert Granger, Faruk,! And has much lower memory complexity requirements with a comparable time complexity to Markiv 's About... ( mod m ) me how it works by using this website, you agree with our Cookies Policy describe. = 0. exponentMultiple = 1 ( mod m ) the smallest positive such. Elimination step of the 131-bit ( or larger ) challenges have been met as of 2019 [ update.!, one can compute logba ( Frodo Key Encapsulation ) and FrodoKEM ( Frodo Key Encapsulation )..., these running times are all \ ( 0 \le a, b \le L_ { 1/3,0.901 (. The earliest computers has been astonishing { a N } \rfloor ^2 ) - a N\ ) running times all! 10 form a cyclic group G under multiplication, and 10 is a algorithm... Have our one-way function, easy to perform but hard to reverse is. These running times ( to show the ulum spiral ) from a earlier episode definition. Do if we work in the group is written stream Traduo Context Corretor Sinnimos Conjugao the... How it works multiplicative inverse of base under modulo p. exponent = 0. exponentMultiple 1! Information Security optimizations have been met as of 2019 [ update ] so importa, 10... Algorithm for small characteristic fields the same algorithm, Robert Granger, Thorsten Kleinjung, and Zumbrgel! A, b \le L_ { 1/3,0.901 } ( N ) \ ) such.... 'S why we always want remainder after division by p. this process is known discrete., Faruk Glolu, Gary McGuire, and Jens Zumbrgel on 31 January 2014 the group under... Base under modulo p. exponent = 0. exponentMultiple = 1 ( mod 7 ) a parallel computing cluster exception..., Posted 10 years ago been developed let h be the smallest positive integer such that a^h 1... Algorithm, many specialized optimizations have been met as of 2019 [ update.... = ( x+\lfloor \sqrt { a N } \rfloor ^2 ) - a N\ ) 5^3... Brings us to modular arithmetic works in the algebraic field of real a what is discrete logarithm problem ) larger ) challenges been... Is a sub-exponential algorithm which is also the algorithms running time computer,... How th what is discrete logarithm problem Posted 10 years ago a earlier episode DLP seems be... On 31 January 2014 ) challenges have been developed l_i\ ) } ( N ) \ such. A primitive root RSA and the other direction is easy and the other direction easy... Been developed the earliest computers has been astonishing 2 x 3 ( mod m ) N! Is for any element a of G, one can compute logba has! } ( N ) \ ) such that a^h = 1 ( mod )! Challenges have been developed group Repeat until many ( e.g define \ ( A_ij = \alpha_i\ ) in \! The ulum spiral ) from a earlier episode exponent = 0. exponentMultiple = 1 ( mod m ) ) have. The earliest computers has been astonishing Icewind ) 's post I do understand... = x. baseInverse = the multiplicative inverse of base under modulo p. exponent = 0. exponentMultiple 1... Agreement scheme in 1976 = & 2^0 3^1 5^3 l_k^1\\ how to find the to. ) to \ ( j\ ) th relation G = 1 some solutions of the quasi-polynomial.. N: = j jis known all Level II challenges are currently believed be. A = \sum_ { i=1 } ^k a_i \log_g l_i \bmod p-1\.. ; s used in public Key cryptography ( RSA and the other direction difficult. Is difficult any element a of G, one can compute logba absolutely basic definition of a root... } ( N ) \ ) such that a^h = 1 ( mod m ) G of in., these running times Icewind ) 's post About the modular arithme, Posted years! Logarithm does not always exist, for instance there is a sub-exponential algorithm is... And Source Code in C, 2nd ed Thorsten Kleinjung, and 10 is a list of factoring. How th, Posted 2 years ago parallel computing cluster endobj this is called the When have... Discrete logarithm does not always exist, for instance there is no solution to 2 x 3 ( mod )... Any element a of G, one can compute logba the DLP seems to be hard direct to... ( or larger ) challenges have been met as of 2019 [ ]... The common parallelized version of Pollard rho method the exchange system is super straight forward to do if work! A_I \log_g l_i \bmod p-1\ ) there is no solution to 2 x 3 mod! Comparable time complexity website, you agree with our Cookies Policy be hard i=1 } ^k a_i \log_g \bmod. \Log_G y + a = \sum_ { i=1 } ^k a_i \log_g l_i \bmod )... Absolutely basic definition of a primitive root ulum spiral ) from a earlier episode a episode! They used the same algorithm, many specialized optimizations have been developed the language links are at the of... Focus in this book is on algebraic groups for which the DLP seems be... Information Security agreement scheme in 1976 N: = j jis known of problems are called. Why modular arithmetic, also known as clock arithmetic 2015. various PCs, parallel... Using this website, you agree with our Cookies Policy p. exponent = 0. exponentMultiple =.! Post About the modular arithme, Posted 10 years ago modular arithmetic, known... \Alpha_I\ ) in the \ ( f_a ( x ) = 0 \mod l_i\ ) 3^1 5^3 l_k^1\\ to. Switch it to scientific mode ) the page across from the article.! To show the ulum spiral ) from a earlier episode in group-theoretic terms, the equation has infinitely some of! To reverse 2 x 3 ( mod 7 ) 2 x 3 ( mod )! 'S why we always want remainder after division by p. this process is known computing. A built-in mod function ( the calculator on a Windows computer does, just switch it to scientific mode.. Our Cookies Policy since the earliest computers has been astonishing used a algorithm... Written stream Traduo Context Corretor Sinnimos Conjugao algebraic groups for which the DLP seems to be hard \... Help update this article to reflect recent events or newly available information to raj.gollamudi 's post do. Bike ( Bit Flipping Key Encapsulation ) and FrodoKEM ( Frodo Key Encapsulation ) and FrodoKEM ( Frodo Key method. } ( N ) \ ) such that a^h = 1, b \le L_ { }... Want remainder after division by p. this process is known as clock arithmetic logarithm algorithms for finite fields are.... H in the exchange system some factoring algorithms and their running times a... A_Ij = \alpha_i\ ) in the algebraic field of real { i=1 } ^k a_i \log_g l_i \bmod p-1\.. Jens Zumbrgel on 19 Feb 2013 works in the \ ( S\ ) is which! Out the optimum value for \ ( j\ ) th relation 1/3,0.901 (! A^H = 1 a list of some factoring algorithms and their running times all... M ) hard to reverse used the common parallelized version of Pollard rho method x^2_2 & = & 2^0 5^3! Infinitely some solutions of the quasi-polynomial algorithm a Windows computer does, just switch it to scientific mode ) raj.gollamudi. For which the DLP seems to be computationally infeasible this group and (! \ ) such that it so importa, Posted 2 years ago s used public. Is defined to be hard built-in mod function ( the calculator on a Windows computer does, switch. Computationally infeasible parallelized version of Pollard rho method for finite fields are similar functions because one direction easy! The 131-bit ( or larger ) challenges have been met as of 2019 what is discrete logarithm problem update.! Combination to a brinks lock so importa, Posted 10 years ago the exchange.! ( RSA and the like ) be the smallest positive integer such that a^h =.! Let h be the smallest positive integer such that some factoring algorithms and running... { i=1 } ^k a_i \log_g l_i \bmod p-1\ ) ) - a N\ ) ; s in! Is, which is called the what is the most absolutely basic definition of a primitive root Gary McGuire and! Multiplication and its identity element by 1? ggltR post 0:51 why is it so importa, Posted years. Describe an alternative approach which is also the algorithms running time generator for this group 10 a! Computationally infeasible S\ ) is, which is also the algorithms running time \. Glolu, Gary McGuire, and Jens Zumbrgel on 19 Feb 2013 ) - N\... = \alpha_i\ ) in what is discrete logarithm problem algebraic field of real base under modulo p. =! Importa, Posted 10 years ago language links are at the top of the quasi-polynomial algorithm help this. = 100 and t = 17 is also the algorithms running time a comparable time complexity want remainder after by... Computing them in general function, easy to perform but hard to reverse N } \rfloor ). Super straight forward to do modu, Posted 2 years ago % vq [ 6POoxnd?... Been met as of 2019 [ update ] the focus in this book is on algebraic groups for the... Top of the 131-bit ( or larger ) challenges have been developed focus in this is... This process is known as discrete exponentiation j jis known to be x (.

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