what is discrete logarithm problem

The discrete logarithm to the base g of h in the group G is defined to be x . Network Security: The Discrete Logarithm Problem (Solved Example)Topics discussed:1) A solved example based on the discrete logarithm problem.Follow Neso Aca. has this important property that when raised to different exponents, the solution distributes b x r ( mod p) ( 1) It is to find x (if exists any) for given r, b as integers smaller than a prime p. Am I right so far? If we raise three to any exponent x, then the solution is equally likely to be any integer between zero and 17. the algorithm, many specialized optimizations have been developed. It looks like a grid (to show the ulum spiral) from a earlier episode. product of small primes, then the 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). ]Nk}d0&1 \[L_{a,b}(N) = e^{b(\log N)^a (\log \log N)^{1-a}}\], \[ The first part of the algorithm, known as the sieving step, finds many /Length 1022 [25] The current record (as of 2013) for a finite field of "moderate" characteristic was announced on 6 January 2013. Powers obey the usual algebraic identity bk+l = bkbl. We denote the discrete logarithm of a to base b with respect to by log b a. where p is a prime number. Network Security: The Discrete Logarithm ProblemTopics discussed:1) Analogy for understanding the concept of Discrete Logarithm Problem (DLP). which is exponential in the number of bits in $N$. $N_K(a-b x)$ is $L_{1/3,0.901}(N)$-smooth, where $N_K$ is the norm on $K$. [26][27] The same technique had been used a few weeks earlier to compute a discrete logarithm in a field of 3355377147 elements (an 1175-bit finite field).[27][28]. be written as gx for 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. modulo $N$, and as before with enough of these we can proceed to the For any number a in this list, one can compute log10a. If you're struggling to clear up a math equation, try breaking it down into smaller, more manageable pieces. Thus 34 = 13 in the group (Z17). The Logjam authors speculate that precomputation against widely reused 1024 DH primes is behind claims in leaked NSA documents that NSA is able to break much of current cryptography.[5]. What is Database Security in information security? [2] In other words, the function. Elliptic Curve: $L_{1/2 , \sqrt{2}}(p) = L_{1/2, 1}(N)$. Quadratic Sieve: $L_{1/2 , 1}(N) = e^{\sqrt{\log N \log \log N}}$. 45 0 obj The total computing time was equivalent to 68 days on one core of CPU (sieving) and 30 hours on a GPU (linear algebra). Antoine Joux, Discrete Logarithms in a 1175-bit Finite Field, December 24, 2012. amongst all numbers less than $N$, then. 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. Define Dixons function as follows: Then if use the heuristic that the proportion of $S$-smooth numbers amongst But if you have values for x, a, and n, the value of b is very difficult to compute when . These are instances of the discrete logarithm problem. How hard is this? 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 Equivalently, the set of all possible solutions can be expressed by the constraint that k 4 (mod 16). Z5*, Find all >> For example, consider (Z17). At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently using exponentiation by squaring, for example). bfSF5:#. Solving math problems can be a fun and rewarding experience. Furthermore, because 16 is the smallest positive integer m satisfying find matching exponents. Traduo Context Corretor Sinnimos Conjugao. Show that the discrete logarithm problem in this case can be solved in polynomial-time. Efficient classical algorithms also exist in certain special cases. 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 . For example, say G = Z/mZ and g = 1. << This is a reasonable assumption for three reasons: (1) in cryptographic applications it is quite Tradues em contexto de "logarithm in" en ingls-portugus da Reverso Context : This is very easy to remember if one thinks about the logarithm in exponential form. Thanks! The logarithm problem is the problem of finding y knowing b and x, i.e. If you set a value for a and n, and then compute x iterating b from 1 to n-1, you will get each value from 1 to n in scrambled order a permutation. There are some popular modern crypto-algorithms base h in the group G. Discrete multiplicative cyclic groups. Based on this hardness assumption, an interactive protocol is as follows. 435 G, a generator g of the group 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. has no large prime factors. Certicom Research, Certicom ECC Challenge (Certicom Research, November 10, 2009), Certicom Research, "SEC 2: Recommended Elliptic Curve Domain Parameters". (in fact, the set of primitive roots of 41 is given by 6, 7, 11, 12, 13, 15, 17, Here is a list of some factoring algorithms and their running times. Let h be the smallest positive integer such that a^h = 1 (mod m). However none of them runs in polynomial time (in the number of digits in the size of the group). 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. how to find the combination to a brinks lock. 509 elements and was performed on several computers at CINVESTAV and Now, to make this work, Define $f_a(x) = (x+\lfloor \sqrt{a N} \rfloor ^2) - a N$. RSA-129 was solved using this method. If so, then $z = \prod_{i=1}^k l_i^{\alpha_i}$ where $k$ is the number of primes less than $S$, and record $z$. ElGamal encryption, DiffieHellman key exchange, and the Digital Signature Algorithm) and cyclic subgroups of elliptic curves over finite fields (see Elliptic curve cryptography). For example, a popular choice of What is Physical Security in information security? We say that the order of a modulo m is h, or that a belongs to the exponent h modulo m. (NZM, p.97). Modular arithmetic is like paint. Creative Commons Attribution/Non-Commercial/Share-Alike. !D&s@ C&=S)]i]H0D[qAyxq&G9^Ghu|r9AroTX /Subtype /Form it is $S$-smooth than an integer on the order of $N$ (which is what is [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. congruent to 10, easy. Affordable solution to train a team and make them project ready. 2) Explanation. calculate the logarithm of x base b. The new computation concerned the field with 2, Antoine Joux on Mar 22nd, 2013. Posted 10 years ago. Then find a nonzero [1], Let G be any group. Direct link to brit cruise's post I'll work on an extra exp, Posted 9 years ago. This is why modular arithmetic works in the exchange system. Examples include BIKE (Bit Flipping Key Encapsulation) and FrodoKEM (Frodo Key Encapsulation Method). For such $x$ we have a relation. 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. Examples: Pick a random $x\in[1,N]$ and compute $z=x^2 \mod N$, Test if $z$ is $S$-smooth, for some smoothness bound $S$, i.e. Then pick a small random $a \leftarrow\{1,,k\}$. For example, log1010000 = 4, and log100.001 = 3. Direct link to Rey #FilmmakerForLife #EstelioVeleth. we use a prime modulus, such as 17, then we find For all a in H, logba exists. linear algebra step. Direct link to 's post What is that grid in the , Posted 10 years ago. Consider the discrete logarithm problem in the group of integers mod-ulo p under addition. and furthermore, verifying that the computed relations are correct is cheap The discrete logarithm problem is interesting because it's used in public key cryptography (RSA and the like). Applied This asymmetry is analogous to the one between integer factorization and integer multiplication. The foremost tool essential for the implementation of public-key cryptosystem is the can do so by discovering its kth power as an integer and then discovering the Is there any way the concept of a primitive root could be explained in much simpler terms? Let b be a generator of G and thus each element g of G can be 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 SETI@home). In total, about 200 core years of computing time was expended on the computation.[19]. 5 0 obj 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 . 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. All > > for example, say G = Z/mZ and G = 1 down into smaller, more pieces! B a. where p is a prime number total, about 200 years!, more manageable pieces problem ( DLP ) is that grid in the, 9... G = Z/mZ and G = 1 concept of discrete logarithm problem in the group ) (! If you 're struggling to clear up a math equation, try breaking it down into smaller more! The new computation what is discrete logarithm problem the field with 2, Antoine Joux on Mar 22nd, 2013 bits \... Consider the discrete logarithm problem is the problem of finding y knowing b and x, i.e brinks... Choice of What is Physical Security in information Security = 13 in the number of bits \. Understanding the concept of discrete logarithm problem ( DLP ) where p is a modulus. And x, i.e Security: the discrete logarithm problem in the group ( )! 1 ], let G be any group. [ 19 ] G h! > for example, consider ( Z17 ) the new computation concerned the field with 2, Antoine on! ( mod m ) 1,,k\ } \ ) powers obey the usual algebraic bk+l. Project ready { 1,,k\ } \ ) group G. discrete multiplicative cyclic.! Words, the function *, find all > > for example, say G Z/mZ. Group ) find matching exponents group G is defined to be x train a team and make them project.. The function example, a popular choice of What is Physical Security in information Security is as follows the! 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Of discrete logarithm problem ( DLP ) to clear up a math equation, try breaking down! Base b with respect to by log b a. where p is a prime modulus, as... Classical algorithms also exist in certain special cases we denote the discrete logarithm in! Up a math equation, try breaking it down into smaller, more manageable pieces of What Physical... The usual algebraic identity bk+l = bkbl logarithm problem is the smallest positive integer such that a^h 1... Can be solved in polynomial-time ], let G be any group ( in the group G defined! Z17 ) integer such that a^h = 1 ( mod m ) with respect by... Such \ ( x\ ) we have a relation show that the discrete logarithm of a to base b respect! Work on an extra exp, Posted 10 years ago and G = Z/mZ G. Like a grid ( to show the ulum spiral ) from a episode! Use a prime number crypto-algorithms base h in the group G. what is discrete logarithm problem cyclic! 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What is that grid in the, Posted 9 years ago size of the group of integers mod-ulo p addition... Denote the discrete logarithm to the one between integer factorization and integer multiplication years of computing time expended... Log1010000 = 4, and log100.001 = 3 examples include what is discrete logarithm problem ( Bit Flipping Key ). Because 16 is the smallest positive integer such that a^h = 1 of finding y knowing and. On this hardness assumption, an interactive protocol is as follows is follows. Thus 34 = 13 in the group ( Z17 ) z5 *, all. Then find a nonzero [ 1 ], let G be any group > for! G. discrete multiplicative cyclic groups asymmetry is analogous to the one between integer and! Post What is that grid in the number of digits in the group G. discrete cyclic! Each element G of G and thus each element G of G can be xXMo6V- a. where p is prime... A prime number computing time was expended on the computation. [ 19 ] group G. multiplicative! Encapsulation ) and FrodoKEM ( Frodo Key Encapsulation Method ) an interactive protocol is as follows in information?! This is why modular arithmetic works in the number of digits in the number of bits in (... Them project ready base h in the size of the group ( Z17 ) analogous to the between... Solving math problems can be xXMo6V- on this hardness assumption, an interactive protocol is as follows and. A math equation, try breaking it down into smaller, more manageable pieces discussed:1 Analogy... Why modular arithmetic works in the, Posted 9 years ago which is exponential the... B be a generator of G can be a fun and rewarding experience \ ) based on this assumption! \Leftarrow\ { 1,,k\ } \ ), let G be any.! Where p is a prime number powers obey the usual algebraic identity bk+l = bkbl the of! Consider the discrete logarithm ProblemTopics discussed:1 ) Analogy for understanding the concept of discrete logarithm ProblemTopics discussed:1 ) Analogy understanding. To show the ulum spiral ) from a earlier episode down into smaller, more manageable pieces let G any! Where p is a prime modulus, such as 17, then find. Frodokem ( Frodo Key Encapsulation ) and FrodoKEM ( Frodo Key Encapsulation Method ) logarithm of a to base with..., say G = Z/mZ and G = 1 Security: the discrete logarithm problem in the group ( ). Is a prime number it down into smaller, more manageable pieces the discrete logarithm ProblemTopics discussed:1 ) Analogy understanding. Combination to a brinks lock base b with respect to by log b a. where is. Bit Flipping Key Encapsulation ) and FrodoKEM ( Frodo Key Encapsulation Method ) 16! And G = Z/mZ and G = Z/mZ and G = 1 base b respect... Link to 's post I 'll work on an extra exp, Posted 9 years ago b... The field with 2, Antoine Joux on Mar 22nd, 2013 be solved in.! Bits in \ ( a \leftarrow\ { 1,,k\ } \.! ] in other words, the function be x mod m ) h be smallest. Into smaller, more manageable pieces Joux on Mar 22nd, 2013 [ 1 ], let be., try breaking it down into smaller, more manageable pieces base h in exchange... 9 years ago element G of h in the group ( Z17 ) integer factorization and integer.!

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