Talk:Discrete logarithm

message: I think (Zp)× is {0,1, …, p − 1}, not {1, …, p − 1}
No, it isn't Z_p is {0..p-1} (additive group), but (Z_p)^× is {1..p-1}, without zero (multiplicative group -- that × in the exponent connotes multiplication). Veky 09:30, 10 January 2006 (UTC)
thread_title:  {0,1,...,p-1} 

message: Should we add a few sentences about quantum computing and Shor's efficient quantum solution of the discrete log problem? WindowMaker 22:24, 18 April 2006 (UTC)
replies: text: Sure; please do. I'm not familiar with it. If it's a modification of Shor's algorithm for factoring, perhaps it would be best treated there, and just cited here? Joshua Davis 00:34, 19 April 2006 (UTC)
thread_title:  Quantum Computing 

message: "This algorithm 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"
Other way 'round. No?
:No, I don't think so. If the size of the group is 256, then the algorithm takes (proportional to) 256 steps, so an exponential number of steps in log 256 = 8, the number of bits needed to represent 256. (I'm using base 2 for exponentials and logs.) Make sense? Joshua Davis 13:09, 1 November 2006 (UTC)
thread_title:  Typo? 

message: Hasn't it been proven that calculating discrete logarithms is Turing equivalent to integer factorization (i.e., a polynomial-time algorithm for one implies the existence of a polynomial-time algorithm to solve the other?)  —Preceding unsigned comment added by 24.254.224.181 (talk) 22:25, 4 January 2008 (UTC)
replies: text: Nevermind, I was wrong.  There are some obstacles to showing this, or at least that's what I've read. Simphilip (talk) 02:50, 5 January 2008 (UTC)

text: You might try your question at Wikipedia:Reference desk/Mathematics. Mgnbar (talk) 03:39, 19 August 2011 (UTC)
thread_title:  Integer Factorization 

message: I think the last "4" in the sentence "Since 316 ≡ 1 (mod 17) it also follows that if n is an integer then 34+16 n ≡ 13 x 1n  ≡ 4 (mod 17)." should be "13".  So the sentence should read "Since 316 ≡ 1 (mod 17) it also follows that if n is an integer then 34+16 n ≡ 13 x 1n  ≡ 13 (mod 17)."  Anyone disagree?
Greg McFarlane (talk) 05:55, 30 April 2008 (UTC)
replies: text: Agreed. It should be correct now. 85.2.113.214 (talk) 06:04, 30 April 2008 (UTC)
thread_title:  Error in example? 

message: "Computing discrete logarithms is apparently difficult. Not only is no efficient algorithm known for the worst case, but the average-case complexity can be shown to be at least as hard as the worst case using random self-reducibility."
"at least as"? It's not the worst case if the average case is worse. (VillemVillemVillem (talk) 19:34, 16 July 2010 (UTC)
thread_title:  Worst case 

message: A "too technical" tag was just placed on this article, saying that the presentation is too difficult for an elementary concept. The concept is not really elementary, in that the non-mathematical layperson will never encounter it. It's typically studied at the advanced undergraduate level, in courses in abstract algebra and cryptography. The treatment here seems appropriate to me. In fact, the article seems to do a good job of relating discrete logs to ordinary logs in the real numbers. So I vote against the tag. Mgnbar (talk) 11:53, 14 August 2013 (UTC)
replies: text: I generally agree, though the lede could be a little less jargony. Before i edit it, I noticed it refers to finite cyclic groups. That restriction can't be right or am I missing something? --agr (talk) 15:25, 14 August 2013 (UTC)
replies: text: I think that it was added in an attempt to make the logarithm always unique and well-defined. But it doesn't succeed, and infinite cyclic groups behave just as well as finite cyclic groups with respect to this issue. My vote is that we remove "finite cyclic", just defining logs in arbitrary groups, but that we also make clear that logs do not always exist and may not always be unique. Mgnbar (talk) 15:42, 14 August 2013 (UTC)
replies: text: I'll go with all the above.  I'm no expert, but is this concept not defined for any group?  (Obviously with non-uniqueness, but then this is also the case for finite groups: they are periodic).  Should we not also change "is a solution" in the lead to "is an integer solution"? After all, there are contexts in which non-integers make sense, or might seem to make sense. — Quondum 11:54, 15 August 2013 (UTC)
replies: text: I've made the changes. I've also tried to make the notation more consistent. Mgnbar (talk) 12:35, 15 August 2013 (UTC)
thread_title:  "Too technical" 

message: An improved algorithm for computing discrete logarithms was published just now (mid-May 2014).  See this article in Science Daily.  The paper itself is available here via Springer Link (subscription required).  The issue has been mentioned on Bruce Schneier's blog (see here), and it's also being talked about on Slashdot (see here), but I haven't found any reliable sources yet listing any specific crypto systems that are no longer secure as a result.  People should probably keep an eye on this and see how it develops.  — Richwales (no relation to Jimbo) 02:34, 19 May 2014 (UTC)
thread_title:  New (May 2014) algorithm and its impact on crypto? 

message: Is it worth mentioning in the article that the difficulty of the discrete logarithm problem depends on the group representation?  For example, while discrete logarithms in modular multiplication groups are hard, discrete logarithms in modular addition groups are easy—calculating such discrete logarithms is simply the Extended Euclidean GCD algorithm. -- Myria (talk) 22:35, 10 December 2015 (UTC)
replies: text: Those are different groups. You can argue that they are isomorphic to each other, but then the computational difficulty shifts to the computation of the isomorphism. Traditionally this article has not dealt with the easy cases, because the hard cases are more mathematically interesting and cryptographically useful. But I would not be opposed to a section on "Easy special cases" or something like that. Mgnbar (talk) 16:48, 11 December 2015 (UTC)

text: Today's edit to the Algorithms section attempts to address this complaint. Regards. Mgnbar (talk) 16:39, 22 January 2016 (UTC)
thread_title:  Easy in some groups 

message: The first sentence currently reads: "In mathematics, a discrete logarithm is an integer k exponent solving the equation , where b and g are elements of a finite group" (emphasis added), but an example in the first section is based on an infinte group of powers of 10. Is the first sentence wrong? Should we drop or change the word "finite"? Or is the example somewhat misleading? Chrisahn (talk) 20:19, 11 May 2017 (UTC)
replies: text: The opening sentence should not include the word "finite". Neither should the second sentence. Please remove those two occurrences. Thanks for noticing. Mgnbar (talk) 22:05, 11 May 2017 (UTC)
thread_title:  '''finite''' group in first sentence, '''infinite''' group in first example 

message: I notice that this type of logarithm lacks a notation encountered in the case of ordinary logaritms.--5.2.200.163 (talk) 16:26, 7 September 2017 (UTC)
replies: text: And what notation is that? Ordinary logarithms are denoted logb g, and so are discrete logarithms. Mgnbar (talk) 20:34, 7 September 2017 (UTC)
replies: text: Perhaps something (like *,m(od) exponent or index) for distinction from ordinary logarithms. I have opened this section due to not noticing the ordinary-like notation in the introduction of the article.--5.2.200.163 (talk) 12:59, 11 September 2017 (UTC)
replies: text: Not all discrete logarithms arise from modular arithmetic. So a notation involving "mod" does not apply in general. But I have added the notation to the introduction of the article (although the lack of uniqueness has not been discussed yet). Mgnbar (talk) 16:44, 11 September 2017 (UTC)
thread_title:  Notation (or lack of it) for discrete modular logarithm 

message: It's long overdue, but let's list what some reliable sources say about discrete logarithms.
* Graham Ellis, "Rings and Fields": Defines the problem in GF(p^k)^*. Does concrete example in GF(2^4)^*. Describes difficulty of general problem and cryptographic applications (p. 156).
* Neal Koblitz, "A Course in Number Theory and Cryptography", second edition: Draws an analogy with real logarithms. Defines the problem in an arbitrary finite group. Says, "the word 'discrete' distinguishes the finite group situation from the classical (continuous) situation". Mentions (Z/nZ)^* (n not specified) and GF(p^k)^* as examples. Does example in (Z/19Z)^*. Does example in GF(3^2)^*. Describes cryptographic applications. Discusses algorithms for GF(p^k)^* (p. 97-106). Exercises show that discrete logs help expedite computations in finite fields (p. 106-107) and that there is a polynomial time algorithm for (Z/pZ)^* where p is a Fermat prime (p. 109). Later, defines the problem in the group of points on an elliptic curve over GF(p^k). The group operation is written additively, so powers are written as multiples and logarithms look like divisions. Describes cryptographic algorithms (p. 180-185).
* Kenneth H. Rosen, "Elementary Number Theory and its Applications", third edition: Defines the problem in (Z/pZ)^*, where p is prime. Discusses algorithmic difficulty and cryptographic applications (p. 255).
* Adleman and Huang, "Function field sieve method for discrete logarithms over finite fields", Information and Computation 151 (1999): Defines the problem in GF(p^k)^*. Mentions intense research and cryptographic relevance. Relates discrete log algorithms to factoring algorithms. Discusses difficulty.
* MathWorld, Discrete logarithm article: Defines the problem in (Z/nZ)^*, where n is not necessarily prime. Gives example with n = 41. Discusses some alternative terminology and one television appearance.
In the near future I intend to use these references as a guide to revising the article in a better-cited way. Mgnbar (talk) 22:55, 28 October 2017 (UTC)
replies: text: Maybe it's only my, rare to find elsewhere, personal opinion that it is a bit far fetched to report the solitary, parenthesized remark by Koblitz [here "log" has a different but analogous meaning than before], immediately referring to the notation y=logbx, as "Koblitz drawing an analogy to real logarithm". In any case, I think Koblitz mostly works in fields, extending results to groups, and this sheds some light on earlier reservations of mine. As usual, I just want to document my opinion. Purgy (talk) 08:09, 29 October 2017 (UTC)
replies: text: That parenthetical remark is not solitary. It's part of an entire paragraph of analogy: "When working with the real numbers, exponentiation (finding b^x to a prescribed accuracy) is not significantly easier than the inverse operation (finding log_b x to a prescribed accuracy). But now suppose that we have a finite group...One can compute b^x for large x rather rapidly...But...how can we compute x = log_b y (where here 'log' has a different but analogous meaning than before)?" Mgnbar (talk) 12:11, 29 October 2017 (UTC)
replies: text: Haven't I pointed to the possibility of me being solitary in my opinion? Certainly you have cited the one and only (solitary?) occurrence in a perfectly reliable source for the notations of discrete and real logarithms being analogous. I apologize, for always having had their semantics, and not the syntax of "log" in mind. BTW, the cited paragraph seems to emphasize the most relevant disanalogy of the notions, and I could not make me aware of any other remarks on logarithmic analogy. I think I better remove this page from my watchlist. Purgy (talk) 16:28, 29 October 2017 (UTC)
thread_title:  Reliable sources