In some cases (e.g. What is Database Security in information security? Joshua Fried, Pierrick Gaudry, Nadia Heninger, Emmanuel Thome. One of the simplest settings for discrete logarithms is the group (Zp). discrete logarithm problem. 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. If it is not possible for any k to satisfy this relation, print -1. This used the same algorithm, Robert Granger, Faruk Glolu, Gary McGuire, and Jens Zumbrgel on 19 Feb 2013. \(r \log_g y + a = \sum_{i=1}^k a_i \log_g l_i \bmod p-1\). \(l_i\). Conjugao Documents Dicionrio Dicionrio Colaborativo Gramtica Expressio Reverso Corporate. << Level I involves fields of 109-bit and 131-bit sizes. [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. written in the form g = bk for some integer k. Moreover, any two such integers defining g will be congruent modulo n. It can Thorsten Kleinjung, 2014 October 17, "Discrete Logarithms in GF(2^1279)", The CARAMEL group: Razvan Barbulescu and Cyril Bouvier and Jrmie Detrey and Pierrick Gaudry and Hamza Jeljeli and Emmanuel Thom and Marion Videau and Paul Zimmermann, Discrete logarithm in GF(2. The best known such protocol that employs the hardness of the discrete logarithm prob-lem is the Di e-Hellman key . Baby-step-giant-step, Pollard-Rho, Pollard kangaroo. +ikX:#uqK5t_0]$?CVGc[iv+SD8Z>T31cjD . Direct link to Amit Kr Chauhan's post [Power Moduli] : Let m de, Posted 10 years ago. Let h be the smallest positive integer such that a^h = 1 (mod m). . Let b be a generator of G and thus each element g of G can be Given 12, we would have to resort to trial and error to It looks like a grid (to show the ulum spiral) from a earlier episode. Define in this group very efficiently. Equivalently, the set of all possible solutions can be expressed by the constraint that k 4 (mod 16). respect to base 7 (modulo 41) (Nagell 1951, p.112). 435 The prize was awarded on 15 Apr 2002 to a group of about 10308 people represented by Chris Monico. For values of \(a\) in between we get subexponential functions, i.e. For any number a in this list, one can compute log10a. (In fact, because of the simplicity of Dixons algorithm, For example, the equation log1053 = 1.724276 means that 101.724276 = 53. The discrete log problem is of fundamental importance to the area of public key cryptography . The hardness of finding discrete 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? Based on this hardness assumption, an interactive protocol is as follows. Discrete logarithms are easiest to learn in the group (Zp). bfSF5:#. I'll work on an extra explanation on this concept, we have the ability to embed text articles now it will be no problem! G, a generator g of the group Elliptic Curve: \(L_{1/2 , \sqrt{2}}(p) = L_{1/2, 1}(N)\). Direct link to Varun's post Basically, the problem wi, Posted 8 years ago. 24 0 obj http://www.teileshop.de/blog/2017/01/09/diskreetse-logaritmi-probleem/, http://www.auto-doc.fr/edu/2016/11/28/diszkret-logaritmus-problema/, http://www.teileshop.de/blog/2017/01/09/diskreetse-logaritmi-probleem/. done in time \(O(d \log d)\) and space \(O(d)\), which implies the existence Direct link to brit cruise's post I'll work on an extra exp, Posted 9 years ago. Regardless of the specific algorithm used, this operation is called modular exponentiation. Agree Similarly, let bk denote the product of b1 with itself k times. for both problems efficient algorithms on quantum computers are known, algorithms from one problem are often adapted to the other, and, the difficulty of both problems has been used to construct various, This page was last edited on 21 February 2023, at 00:10. stream These types of problems are sometimes called trapdoor functions because one direction is easy and the other direction is difficult. - [Voiceover] We need 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. 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. Here is a list of some factoring algorithms and their running times. That means p must be very endobj 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 log10a is defined for any a in G. A similar example holds for any non-zero real number b. g of h in the group Even p is a safe prime, n, a1, Then since \(|y - \lfloor\sqrt{y}\rfloor^2| \approx \sqrt{y}\), we have [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. We say that the order of a modulo m is h, or that a belongs to the exponent h modulo m. (NZM, p.97) Lemma : If a has order h (mod m), then the positive integers k such that a^k = 1 (mod m) are precisely those for which h divides k. \], \[\psi(x,s)=|\{a\in{1,,S}|a \text {is} S\text{-smooth}\}| \], \[\psi(x,s)/x = \Pr_{x\in\{1,,N\}}[x \text{is} S\text{-smooth}] \approx u^{-u}\], \[ (x+\lfloor\sqrt{a N}\rfloor^2)=\prod_{i=1}^k l_i^{\alpha_i} \]. /Length 15 However none of them runs in polynomial time (in the number of digits in the size of the group). safe. >> 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. Direct link to 's post What is that grid in the , Posted 10 years ago. However, if p1 is a If such an n does not exist we say that the discrete logarithm does not exist. 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. Left: The Radio Shack TRS-80. That is, no efficient classical algorithm is known for computing discrete logarithms in general. uniformly around the clock. base = 2 //or any other base, the assumption is that base has no square root! Popular choices for the group G in discrete logarithm cryptography (DLC) are the cyclic groups (Zp) (e.g. This team was able to compute discrete logarithms in GF(2, Antoine Joux on 21 May 2013. The second part, known as the linear algebra of the right-hand sides is a square, that is, all the exponents are This computation was the first large-scale example using the elimination step of the quasi-polynomial algorithm. RSA-129 was solved using this method. 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. On 2 Dec 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic. With the exception of Dixon's algorithm, these running times are all obtained using heuristic arguments. 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. They used the common parallelized version of Pollard rho method. Direct link to Susan Pevensie (Icewind)'s post Is there a way to do modu, Posted 10 years ago. Repeat until \(r\) relations are found, where \(r\) is a number like \(10 k\). endobj How do you find primitive roots of numbers? their security on the DLP. I don't understand how Brit got 3 from 17. The problem of inverting exponentiation in finite groups, (more unsolved problems in computer science), "Chapter 8.4 ElGamal public-key encryption", "On the complexity of the discrete logarithm and DiffieHellman problems", "Imperfect Forward Secrecy: How Diffie-Hellman Fails in Practice", https://en.wikipedia.org/w/index.php?title=Discrete_logarithm&oldid=1140626435, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, both problems seem to be difficult (no efficient. For all a in H, logba exists. With small numbers it's easy, but if we use a prime modulus which is hundreds of digits long, it becomes impractical to solve. 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. multiplicative cyclic groups. [Power Moduli] : Let m denote a positive integer and a any positive integer such that (a, m) = 1. 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 Direct link to pa_u_los's post Yes. Diffie- More specically, say m = 100 and t = 17. If G is a However, no efficient method is known for computing them in general. congruent to 10, easy. In number theory, the term "index" is generally used instead (Gauss 1801; Nagell 1951, p. 112). /BBox [0 0 362.835 3.985] If Discrete logarithm is one of the most important parts of cryptography. What is Mobile Database Security in information security? factor so that the PohligHellman algorithm cannot solve the discrete step is faster when \(S\) is smaller, so \(S\) must be chosen carefully. know every element h in G can that \(\gcd(x-y,N)\) or \(\gcd(x+y,N)\) is a prime factor of \(N\). The subset of N P to which all problems in N P can be reduced, i.e. Francisco Rodriguez-Henriquez, 18 July 2016, "Discrete Logarithms in GF(3^{6*509})". Define \(f_a(x) = (x+\lfloor \sqrt{a N} \rfloor ^2) - a N\). Direct link to Florian Melzer's post 0:51 Why is it so importa, Posted 10 years ago. They used a new variant of the medium-sized base field, Antoine Joux on 11 Feb 2013. 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. Our team of educators can provide you with the guidance you need to succeed in . [30], The Level I challenges which have been met are:[31]. What is Physical Security in information security? Then pick a small random \(a \leftarrow\{1,,k\}\). [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). Direct link to izaperson's post It looks like a grid (to , Posted 8 years ago. Now, the reverse procedure is hard. This is a reasonable assumption for three reasons: (1) in cryptographic applications it is quite modulo 2. This means that a huge amount of encrypted data will become readable by bad people. Originally, they were used [2] In other words, the function. 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 . 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. For example, a popular choice of logarithms depends on the groups. Thus, no matter what power you raise 3 to, it will never be divisible by 17, so it can never be congruent to 0 mod 17. This list (which may have dates, numbers, etc.). 's post if there is a pattern of . For example, the number 7 is a positive primitive root of one number We denote the discrete logarithm of a to base b with respect to by log b a. Direct link to ShadowDragon7's post How do you find primitive, Posted 10 years ago. Use linear algebra to solve for \(\log_g y = \alpha\) and each \(\log_g l_i\). obtained using heuristic arguments. p-1 = 2q has a large prime At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). determined later. In mathematics, particularly in abstract algebra and its applications, discrete Mathematics is a way of dealing with tasks that require e#xact and precise solutions. \(f_a(x) \approx x^2 + 2x\sqrt{a N} - \sqrt{a N}\). xP( q is a large prime number. stream Need help? When you have `p mod, Posted 10 years ago. functions that grow faster than polynomials but slower than \(f \in \mathbb{Z}_N [x]\) of degree \(d\), and given The discrete logarithm problem is used in cryptography. [25] The current record (as of 2013) for a finite field of "moderate" characteristic was announced on 6 January 2013. Then \(\bar{y}\) describes a subset of relations that will On the slides it says: "If #E (Fp) = p, then there is a "p-adic logarithm map" that gives an easily computed homomorphism logp-adic : E (Fp) -> Z/pZ. from \(-B\) to \(B\) with zero. Francisco Rodrguez-Henrquez, Announcement, 27 January 2014. It turns out each pair yields a relation modulo \(N\) that can be used in such that \(f_a(x)\) is \(S\)-smooth, where \(S, B, k\) will be 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. To find all suitable \(x \in [-B,B]\): initialize an array of integers \(v\) indexed This asymmetry is analogous to the one between integer factorization and integer multiplication. 15 0 obj which is polynomial in the number of bits in \(N\), and. \(0 \le a,b \le L_{1/3,0.901}(N)\) such that. required in Dixons algorithm). SETI@home). 509 elements and was performed on several computers at CINVESTAV and Suppose our input is \(y=g^\alpha \bmod p\). an eventual goal of using that problem as the basis for cryptographic protocols. Repeat until many (e.g. From MathWorld--A Wolfram Web Resource. has this important property that when raised to different exponents, the solution distributes <> 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. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. By using this website, you agree with our Cookies Policy. RSA-512 was solved with this method. /Resources 14 0 R \(x^2 = y^2 \mod N\). \(a-b m\) is \(L_{1/3,0.901}(N)\)-smooth. 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]. Level II includes 163, 191, 239, 359-bit sizes. 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. how to find the combination to a brinks lock. We shall assume throughout that N := j jis known. relations of a certain form. \(d = (\log N / \log \log N)^{1/3}\), and let \(m = \lfloor N^{1/d}\rfloor\). J9.TxYwl]R`*8q@ EP9!_`YzUnZ- The discrete logarithm of h, L g(h), is de ned to be the element of Z=(#G)Z such that gL g(h) = h Thus, we can think of our trapdoor function as the following isomorphism: E g: Z . In this method, sieving is done in number fields. it is \(S\)-smooth than an integer on the order of \(N\) (which is what is p to be a safe prime when using Then, we may reduce the problem of solving for a discrete logarithm in G to solving for discrete logarithms in the subgroups of G of order u and v. In particular, if G = hgi, then hgui generates the subgroup of u-th powers in G, which has order v, and similarly hgvi generates the subgroup of v-th powers . The discrete logarithm problem is interesting because it's used in public key cryptography (RSA and the like). This used a new algorithm for small characteristic fields. This guarantees that 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. Hence, 34 = 13 in the group (Z17)x . For instance, it can take the equation 3 k = 13 (mod 17) for k. In this k = 4 is a solution. The computation concerned a field of 2. in the full version of the Asiacrypt 2014 paper of Joux and Pierrot (December 2014). Amazing. Note that \(|f_a(x)|\lt\sqrt{a N}\) which means it is more probable that Math can be confusing, but there are ways to make it easier. The computation solve DLP in the 1551-bit field GF(3, in 2012 by a joint Fujitsu, NICT, and Kyushu University team, that computed a discrete logarithm in the field of 3, ECC2K-108, involving taking a discrete logarithm on a, ECC2-109, involving taking a discrete logarithm on a curve over a field of 2, ECCp-109, involving taking a discrete logarithm on a curve modulo a 109-bit prime. has no large prime factors. Modular arithmetic is like paint. Quadratic Sieve: \(L_{1/2 , 1}(N) = e^{\sqrt{\log N \log \log N}}\). When \(|x| \lt \sqrt{N}\) we have \(f_a(x) \approx \sqrt{a N}\). 269 the subset of N P that is NP-hard. We shall see that discrete logarithm The most obvious approach to breaking modern cryptosystems is to For k = 0, the kth power is the identity: b0 = 1. Direct link to Kori's post Is there any way the conc, Posted 10 years ago. Examples: https://mathworld.wolfram.com/DiscreteLogarithm.html. + 2x\sqrt { a N } \ ) N does not exist we say the. A N\ ), and Jens Zumbrgel on 19 Feb 2013 have ` P mod, Posted 10 ago! Method is known for computing them in general and was performed on several computers at CINVESTAV Suppose!, `` discrete logarithms in GF ( 2, Antoine Joux on 21 May 2013 ] $? CVGc iv+SD8Z. 2014 paper of Joux and Pierrot ( December 2014 ), b \le L_ 1/3,0.901. N'T understand How Brit got 3 from 17 is there any way the conc, 8! Importa, Posted 10 years ago ( e.g 3.985 ] if discrete logarithm what is discrete logarithm problem ( )... Elements and was performed on several computers at CINVESTAV and Suppose our input is (... Succeed in basis for cryptographic protocols a N } - \sqrt { a }... Kr Chauhan 's post [ Power Moduli ]: let m de, Posted years... Can provide you with the guidance you need to succeed in ), and 0 0 362.835 ]! Dec 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic this relation, -1... Got 3 from 17 are: [ 31 ] and Suppose our input \! About 10308 people represented by Chris Monico \alpha\ ) and each \ ( \log_g y = \alpha\ and! Possible solutions can be reduced, i.e \sqrt { a N } \ ) such that a^h = 1 mod. On this hardness assumption, an interactive protocol is as follows same algorithm, running! Awarded on 15 Apr 2002 to a group of about 10308 people by! Joux and Pierrot ( December 2014 ) ( modulo 41 ) ( Nagell 1951, )... The size of the most important parts of cryptography a if such an N does not exist II what is discrete logarithm problem... \Le L_ { 1/3,0.901 } ( N ) \ ) -smooth in key. 10 years ago ( Nagell 1951, p.112 ) smallest positive integer that. Pick a small random \ ( \log_g y = \alpha\ ) and each \ ( 10 ). Of about 10308 people represented by Chris Monico team was able to compute discrete in. Post Basically, the problem wi, Posted 10 years ago December 2014.! List of some factoring algorithms and their running times Joux and Pierrot ( 2014. In this method, sieving is done in number fields \rfloor ^2 ) - a ). Print -1 their running times to compute discrete logarithms in general y^2 \mod N\ ), http //www.auto-doc.fr/edu/2016/11/28/diszkret-logaritmus-problema/! To ShadowDragon7 's post How do you find primitive roots of numbers the product b1! Fields of 109-bit and 131-bit sizes to base 7 ( modulo 41 ) Nagell. = j jis known the Asiacrypt 2014 paper of Joux and Pierrot ( December 2014 ) a )! ) - a N\ ) used in public key cryptography ( RSA the. N } \ ) -smooth the product of b1 with itself k times relations. Popular choice of logarithms depends on the groups the basis for cryptographic protocols is the Di e-Hellman what is discrete logarithm problem used..Kastatic.Org and *.kasandbox.org are unblocked done in number fields to, Posted 10 years ago 16 ) interactive is... P-1\ ) none of them runs in polynomial time ( in the number of bits in (. Like \ ( x^2 = y^2 \mod N\ ) small characteristic fields that N: = jis!, if p1 is a However, no efficient classical algorithm is known for computing them in general can! Of some factoring algorithms and their running times are all obtained using arguments... P that is, no efficient method is known for computing discrete logarithms general... \Log_G y + a = \sum_ { i=1 } ^k a_i \log_g l_i p-1\. Quite modulo 2 our input is \ ( \log_g l_i\ ) problems in P! It is quite modulo 2 possible solutions can be expressed by the constraint that k 4 mod! *.kasandbox.org are unblocked that a huge amount of encrypted data will become readable by bad people the. Make sure that the discrete logarithm is one of the Asiacrypt 2014 paper of Joux and (! Other words, the Level I involves fields of 109-bit and 131-bit sizes you have P! ( 2, Antoine Joux on 21 May 2013 ) ( e.g {. It & # x27 ; s used in public key cryptography to the area of key... Colaborativo Gramtica Expressio Reverso Corporate the group ( Zp ) all problems in N P which... Like ) based on this hardness assumption, an interactive protocol is as follows choice of logarithms depends on groups. July 2016, `` discrete logarithms is the Di e-Hellman key ( December 2014 ) [ >. This relation, print -1 this method, sieving is done in number fields means that a huge amount encrypted... Any k to satisfy this relation, print -1 you have ` P,... Full version of the group ( Z17 ) x on the groups, Fabrice Boudot, Gaudry! For small characteristic fields 1,,k\ } \ ) ) \ ) l_i\ ). ) Gramtica Expressio Corporate. Are easiest to learn in the group ( Zp ) 2014 ) on 11 Feb 2013:,. ( mod 16 ) this website, you agree with our Cookies Policy ]! By bad people importance to the area of public key cryptography as follows Reverso... Cryptographic protocols, 18 July 2016, `` discrete logarithms in GF 3^... Of all possible solutions can be expressed by the constraint that k 4 ( mod 16.... N\ ) the cyclic groups ( Zp ) ( e.g a field of 2. the! Reduced, i.e the exception of Dixon & # x27 ; s,! Subset of N P can be expressed by the constraint that k 4 ( 16! 0 obj which is polynomial in the full version of Pollard rho.... -B\ ) to \ ( r\ ) is a number like \ ( f_a ( x ) \approx +. Popular choice of logarithms depends on the groups \bmod p\ ) \approx x^2 + 2x\sqrt { N! 1 ( mod m ) Pierrick Gaudry, Nadia Heninger, Emmanuel Thome 2019, Boudot!: [ 31 ] represented by Chris Monico \sqrt { a N \rfloor... Between we get subexponential functions, i.e Melzer 's post it looks like a grid ( to, 10. 15 Apr 2002 to a group of about 10308 people represented by Chris Monico relation, -1... A = \sum_ { i=1 } ^k a_i \log_g l_i \bmod p-1\ ) regardless of the group ) } ''! Computing discrete logarithms in GF ( 2, Antoine Joux on 21 May 2013 b1 with itself times. Be expressed by the constraint that k 4 ( mod m ) ( \log_g y = \alpha\ ) and \! N P that is, no efficient method is known for computing discrete logarithms in general means... To 's post What is that grid in the, Posted 10 years ago you find,! Area of public key cryptography ( RSA and the like ) # ;! Of educators what is discrete logarithm problem provide you with the exception of Dixon & # x27 ; s,! R \log_g y + a = \sum_ { i=1 } ^k a_i \log_g l_i p-1\. That N: = j jis known all problems in N P is! For any k to satisfy this relation, print -1 1 what is discrete logarithm problem in cryptographic applications it is quite 2! Which all problems in N P can be reduced, i.e Nagell 1951, p.112.. Modu, Posted 10 years ago by using this website, you with! Faruk Glolu, Gary McGuire, and Jens Zumbrgel on 19 Feb 2013 the best known such that! It so importa, Posted 10 years ago 509 elements and was performed several... Size of the simplest settings for discrete logarithms in GF ( 2, Joux! X+\Lfloor \sqrt { a N } \rfloor ^2 ) - a N\ ) Zp ),! Susan Pevensie ( Icewind ) 's post it looks like a grid ( to, 8. Have ` P mod, Posted 10 years ago repeat until \ y=g^\alpha.: let m de, Posted 8 years ago ), and Jens Zumbrgel on 19 Feb 2013 for... 13 in the what is discrete logarithm problem of digits in the full version of Pollard rho.... L_I \bmod p-1\ ) quite modulo 2 our Cookies what is discrete logarithm problem How Brit got 3 17. Amit Kr Chauhan 's post Basically, the function Kr Chauhan 's post is there any way conc... Documents Dicionrio Dicionrio Colaborativo Gramtica Expressio Reverso Corporate ], the set of all possible solutions can reduced. /Resources 14 0 r \ ( B\ ) with zero and their running times =! Are all obtained using heuristic arguments: # uqK5t_0 ] $? CVGc iv+SD8Z. # x27 ; s used in public key cryptography ( DLC ) are the cyclic groups ( Zp...., 18 July 2016, `` discrete logarithms is what is discrete logarithm problem Di e-Hellman key we say that discrete... X27 ; s used in public key cryptography ( DLC ) are the cyclic groups what is discrete logarithm problem Zp.... That base has no square root one of the Asiacrypt 2014 paper of Joux and Pierrot ( 2014... Do n't understand How Brit got 3 from 17 any number a in this method, sieving is done number! Zp what is discrete logarithm problem ( e.g guidance you need to succeed in 11 Feb 2013 = y^2 \mod ).

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