Talk:Euclidean algorithm

The RSA Section is Confused and Confusing

The section "Multiplicative inverses and the RSA algorithm" starts out with a long paragraph defining what a finite field is, but then in the third paragraph admits that RSA does not use finite fields, but claims that it uses rings. Actually, it doesn't really use rings either, it just uses finite abelian groups.

The Extended Euclidean Algorithm is used during RSA key generation to (1) verify that a component (call it e) of one of the keys is a member of the multiplicative group of integers modulo (p-1)·(q-1) where p and q are primes; and to (2) find the inverse of e in that group, which will be a component of the other key. See https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29#Key_generation

LaQuilla (talk) 10:31, 6 December 2014 (UTC)

I agree. This is better explained (in my opinion) in Extended Euclidean algorithm. However the problem addressed here is the problem of "modular multiplicative inverse". This is not just a problem of group theory, as addition is used by the algorithm for computing the inverse.

The section has other issues: modular inverses are computed in many other widely used cryptographic schemes, such as ECDHE, which is used in the secure connexions to Wikipedia. Also RSA is not specifically used in e-commerce, it is used is almost all secure web connections, such as (again) secure connexions to Wikipedia.

Thus the section has not only to be rewritten, but also to be renamed "Modular inverse". D.Lazard (talk) 11:38, 6 December 2014 (UTC)

There is a more straightforward way to describe the calculation of multiplicative inverses in finite fields and the methods extension to rings. LACornell (talk) 15:54, 29 July 2016 (UTC)

Invitation to discourse on alternative formulations of some sections and related algorithms.

I have ways of describing some of the applications here that are different than these relying more heavily and more transparently on abstract algebra. Also I have a factorization method algorithm that has the Euclidean Algorithm as its starting point which I think deserve a page of its own linked to this one. Can anyone speak to these things? LACornell (talk) 15:49, 29 July 2016 (UTC)

Whether something is appropriate for inclusion here depends on whether it is covered by third-party reliable sources, typically peer-reviewed publications from established mathematics journals, or textbooks on the subject published by recognized scientific publishing houses. Generally speaking, we should not include anything that is exclusively self-published or original. The same holds for whether it is appropriate to have a separate article on a topic. Sławomir Biały (talk) 17:15, 29 July 2016 (UTC)

Proof of worst case

Under the subsection "worst case" under the the section "efficiency" an argument is given that the Fibonacci numbers F_k+1 and F_k+2 are the smallest numbers that take k steps. However, the author never defines what it means for a pair to be smallest. Given the argument about bounding the number of steps that follows, it seems that the implicit definition being used is (a, b) is "smaller" than (c, d) if b < d and (presumably) if b = d a < c. However, this makes the induction argument unclear. If we are just trying to get b to be small, how do we know that we can you use the induction hypothesis on the M-1 step where b = q₁ r₀ + r₁. Couldn't we have r₀ be bigger than F_M, but b be smaller than F_M+1? — Preceding unsigned comment added by Narjul (talk • contribs) 16:18, 30 January 2017 (UTC)

"Smallest" has to be taken in its strongest meaning. That is: if the pair a > b requires n steps for the Euclidean algorithm, then we have both a ≥ F_n+2 and b ≥ F_n+1. Proof: Let a = bq + r be the first division step. As the pair q > r requires n − 1 steps, the induction hypothesis is b ≥ F_{n + 1} and r ≥ F_n. Then a = bq + r ≥ b + r ≥ F_{n + 1} + F_n = F_n+2, qed. I have edited the article for clarifying this point (and simplifying the proof). D.Lazard (talk) 17:46, 30 January 2017 (UTC)

Bibliography

Would there be any objections to moving the "Bibliography" section to a subsection of "References". This is not a major move but there is more than one reason why it would make sense.

• On Wikipedia a bibliography section is normally the first section in the appendices, usually for biographies, or otherwise "Works" per Works or publications.
• A bibliography of sources, along with a references or notes section providing text-source integrity, constitute the citations so are certainly related.
• The number of GA and FA articles that use a separate sourcing "Bibliography" section is small so it becomes an "exception".

This has nothing to do with any article rating but as "approaching" (GA) and attaining the "best articles Wikipedia has to offer" (FA) other articles are often styled accordingly. The use of "Bibliography" sections in more than one place is confusing and this would remove that aspect and show the relationship as we normally would in subsections. Otr500 (talk) 18:53, 24 July 2018 (UTC)

I see no value in complicating a simple structure, comprising two distinct sections:

1. (hundreds of) specific references, each supporting specific parts of the article, in one (References) section; and
2. a list of (more than a dozen) relevant books for further reading in another (Bibliography) section.

Your proposed alternative puts the books of the bibliography under the heading "References", which implies that they each support some specific assertions made in the article. But that's not necessarily so; and it's not the purpose of a bibliography to do that. yoyo (talk) 12:14, 6 November 2018 (UTC)

Integers: ordinary, normal, usual, real, Gaussian

In the § Gaussian integers section, we find these expressions:

• "ordinary integers"
• "normal integers".

I found these terms confusing, since

1. I don't recall seeing them used, in decades of reading maths, before this; and
2. The phrase "normal integers" occurs in close proximity to a discussion of a norm.

Perhaps both of the above phrases were meant to specify the usual or everyday integers, namely those in (or isomorphic to those in) the real numbers? In any case, the sense of the section wouldn't suffer, and accuracy would improve, if we were to replace both "ordinary integers" and "normal integers" by "real integers".

Before I make such a change, I'd like to hear other opinions.

yoyo (talk) 11:55, 6 November 2018 (UTC)

The phrase "ordinary integer" is commonly used for distinguishing usual integers from Gaussian integers, and more generally from algebraic integers. The phrase "normal integer" is less common and must changed into "ordinary integer". "Real integer" is not a good idea, because it would be WP:OR, and also because ${\sqrt {2}}$ (for example) is an algebraic integer and a real number. I'll change "normal" to "ordinary", and add an explanatory footnote after the first use of "ordinary". D.Lazard (talk) 12:35, 6 November 2018 (UTC)

I agree with D.Lazard. "Ordinary integers" is not at all confusing, "normal integers" may be slightly confusing, but "real integers" is definitely confusing. Maproom (talk) 14:28, 6 November 2018 (UTC)

Inconsistent history

The present text of the article says that the Euclidean algorithm was first described in Europe by Bachet in 1624. This can hardly be true if it was already described in Euclid's Elements, which was known in Europe in various editions and translations long before Bachet.109.149.2.98 (talk) 13:49, 7 April 2019 (UTC)

Point well taken. The source only says that Bachet gave the first numerical description of the algorithm in Europe. I'll edit this. --Bill Cherowitzo (talk) 19:02, 7 April 2019 (UTC)