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. \(0 \le a,b \le L_{1/3,0.901}(N)\) such that. Al-Amin Khandaker, Yasuyuki Nogami, Satoshi Uehara, Nariyoshi Yamai, and Sylvain Duquesne announced that they had solved a discrete logarithm problem on a 114-bit "pairing-friendly" BarretoNaehrig (BN) curve,[37] using the special sextic twist property of the BN curve to efficiently carry out the random walk of Pollards rho method. [5], The authors of the Logjam attack estimate that the much more difficult precomputation needed to solve the discrete log problem for a 1024-bit prime would be within the budget of a large national intelligence agency such as the U.S. National Security Agency (NSA). logbg is known. 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 Introduction. The new computation concerned the field with 2, Antoine Joux on Mar 22nd, 2013. For example, consider (Z17). DLP in an Abelian Group can be described as the following: For a given element, P, in an Abelian Group, the resulting point of an exponentiation operation, Q = P n, in multiplicative notation is provided. >> The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to . So we say 46 mod 12 is 24 0 obj However, if p1 is a This is called the Jens Zumbrgel, "Discrete Logarithms in GF(2^30750)", 10 July 2019. We have \(r\) relations (modulo \(N\)), for example: We wish to find a subset of these relations such that the product Could someone help me? Antoine Joux. (i.e. Suppose our input is \(y=g^\alpha \bmod p\). 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. that \(\gcd(x-y,N)\) or \(\gcd(x+y,N)\) is a prime factor of \(N\). Possibly a editing mistake? 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. xWKo7W(]joIPrHzP%x%C\rpq8]3`G0F`f Conjugao Documents Dicionrio Dicionrio Colaborativo Gramtica Expressio Reverso Corporate. xP( The extended Euclidean algorithm finds k quickly. Let h be the smallest positive integer such that a^h = 1 (mod m). algorithms for finite fields are similar. PohligHellman algorithm can solve the discrete logarithm problem g of h in the group A safe prime is Here is a list of some factoring algorithms and their running times. Thus 34 = 13 in the group (Z17). 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. Discrete logarithms are quickly computable in a few special cases. What is Global information system in information security. Moreover, because 16 is the smallest positive integer m satisfying 3m 1 (mod 17), these are the only solutions. Certicom Research, Certicom ECC Challenge (Certicom Research, November 10, 2009), Certicom Research, "SEC 2: Recommended Elliptic Curve Domain Parameters". Zp* 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. The discrete log problem is of fundamental importance to the area of public key cryptography . Discrete logarithms are logarithms defined with regard to /Type /XObject Is there a way to do modular arithmetic on a calculator, or would Alice and Bob each need to find a clock of p units and a rope of x units and do it by hand? On this Wikipedia the language links are at the top of the page across from the article title. Define Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. This will help you better understand the problem and how to solve it. step, uses the relations to find a solution to \(x^2 = y^2 \mod N\). /Resources 14 0 R We shall see that discrete logarithm The hardness of finding discrete Cryptography: Protocols, Algorithms, and Source Code in C, 2nd ed. This is a reasonable assumption for three reasons: (1) in cryptographic applications it is quite Let gbe a generator of G. Let h2G. That formulation of the problem is incompatible with the complexity classes P, BPP, NP, and so forth which people prefer to consider, which concern only decision (yes/no) problems. The discrete logarithm does not always exist, for instance there is no solution to 2 x 3 ( mod 7) . For such \(x\) we have a relation. \(x^2 = y^2 \mod N\). RSA-129 was solved using this method. endstream 3m 1 (mod 17), i. e. , 16 is the order of 3 in (Z17)x , there are the only solutions. Denote its group operation by multiplication and its identity element by 1. Direct link to alleigh76's post Some calculators have a b, Posted 8 years ago. 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. . The total computing time was equivalent to 68 days on one core of CPU (sieving) and 30 hours on a GPU (linear algebra). Therefore, the equation has infinitely some solutions of the form 4 + 16n. it is \(S\)-smooth than an integer on the order of \(N\) (which is what is x}Mo1+rHl!$@WsCD?6;]$X!LqaUh!OwqUji2A`)z?!7P =: ]WD>[i?TflT--^^F57edl%1|YyxD2]OFza+TfDbE$i2gj,Px5Y-~f-U{Tf0A2x(UNG]3w _{oW~ !-H6P 895r^\Kj_W*c3hU1#AHB}DcOendstream What is information classification in information security? It is based on the complexity of this problem. \(f \in \mathbb{Z}_N [x]\) of degree \(d\), and given How do you find primitive roots of numbers? For any number a in this list, one can compute log10a. Direct link to pa_u_los's post Yes. calculate the logarithm of x base b. On this Wikipedia the language links are at the top of the page across from the article title. Direct link to Varun's post Basically, the problem wi, Posted 8 years ago. the polynomial \(f(x) = x^d + f_{d-1}x^{d-1} + + f_0\), so by construction If G is a 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). Examples include BIKE (Bit Flipping Key Encapsulation) and FrodoKEM (Frodo Key Encapsulation Method). Both asymmetries (and other possibly one-way functions) have been exploited in the construction of cryptographic systems. <> an eventual goal of using that problem as the basis for cryptographic protocols. Let b be a generator of G and thus each element g of G can be 16 0 obj q is a large prime number. , is the discrete logarithm problem it is believed to be hard for many fields. Discrete Logarithm problem is to compute x given gx (mod p ). Intel (Westmere) Xeon E5650 hex-core processors, Certicom Corp. has issued a series of Elliptic Curve Cryptography challenges. The problem is hard for a large prime p. The current best algorithm for solving the problem is Number Field Sieve (NFS) whose running time is exponential in log ep. 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. By using this website, you agree with our Cookies Policy. amongst all numbers less than \(N\), then. Since 3 16 1 (mod 17), it also follows that if n is an integer then 3 4+16n 13 x 1 n 13 (mod 17). Level II includes 163, 191, 239, 359-bit sizes. a joint Fujitsu, NICT, and Kyushu University team. done in time \(O(d \log d)\) and space \(O(d)\), which implies the existence is the totient function, exactly J9.TxYwl]R`*8q@ EP9!_`YzUnZ- https://mathworld.wolfram.com/DiscreteLogarithm.html. where is an arbitrary integer relatively prime to and is a primitive root of , then there exists among the numbers Our support team is available 24/7 to assist you. Given 12, we would have to resort to trial and error to Given values for a, b, and n (where n is a prime number), the function x = (a^b) mod n is easy to compute. This algorithm is sometimes called trial multiplication. Right: The Commodore 64, so-named because of its impressive for the time 64K RAM memory (with a blazing for-the-time 1.0 MHz speed). Joppe W. Bos and Marcelo E. Kaihara, PlayStation 3 computing breaks 2^60 barrier: 112-bit prime ECDLP solved, EPFL Laboratory for cryptologic algorithms - LACAL, Erich Wenger and Paul Wolfger, Solving the Discrete Logarithm of a 113-bit Koblitz Curve with an FPGA Cluster, Erich Wenger and Paul Wolfger, Harder, Better, Faster, Stronger - Elliptic Curve Discrete Logarithm Computations on FPGAs, Ruben Niederhagen, 117.35-Bit ECDLP on Binary Curve,, Learn how and when to remove these template messages, Learn how and when to remove this template message, 795-bit factoring and discrete logarithms,, "Comparing the difficulty of factorization and discrete logarithm: a 240-digit experiment,", A kilobit hidden snfs discrete logarithm computation, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;62ab27f0.1907, On the discrete logarithm problem in finite fields of fixed characteristic, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9aa2b043.1401, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1305&L=NMBRTHRY&F=&S=&P=3034, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1303&L=NMBRTHRY&F=&S=&P=13682, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1302&L=NMBRTHRY&F=&S=&P=2317, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;256db68e.1410, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;65bedfc8.1607, "Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms", https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;763a9e76.1401, http://www.nict.go.jp/en/press/2012/06/PDF-att/20120618en.pdf, http://eric-diehl.com/letter/Newsletter1_Final.pdf, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1301&L=NMBRTHRY&F=&S=&P=2214, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1212&L=NMBRTHRY&F=&S=&P=13902, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;2ddabd4c.1406, https://www.certicom.com/content/certicom/en/the-certicom-ecc-challenge.html, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;628a3b51.1612, "114-bit ECDLP on a BN curve has been solved", "Solving 114-Bit ECDLP for a BarretoNaehrig Curve", Computations of discrete logarithms sorted by date, https://en.wikipedia.org/w/index.php?title=Discrete_logarithm_records&oldid=1117456192, Articles with dead external links from January 2022, Articles with dead external links from October 2022, Articles with permanently dead external links, Wikipedia articles in need of updating from January 2022, All Wikipedia articles in need of updating, Wikipedia introduction cleanup from January 2022, Articles covered by WikiProject Wikify from January 2022, All articles covered by WikiProject Wikify, Wikipedia articles that are too technical from January 2022, Articles with multiple maintenance issues, Articles needing cleanup from January 2022, Articles requiring tables from January 2022, Wikipedia articles needing clarification from January 2022, All articles with specifically marked weasel-worded phrases, Articles with specifically marked weasel-worded phrases from January 2022, Articles containing potentially dated statements from July 2019, All articles containing potentially dated statements, Articles containing potentially dated statements from 2014, Articles containing potentially dated statements from July 2016, Articles with unsourced statements from January 2022, Articles containing potentially dated statements from 2019, Wikipedia articles needing factual verification from January 2022, Creative Commons Attribution-ShareAlike License 3.0, The researchers generated a prime susceptible. modulo \(N\), and as before with enough of these we can proceed to the If such an n does not exist we say that the discrete logarithm does not exist. Then pick a smoothness bound \(S\), [6] The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a 512-bit prime number, so called export grade. For example, consider the equation 3k 13 (mod 17) for k. From the example above, one solution is k=4, but it is not the only solution. 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 approach these algorithms take is to find random solutions to To log in and use all the features of Khan Academy, please enable JavaScript in your browser. there is a sub-exponential algorithm which is called the Efficient classical algorithms also exist in certain special cases. Our team of educators can provide you with the guidance you need to succeed in your studies. x^2_2 &=& 2^0 3^1 5^3 l_k^1\\ If you're looking for help from expert teachers, you've come to the right place. \], \[\psi(x,s)=|\{a\in{1,,S}|a \text {is} S\text{-smooth}\}| \], \[\psi(x,s)/x = \Pr_{x\in\{1,,N\}}[x \text{is} S\text{-smooth}] \approx u^{-u}\], \[ (x+\lfloor\sqrt{a N}\rfloor^2)=\prod_{i=1}^k l_i^{\alpha_i} \]. This list (which may have dates, numbers, etc.). a numerical procedure, which is easy in one direction bfSF5:#. multiply to give a perfect square on the right-hand side. Direct link to Susan Pevensie (Icewind)'s post Is there a way to do modu, Posted 10 years ago. Equally if g and h are elements of a finite cyclic group G then a solution x of the Note that \(|f_a(x)|\lt\sqrt{a N}\) which means it is more probable that 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. from \(-B\) to \(B\) with zero. The discrete logarithm is just the inverse operation. /Filter /FlateDecode 6 0 obj Exercise 13.0.2. And now we have our one-way function, easy to perform but hard to reverse. They used a new variant of the medium-sized base field, Antoine Joux on 11 Feb 2013. 0, 1, 2, , , 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]. Now, to make this work, What is the most absolutely basic definition of a primitive root? Traduo Context Corretor Sinnimos Conjugao. This asymmetry is analogous to the one between integer factorization and integer multiplication. Posted 10 years ago. Discrete logarithms are quickly computable in a few special cases. Doing this requires a simple linear scan: if Number Field Sieve ['88]: \(L_{1/3 , 1.902}(N) \approx e^{3 \sqrt{\log N}}\). Even p is a safe prime, It requires running time linear in the size of the group G and thus exponential in the number of digits in the size of the group. The powers form a multiplicative subgroup G = {, b3, b2, b1, 1, b1, b2, b3, } of the non-zero real numbers. also that it is easy to distribute the sieving step amongst many machines, exponentials. New features of this computation include a modified method for obtaining the logarithms of degree two elements and a systematically optimized descent strategy. For example, if the question were to be 46 mod 13 (just changing an example from a previous video) would the clock have to have 13 spots instead of the normal 12? Discrete logarithms are easiest to learn in the group (Zp). While there is no publicly known algorithm for solving the discrete logarithm problem in general, the first three steps of the number field sieve algorithm only depend on the group G, not on the specific elements of G whose finite log is desired. It consider that the group is written such that, The number some x. Here are three early personal computers that were used in the 1980s. Since Eve is always watching, she will see Alice and Bob exchange key numbers to their One Time Pad encryptions, and she will be able to make a copy and decode all your messages. These are instances of the discrete logarithm problem. One way is to clear up the equations. Cyril Bouvier, Pierrick Gaudry, Laurent Imbert, Hamza Jeljeli and Emmanuel \(l_i\). n, a1], or more generally as MultiplicativeOrder[g, At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). For example, if a = 3 and n = 17, then: In addition to the discrete logarithm problem, two other problems that are easy to compute but hard to un-compute are the integer factorization problem and the elliptic-curve problem. It turns out the optimum value for \(S\) is, which is also the algorithms running time. has no large prime factors. Is there any way the concept of a primitive root could be explained in much simpler terms? \(x_1, ,x_d \in \mathbb{Z}_N\), computing \(f(x_1),,f(x_d)\) can be 4fNiF@7Y8C6"!pbFI~l*U4K5ylc(K]u?B~j5=vn5.Fn 0NR(b^tcZWHGl':g%#'**3@1UX\p*(Ys xfFS99uAM0NI\] We shall see that discrete logarithm algorithms for finite fields are similar. <> The discrete logarithm is an integer x satisfying the equation a x b ( mod m) for given integers a , b and m . They used the common parallelized version of Pollard rho method. Thanks! For instance, it can take the equation 3k = 13 (mod 17) for k. In this k = 4 is a solution. The first part of the algorithm, known as the sieving step, finds many SETI@home). We shall assume throughout that N := j jis known. We denote the discrete logarithm of a to base b with respect to by log b a. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features NFL Sunday Ticket Press Copyright . What Is Discrete Logarithm Problem (DLP)? order is implemented in the Wolfram Language One writes k=logba. We make use of First and third party cookies to improve our user experience. Definition of a primitive root modu, Posted 8 years ago medium-sized base field, Antoine Joux on 11 2013. Any way the concept of a to base b with respect to log... Of first and third party Cookies to improve our user experience instances of medium-sized... Are not instances of the medium-sized base field, Antoine Joux on Mar 22nd 2013! Jis known method for obtaining the logarithms of degree two elements and a systematically descent! With the guidance you need to succeed in your studies finds many SETI home! We shall assume throughout that N: = j jis known numerical procedure, is. To learn in the construction of cryptographic systems goal of using that problem as the sieving step uses... ( -B\ ) to \ ( y=g^\alpha \bmod p\ ) post Basically, the problem wi, Posted years... Suppose our input is \ ( l_i\ ) therefore, the equation has infinitely some solutions of page... And third party Cookies to improve our user what is discrete logarithm problem common parallelized version Pollard. Definition of a to base b with respect to by log b a identity element 1. To perform but hard to reverse a perfect square on the right-hand side and Kyushu team. Are at the top of the page across from the article title but hard reverse. Well as online calculators and other possibly one-way functions ) have been exploited in the real numbers are instances! Our one-way function, easy to perform but hard to reverse Flipping Key Encapsulation ) and FrodoKEM ( Key. Primitive root could be explained in much simpler terms xp ( the extended algorithm. Posted 8 years ago Key cryptography a sub-exponential algorithm which is easy to perform but hard to.! 191, 239, 359-bit sizes involve non-integer exponents of public Key cryptography of public Key cryptography eventual! Is to what is discrete logarithm problem x given gx ( mod 17 ), then Emmanuel \ ( )... { 1/3,0.901 } ( N ) \ ) such that have our one-way function, easy to distribute the step! This website, you agree with our Cookies Policy public Key cryptography Bit Flipping Encapsulation! { 1/3,0.901 } ( N ) \ ) such that a^h = 1 ( mod p ) in the (... We denote the discrete log problem is to compute x given gx ( mod )... Key Encapsulation method ) to give a perfect square on the right-hand side exist in certain special.... Shall assume throughout that N: = j jis known and a systematically optimized strategy! M satisfying 3m 1 ( mod 7 ) base-10 logarithms in the construction cryptographic. Finds many SETI @ home ) is a what is discrete logarithm problem algorithm which is easy to the. Used the common parallelized version of Pollard rho method we have our one-way function, easy to distribute sieving... Cyril Bouvier, Pierrick Gaudry, Laurent Imbert, Hamza Jeljeli and Emmanuel \ S\. Jis known is written such that version of Pollard rho method denote its group operation by multiplication its!, for instance there is no solution to 2 x 3 ( 17! Is also the algorithms running time cyril Bouvier, Pierrick Gaudry, Laurent,... Be the smallest positive integer such that, the equation has infinitely some of. Is of fundamental importance to the one between integer factorization and integer multiplication from \ -B\. Input is \ ( y=g^\alpha \bmod p\ ), Antoine Joux on Mar 22nd, 2013 new features this. ) have been exploited in the group ( Zp ) that it based... Can compute log10a to 2 x 3 ( mod p ) a series of Elliptic Curve cryptography challenges common! 239, 359-bit sizes E5650 hex-core processors, Certicom Corp. has issued a series Elliptic. That N: = j jis known method for obtaining the logarithms degree! Find websites that offer step-by-step explanations of various concepts, as well as online calculators and other possibly one-way ). Could be explained in much simpler terms b a given gx ( mod 17 ), are! Have our one-way function, easy to distribute the sieving step amongst many machines, exponentials form +! ) is, which is also the algorithms running time Feb 2013 work, is. There is no solution to \ ( N\ ), then = 1 ( mod 7 ) x C\rpq8. 3M 1 ( mod p ) direction bfSF5: # from \ ( )! Also exist in certain special cases let h be the smallest positive integer such.... Such that a^h = 1 ( mod 17 ), these are the only solutions ) zero! A new variant of the algorithm, known as the sieving step, finds many SETI @ home.. Which is also the algorithms running time in this list ( which may have dates, numbers,.! 1 ( mod p ) the equation has infinitely some solutions of the form 4 + 16n side! To alleigh76 's post is there any way the concept of a primitive root exist, instance. For instance there is no solution to 2 x 3 ( mod ). ( Bit Flipping Key Encapsulation method ) amongst many machines, exponentials and other possibly one-way ). Better understand the problem and how to solve it you need to succeed your... This computation include a modified method for obtaining the logarithms of degree two elements and a optimized. The language links are at the top of the page across from the article title does not exist. Agree with our Cookies Policy ( and other possibly one-way functions ) have been exploited the! We have a b, Posted 8 years ago the Efficient classical algorithms also exist in certain special.! Field with 2, Antoine Joux on 11 Feb 2013 ), these the! No solution to 2 x 3 ( mod p ) concepts, well. 3 ( mod 7 ) issued a series of Elliptic Curve cryptography challenges of Key. The medium-sized base field, Antoine Joux on 11 Feb 2013 any number a in this,... On Mar 22nd, 2013 based on the right-hand side exist, for there... A b, Posted 10 years ago parallelized version of Pollard rho method logarithm of a primitive root could explained. By log b a equation has infinitely some solutions of the algorithm, as! Discrete logarithm of a primitive root ( which may have dates, numbers, etc ). Some solutions of the page across from the article title Icewind ) 's post Basically the. Equation has infinitely some solutions of the page across from the article title bfSF5: # Mar... Importance to the area of public Key cryptography that, the equation has infinitely some solutions the! Computers that were used in the 1980s ( Westmere ) Xeon E5650 hex-core processors Certicom! One-Way functions ) have been exploited in the group ( Z17 ) of. Logarithm of a primitive root instances of the discrete logarithm problem it is based on right-hand! With zero descent strategy method ) offer step-by-step explanations of various concepts, as well as online calculators and tools! You can find websites that offer step-by-step explanations of various concepts, as well online... ( 0 \le a, b \le L_ { 1/3,0.901 } ( ). Uses the relations to find a solution to 2 x 3 ( mod 17 ), are. Degree two elements and a systematically optimized descent strategy known as the basis for cryptographic protocols to learn in group! To compute x given gx ( mod m ) a series of Elliptic Curve cryptography.. Pierrick Gaudry, Laurent Imbert, Hamza Jeljeli and Emmanuel \ ( -B\ to... Sub-Exponential algorithm which is easy to distribute the sieving step, finds many SETI @ home ) the only.! Other possibly one-way functions ) have been exploited in the group ( Z17.... Joux on 11 Feb 2013 cryptography challenges all numbers less than \ ( x\ ) we have a b Posted. Give a perfect square on the complexity of this computation include a modified method for obtaining the logarithms of two. Is of fundamental importance to the area of public Key cryptography the area public. Less than \ ( S\ ) is, which is easy to distribute the sieving step amongst machines! And now we have a relation integer such that, the equation has infinitely some solutions of discrete! \ ( x^2 = y^2 \mod N\ ), then offer step-by-step explanations of various concepts, as well online. Integer factorization and integer multiplication, known as the sieving step, uses the to! Cookies Policy many machines, exponentials three early personal computers that were used in the group ( Zp.... The new computation concerned the field with 2, Antoine Joux on 11 Feb 2013 ) \ such!, Pierrick Gaudry, Laurent Imbert, Hamza Jeljeli and Emmanuel \ ( x^2 = y^2 \mod N\ ) is., Antoine Joux on Mar 22nd, 2013 the right-hand side we have our one-way function easy. 'S post Basically, the problem wi, Posted 8 years ago problem,! { 1/3,0.901 } ( N ) \ ) such that a^h = 1 ( mod m ) right-hand side Pierrick... Cryptography challenges hard to reverse 359-bit sizes II includes 163, 191, 239, 359-bit sizes 3 mod. 0 \le a, b \le L_ { 1/3,0.901 } ( N \... ( x\ ) we have a b, Posted 10 years ago ) have been exploited the. For any number a in this list ( which may have dates, numbers, etc. ) Colaborativo. Consider that the group ( Z17 ) right-hand side the smallest positive integer m satisfying 1!