Talk:Modular multiplicative inverse

early thread
Modified the New examples to show why the repeating numbers terminated the computation

the old explanation is here, which doesn't explain why we terminate

the old examples also use inconsistent notation for the repeating case

assume we want to get x-1 we can do this as following.

x-1 = x * x-2

and

x-2 = x2 * x-4

we continue this expansion until it terminates with either a repeated number, or a one.

NeilUK 20:30, 20 June 2007 (UTC)

New Modular Multiplicative Inverse Algorithm
I'm going to remove this algorithm, because it is badly described and significantly slower than the extended Euclidean algorithm and the modular exponentiation method (even if we have to factor the modulus first). Furthermore it appears to be original research. 85.2.78.131 08:21, 17 July 2007 (UTC)

PHP Script
I think it should be better to use pseudo-code or a more general-purpose language, like C. Squalho (talk) 16:55, 30 January 2009 (UTC)

I'm in complete agreement. The PHP script is absolutely unreadable and seemingly useless. I will work on rewriting. 65.43.176.204 (talk) 20:47, 7 February 2009 (UTC)

Newton's method
According to Hacker's Delight Newton's algorithm, $$x_{n+1} = x_n(2-d\,x_n)$$, apparently also works in modular arithmetic, for powers-of-two anyway. Perhaps this ought to be mentioned? At very least this results in a short and sweet (not to mention efficient) implementation if one is needed for the article. --78.110.85.150 (talk) 23:02, 29 April 2009 (UTC)

Perhaps there should be a mention of negative values for the Modular multiplicative inverse?
Correct me if I'm wrong but doesn't the fact that the solution of Bezout's Identity ax + by = 1 require that either x or y to be negative? If so then this means that its Modular multiplicative inverse is not 1 but - (a - 1) or - (b - 1). Perhaps a small mention of this might be helpful? —Preceding unsigned comment added by 122.106.89.151 (talk) 07:37, 19 March 2010 (UTC)

If a is the modulus then the product of the number to invert and its inverse is -(a - 1) which is equal to -a + 1. However -a + 1 = 1 (mod a). Basically, you forgot we're working over a modulus. —Preceding unsigned comment added by Mindvirus (talk • contribs) 21:21, 5 May 2010 (UTC)


 * I think the last line "return x % m" should read "return (x+m) % m", since in most common programming languages, a modulus of a negative number is negative. --Erel Segal (talk) 14:37, 7 November 2010 (UTC)
 * Not in Python. -- X7q (talk) 15:52, 7 November 2010 (UTC)

Fermat's little theorem algorithm
I am removing this. It is a specific case of direct modular exponentiation by Euler's theorem, already described. —Preceding unsigned comment added by 164.107.227.117 (talk) 21:08, 5 May 2010 (UTC)

What kind of notation is this?
Hi!

The article can be improved by explaining where this notation
 * $$a^{-1} \equiv x \pmod{m}$$         (Equation 1)

comes from? It's supremely horrible and confusing.

Consider this: by looking at this we learn that

if 
 * $$y \equiv x \pmod{m}$$                              (Equation 2)

then there exists an integer k, such that
 * $$y - x \equiv k \pmod{m}$$                               (Equation 3)

So now comparing the first two equations we see that
 * $$y = a^{-1}$$                                                          (Equation 4)

which means that Equation 1 can be converted to the form of Equation 3:
 * $$a^{-1} - x \equiv k \pmod{m}$$         in other words         $$\frac{1}{a} - x  \equiv k \pmod{m}$$                     (Equation 5)

Thus by following standard notation, I have shown that if Equation 1 is true, then Equation 5 must also be true (one can find an integer k such that Equation 5 is true). But of course it is not, since the left-hand side of Equation 5 cannot be an integer (if e.g. a = 5).

So my question: where the hell does the notation used in Equation 1 come from? This definitely needs to be improved in the article, because currently it's just confusing. I'd like to know: who first used that notation? What's its history? Are there any other notations?

Because I'd never use Equation 1, but instead write:
 * $$a x \equiv 1 \pmod{m}$$         (Equation 6)

Math;singular (talk) 20:02, 18 January 2011 (UTC)

What the heck!??
Can someone explain this to me?

I can't get it to turn out the way I'm expecting...

From RSA:
 * 1) Choose two distinct prime numbers, such as: $$p = 61$$ and $$q=53$$. I get this.
 * 2) Compute $$n = p q$$ giving: n = 61 &middot; 53 = 3233. I get this.
 * 3) Compute the totient of the product as $$\phi(n) = (p-1)(q-1)$$ giving: $$\phi(3233) = (61 - 1)(53 - 1) = 3120$$. I get this.
 * 4) Choose any number $$1 < e < 3120$$ that is coprime to 3120. Choosing a prime number for $$e$$ leaves us only to check that $$e$$ is not a divisor of 3120. Let $$e=17$$. I get this.
 * 5) Compute $$d$$, the modular multiplicative inverse of $$e\text{ (mod }\varphi(n)\text{)}$$ yielding: $$d=2753$$. HALP! I don't get this!

I get everything up until "modular multiplicative inverse".

d obviously does not equal, so... What am I missing here? – Ajltalk 07:40, 25 March 2011 (UTC)


 * $$d \times e = 2753 \times 17 = 46801 = 15 \times 3120 + 1$$
 * $$\Rightarrow 2753 \times 17 \equiv 1 \text { mod } 3120$$ Gandalf61 (talk) 15:18, 25 March 2011 (UTC)


 * I must be a complete idiot then, because that still makes no sense at all. I don't know what $$d$$ is!  I only know what it is because someone told me (the RSA article). I guess I should have worded my question better: How do you find the value of $$d$$? – Ajltalk 15:31, 25 March 2011 (UTC)


 * Well, you can use trial and error - you know d has to be less than 3120, so there are a limited number of possibilities. Or you can use the extended Euclidean algorithm, which goes something like this:
 * $$3120 - 183 \times 17 = 9$$
 * $$2*9 - 17 = 1$$
 * $$\Rightarrow 2\times3120 - 2\times 183 \times 17 = 2 \times 9 = 17 + 1$$
 * $$\Rightarrow 2\times3120 - (2\times 183+1) \times 17 = 1$$
 * $$\Rightarrow 2\times3120 - 367 \times 17 = 1$$
 * $$\Rightarrow (2-17)\times3120 + (3120-367) \times 17 = 1$$
 * $$\Rightarrow -15\times3120 + 2753 \times 17 = 1$$
 * $$\Rightarrow 2753 \times 17 = 15\times3120 + 1$$
 * Gandalf61 (talk) 09:42, 26 March 2011 (UTC)

Efficiency of exponentiation under Euler's theorem
I updated the Euler's theorem section to say


 * exponentiation. Though it can be implemented more efficiently using modular exponentiation, an efficient implementation of this operation itself requires the calculation of an inverse mod m in the first place (...)

This was to clarify a previous edit which had been reverted by User:X7q

A read of the modular exponentiation article confirms that the efficient method for modular exponentiation, namely the Montgomery method, does use a multiplicative inverse. Since this section is about the efficiency of the Euler method to calculate the inverse, I believe it is justified to use the most efficient method as the basis for the discussion. But maybe there's an even better way to explain it, I'm certainly open to discuss it.

Even if it isn't, the page before my updates did contain a mistake which I corrected (it said that modular exponentiation is a form of binary exponentiation, which is clearly not true since binary exponentiation is simply one of the methods that can be used to calculate an exponent). 83.183.40.68 (talk) 11:33, 10 July 2011 (UTC)


 * Modular exponentiation can be efficiently performed using binary exponentiation algorithm, which doesn't require modular inverses. So I still think that your edit is wrong. -- X7q (talk) 11:38, 10 July 2011 (UTC)


 * Binary exponentiation without Montgomery multiplication is certainly not efficient when big numbers are involved. It requires a lengthy division at every step in order to reduce the intermediate results mod m. With the Montgomery method the only divisions required are by powers of 2 which are very efficient on any binary computer (and it can be adapted to other bases too I believe). 83.183.40.68 (talk) 11:44, 10 July 2011 (UTC)


 * Your edit made it sound like modular inversion is a required operation in modular exponention, which it isn't. Multiplication can be done without Montgomery reductions just fine, although, yes, it could be slower. -- X7q (talk) 12:01, 10 July 2011 (UTC)


 * It's required for efficiency when a big value of m is involved. But I definitely see your point, so I think the best solution would be to update the page to say this objection applies for big values of m (which is when Montgomery reduction becomes the efficient solution, and normal binary exponentiation is inefficient). Do you agree or do you have another idea? 83.183.40.68 (talk) 12:04, 10 July 2011 (UTC)


 * Agree. Please mention that implementations of modular exponentiation for large moduli typically use Montgomery reductions for efficiency, and that Montgomery reductions can not be used in this case because they require modular inverses. -- X7q (talk) 12:09, 10 July 2011 (UTC)


 * Done. I think the page is definitely better after our discussion, but I'm sure my phrasing is not optimal so feel free to improve it if needed. Thanks for the discussion! 83.183.40.68 (talk) 12:27, 10 July 2011 (UTC)


 * I agree, it looks much better now. Thank you! -- X7q (talk) 12:37, 10 July 2011 (UTC)

Why remove?

 * Example.
 * $$17^{-1} \equiv x \pmod{3120},$$
 * $$17 x \equiv 1 \pmod{3120},$$
 * $$46801 \equiv 1 \pmod{3120},$$
 * $$x=2753,$$
 * $$17\cdot 2753-15\cdot 3120=46801-46800=1.$$ — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 21:38, 11 March 2012 (UTC)
 * There is already an example in the article, much simpler, with smaller numbers and more understandable. I don't think we need another one. It doesn't illustrate anything. -- X7q (talk) 21:52, 11 March 2012 (UTC)
 * BTW, does anybody know how to find x? It seems pretty hard. — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 21:40, 11 March 2012 (UTC)
 * The Computation section describes two methods. -- X7q (talk) 21:52, 11 March 2012 (UTC)
 * OK, I get it,
 * $$9 \equiv 17 \pmod{3120},$$
 * $$(3120-9)/17=[3120-(17 \pmod{3120})]/17=183,$$
 * $$3120-2\cdot 183-1=2753,$$
 * $$x=3120-1-2\cdot [3120-(17 \pmod{3120})]/17=2753.$$ — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 22:13, 11 March 2012 (UTC)
 * On scientific MS Windows calculator need push first "3120", then "Mod", then "17" and you get result "9". So does it correct to write $$9 \equiv 17 \pmod{3120}$$? Maybe need to write $$9 \equiv 3120 \pmod{17}$$? — Preceding unsigned comment added by Versatranitsonlywaytofly (talk • contribs) 22:33, 11 March 2012 (UTC)
 * $$9 \equiv 17 \pmod{3120}$$ is a false statement. $$9 \equiv 3120 \pmod{17}$$ is a true statement. -- X7q (talk) 22:43, 11 March 2012 (UTC)
 * Then need rewrite all "MODed" wikipedia, because here also as you claiming https://netfiles.uiuc.edu/c-blair/www/papers/second.pdf . In this papper even written 7^123 Mod 19 = 1 as a=7 and a^123 = 1 (mod 19). Or also 1 = 11^123 Mod 19 as $$11^{123} \equiv 1 \pmod{19}.$$ So it written as anybody likes. I think after all, that on calculator you must first enter 3120, then 17, then you get 9, but to write you obviously need $$17 \equiv 9 \pmod{3120},$$ because thats how written MOD wikipedia article.
 * Nobody needs to rewrite anything. $$a \equiv b \pmod{m}$$ is a standard (i.e. textbook) math notation, with a simple and very precise meaning: a - b is divisible by m. Nothing else. Like it or not, that's how math is written, deal with it. Let me guess, you probably also wouldn't like how integrals are written, and be shocked discover that dy/dx isn't in fact a fraction :) -- X7q (talk) 09:53, 6 April 2012 (UTC)
 * Regarding 7^123 Mod 19 = 1 - in this example it appears that "Mod" denotes the operation of taking a remainder. Not very widespread notation, but I've seen it. Note that it isn't the same as the other notation. For example 7^123 Mod 19 = 20 is false (left hand side evaluates to 1, but RHS is 20). While $$7^{123} \equiv 20 \pmod{19}$$ denotes the fact that $$7^{123} - 20$$ is divisible by 19, and is a true statement. -- X7q (talk) 09:59, 6 April 2012 (UTC)
 * If you are confused by the notation, you definitely should read a more basic Modular arithmetic article first. -- X7q (talk) 22:47, 11 March 2012 (UTC)

Please do not delete anything (even junk) from talk pages
http://en.wikipedia.org/wiki/Wikipedia:Talk_page_guidelines#Others.27_comments — Preceding unsigned comment added by 60.241.171.231 (talk) 19:33, 8 April 2012 (UTC)

Example wording
First its stated that "and there are no other values of x in \mathbb{Z}_{11} that satisfy this congruence" then, a line later, "Once we have found the inverse of 3 in \mathbb{Z}_{11}, we can find other values of x in \mathbb{Z} that also satisfy the congruence". Isn't that contradictory? — Preceding unsigned comment added by 178.38.79.80 (talk) 13:48, 28 October 2012 (UTC)


 * $$\mathbb{Z}_{11}$$ is the finite group consisting of the 11 equivalence classes of integers modulo 11. Of these 11 equivalence classes, only one satisfies the equation $$3x \equiv 1 \pmod{11}$$, and that is the equivalence class that contains 4. In the set of integers $$\mathbb{Z}$$, this equivalence class corresponds to the infinite set of integers that are congruent to 4 mod 11 i.e. ..., -18, -7, 4, 15, 26, ... So there is no contradiction - the context for the two statements is just different. Gandalf61 (talk) 09:13, 29 October 2012 (UTC)

Messy and terribly frustrating article
The example and computation sections just make the whole concept impossible to understand: 1) What the heck is the "=" sign with 3 bars? 2) What is Z(11)? 3) Why are these equations bitmaps instead of Unicode text? 4) Why don't any of these have a wikipedia hyperlink? 5) Most visitors land in this page because of the RSA article. Why is there no detailed example for 17 (mod 3120)? I tried other numbers than 17 and 3120, to no avail. Searched the whole web, not a single understandable example either. — Preceding unsigned comment added by Chimel31 (talk • contribs) 20:32, 14 February 2013 (UTC)

Explanation of mathematical notation is missing - making the whole article almost useless
The article starts with a term, which has the three-horizontal-lines mathematical sign, which is probably not known to a majority of people, ever to bump into this article. If a reader is familiar with this notation and principles, why would they ever ask Wikipedia? Sorry, Wikipedia is not a encyclopedia for people who already know everything, but for people curious finding out new things. If someone could make this article better? Reference to other existing articles regarding the notation might also be very useful for the beginning. Thank You! — Preceding unsigned comment added by 129.13.72.198 (talk) 12:15, 10 February 2014 (UTC)

why not even more complicated ? not a good article [sarcasm]
Articles in Wiki should basically not just address the already well informed but also the interested in a certain subjects. This article completely fails in that regard. The "three line" math sign is not widely known, therefore it is a hard to understand just that. Let's say you want to find the modular multiplicative inverse i of b in regard to mod n. (i is just a variable and not the mathematical sign for imaginary numbers!!!) Does this not just mean the following equation is correct: (b * i) mod n = 1 ??? This is basically what the result should be?! I solved it with a simple spreadsheet. It is simple. It is even primitive. Not a big deal! I am not a math genius, but I wanted to understand it exactly and this article does not provide the least help. Sometimes articles are just frustrating. Sorry! — Preceding unsigned comment added by 79.227.151.37 (talk • contribs) 00:19, 14 February 2014 (UTC) I agree completely. This article fails the naive reader. — Preceding unsigned comment added by 67.170.225.229 (talk) 06:26, 17 April 2023 (UTC)

Recent activity
While I applaud the new editor 's zeal in editing this page, I have been forced to revert some of these edits on more than one occasion. It takes a while for a new editor to "learn the ropes" and many of the problems I have seen involve not being aware of the style guidelines in Wikipedia. When editors disagree on changes, these need to be discussed on the appropriate talk page until consensus is reached. Details of this procedure can be found at WP:BRD. Writing good mathematical articles in Wikipedia involves several skills; the ability to write clear and precise mathematics, the ability to organize material so that a general reader can be gradually led through an advanced topic, the ability to find, judge and clearly indicate good sources for readers to follow up and some familiarity with the topic in order to provide context and judge what is important. As Wikipedia is a collaborative effort, we learn many of these skills from other editors. My most recent revert removed comments that needed to be discussed on this talk page and should not appear in the article as well as several stylistic problems. I am hoping that this serves to start the dialogue. --Bill Cherowitzo (talk) 06:44, 13 January 2017 (UTC)


 * Yes, I like to see the enthusiasm. I can also remember the frustration of learning that others often disagree with me, and hope that this will not be discouraging to a new editor.  I don't entirely agree with the new edits, but am also not saying that no editing is appropriate.  A simple example is title case, which we do not use here.  —Quondum 07:06, 13 January 2017 (UTC)

Hi there ! This is. Please do not worry about any disagreement/reverting my edits that I did that were editorial. I did not know about talk pages until now, figured that there was something like this and someone would point me in the right direction. I could only get to your personal page, and it did not seem appropriate to comment there. Talk pages are a great feature, but inline editing would be pretty cool! : )

(1) As I mentioned earlier, this article needs to have a solved example for each of the algorithms listed. (2) The reference to computation runtime lacks the appropriate context, meaning it needs to have references to the big O notation in addition to the citation and it needs to tie into the example more clearly. (3) The references of how the concept relates to ring theory and group theory are secondary to the definition, solved examples, and the algorithms used for solving. While I agree that these connections exist, they were not clear last time I read the article. (4) The notation in the article needs to be consistent from start to finish. If one references an external equation, then I think the equation needs to be re-written such that it is consistent with the variables in the article.

Thanks for being patient while I figure out all the various Wiki features.PinkAppleFlower (talk) 17:03, 13 January 2017 (UTC)


 * You've certainly picked up on some things (e.g. signature, edit comments) very quickly . Just a few comments from my side in response:
 * On point (3): This assumes a specific teaching context, not appropriate here. As a mathematical article, definition and membership of mathematical categories (rings, etc.) rank above examples and algorithms.  The latter are more like addenda to the mathematical presentation.  You'll get a feel for the style as you work with multiple articles.
 * On point (4): Notation should be consistent, including with general usage where feasible. Also thought and clarity must be given to whether a variable represents an equivalence class or an integer representing an equivalence class (can vary per variable).  At the moment I find the notation not perfectly consistent.
 * Also, the lead is too cryptic (a reader should not have to follow a link such as modular arithmetic to determine what a congruence class is, and indeed what the congruence relation is in the lead). —Quondum 19:33, 13 January 2017 (UTC)

,I agree with you on your elaboration of point (4) yet, on (3) I disagree with you about the importance of algorithms and solved examples especially on this page, and I am not sure I have a strong opinion for the general case. My reason for disagreeing is that the solution, has many applications and is also a fundamental result. In general, I think 'additive inverses', 'multiplicative inverses' 'multiplicative identities' and etc are topics that are fundamental and have many applications. So, for this page in particular I still think we need to emphasize how to find the solution. I think it is important to include links to the other areas of mathematics that use the concept, but I think the most general audience for this piece are people who need to figure out how to use the multiplicative inverse and what it is. I will think about your last point, but in order to be correct, and consistent in the article, I think we need to use the language that is correct even if it is technical, I will try to make the introduction less reliant on external sources. I will update the example section soon if someone has not done so all ready.


 * Thanks for replying . I fully agree with the points that has made and would at this point just like to add two general remarks. The first is that we are not writing a textbook but rather an encyclopedia (see NOTTEXTBOOK). This can be hard to come to grips with. I know that I have a tendency in my own edits to want to explain everything, provide lots of examples and emphasize important points ... just like a textbook does. This urge has to be controlled; encyclopedic writing is not like textbook writing and I constantly have to remind myself of that. The second point concerns how much freedom we have as editors to express things in new and novel ways for the benefit of our readers. The answer–and this is a bitter pill to swallow–is almost none. As editors we report on what is in the literature, we do not make the stuff up (see Original research). Wikipedia's definition of original research is very broad and not allowed. This of course runs counter to our creative natures, but I understand how important it is to the Wikipedia enterprise to have this policy. I am looking forward to working with you to improve this article, and possibly others. --Bill Cherowitzo  (talk) 21:41, 13 January 2017 (UTC)

introduction
I recently made several changes to one paragraph of the introduction and wish to explain my reasons for doing so in more detail here. The text I edited was It has practical applications to the field of cryptography, i.e. public-key cryptography and the RSA Algorithm, which uses modulo mathematics. It is also a fundamental concept for several mathematical areas of study because it is a member of the category of Inverse Problems and definitions for the additive or multiplicative inverses are important. It is also true that the properties of binary operators are important for these areas of mathematics. Two examples are the associative property and the commutative property. Some such mathematical areas of study are number theory and two branches of abstract algebra, group theory and ring theory. To see an application of this concept to an area of mathematical study look at this page on "multiplicative group of integers modulo n". In the first sentence the link to module is incorrect, that is quite a different topic. In the second sentence there are two problems with using category. First of all it has a technical mathematical meaning (Category theory) and would confuse anyone who was familiar with that term, and secondly, the use of categories here on Wikipedia is an aid to sorting and navigating through pages. It is not definitive and can not be used as a source. Next we have two uses of the word "important". We can not just say that something is important, we have to cite someone who is providing reasons why something is important. The next discussion of binary operators is just not germane to the topic, terms are being thrown around which have no direct relation to what is being discussed. Finally, the "multiplicative group of integers mod n" is not an area of study, it is an algebraic object. I also found the general tone of this paragraph to be below the encyclopedic standard. Finally, I removed some of the "See also" entries, since we do not put things there that have already been linked to in the article, or are too general to be of much use to a reader looking for related concepts.

The rest of the lead also needs work, but we should not try to do too much at one time. --Bill Cherowitzo (talk) 07:05, 16 January 2017 (UTC)

Hi please tag me in any specific comments here. I disagree with your thoughts on the category page discussed here. For one, category pages are a feature that not everyone knows about and they are not used as SOURCES they are used as LINKS. Please understand the difference.

You say "For one, yes, "multiplicative group of integers mod n" is not an area of study, it is an algebraic object"... Yes, absolutely and that was clear from the sentence. I said it was an application, the article needs to show an example of the how Advanced Areas of Mathematical Study applies these concepts.

With respect to the links, the links are there because as it was pointed out on this talk page previously that the introduction did not provide a sufficient introduction to the terminology, but the terminology is correct, minus the =s/equivalence sign...I am still contemplating that. So, it is fine to have creative links that together provide sufficient background for those who need it, and that is given elsewhere on Wikipedia.

I need to revert your changes. You should discuss prior to making changes on the introduction now. The tone is not below standard, it is in fact correct. You do not provide any source for this conclusion. Please discuss prior to reverting and tag me. The comments on this talk page support the changes I have made thus far, and I do not see the same support for your conclusions. Things should be in the see also that are also linked to. Users, including myself often only look at one section, which is why one section needs to be self-contained and also consistent with the rest of the article. Not all linked sources were in the see also section, only the relevant ones were included. For example, I often look at Introductions, Definitions, and See Also sections. PinkAppleFlower (talk) 23:15, 19 January 2017 (UTC)


 * , you seem to be disregarding several valid points made by . Example: the link modulo mathematics is a confusion of two unrelated mathematical concepts.  As a matter of style, we appear to link only to articles, not to WP categories, which you have done (WP categories are a form of metadata not included in the article content).  You have re-introduced capitalization in your revert, which is counter the Manual of Style.  Keep in mind that an article such as this has had attention from a number of mathematicians, Bill being one of them, over a number of years.  As a relatively experienced editor, I have had well-meant changes reverted, and simply had to accept consensus, or even the preference of a single editor whose insight as a mathematician I respect. Modular arithmetic is a field that finds broad application in many fields, from elementary to advanced, with many results. If there is unresolved disagreement between editors, as there is here, the individual points should be resolved.  Bill made fairly constrained changes, though I guess we'll have to enumerate them here. Questions of tone are of course more subjective, and here one needs to cooperate to gain consensus.  If you introduce too many errors or detractions in the opinion of others, it might just be too laborious to work from your version.  —Quondum 00:57, 20 January 2017 (UTC)


 * My comments are embedded below
 * ", you seem to be disregarding several valid points made by ." Okay, I am listening.


 * "Example: the link modulo mathematics is a confusion of two unrelated mathematical concepts." You are correct here, this was a mistake on my part when I was adding in the link.  I did not review the link thoroughly. Thank you for pointing this out.


 * "As a matter of style, we appear to link only to articles, not to WP categories, which you have done (WP categories are a form of metadata not included in the article content)." As I all ready said and explained.  I think it is okay to do this.  Please respond, to my point above that discusses why I think it is important to use this type of link.  Again, Non-Wiki editors likely do not know about many of the features wiki offers.


 * "You have re-introduced capitalization in your revert, which is counter the Manual of Style." Then please just correct the capitalization and do not make content changes in the same edit. Your expertise would be really helpful in this respect, and we could divide the work between all three of us and get this page up to snuff.


 * "Keep in mind that an article such as this has had attention from a number of mathematicians, Bill being one of them, over a number of years." Great, attention does not make the page correct or helpful necessarily. The talk page suggests that their contributions were not helpful, and in general that the page was not helpful, and it was also incorrect prior to the changes I made, and I provided reputable sources for my changes.  There was no definition of the basic mathematics being used or links to define the basic mathematics being used, for example modulo multiplication.


 * "As a relatively experienced editor, I have had well-meant changes reverted, and simply had to accept consensus, or even the preference of a single editor whose insight as a mathematician I respect." Okay, I do not think two people represent consensus and I see consensus on the talk page to support my changes.  I am a respected econometrician which uses the fields of mathematics and statistics.  So I am respected and I do have consensus, so please respond to my point about consensus that I made earlier and again here.


 * "Modular arithmetic is a field that finds broad application in many fields, from elementary to advanced, with many results. If there is unresolved disagreement between editors, as there is here, the individual points should be resolved."  Agreed.  I will change my edit style to be more minor in nature by doing many small changes so that you can fix the editing grammar or edit the content on just one point.  Great idea.  Will you also please do the same?  Does this sound acceptable to you two ?


 * "Bill made fairly constrained changes, though I guess we'll have to enumerate them here." I disagree with this point.  The changes he made resulted in the page being incorrect.  They were not minor, but his correction with respect to one link, the link to modular mathematics is appreciated, I did not mean to link to that page.


 * "Questions of tone are of course more subjective, and here one needs to cooperate to gain consensus." Again, I have consensus from the view of this talk page.  I also work in the field of mathematics and statistics and have given lectures or prepared material for undergraduate and graduate college students.


 * "If you introduce too many errors or detractions in the opinion of others, it might just be too laborious to work from your version." I agree, that working on this page is overly laborious, but I think we may have agreed on a way to make it so we can all work together, correct?


 * PinkAppleFlower (talk) 02:13, 20 January 2017 (UTC)


 * Try to become more familiar with how talk pages are generally used on WP. With extremely limited exceptions, editing another editor's comments in any way is discouraged.  Indentation is used in a specific way; outdenting from the comment you are responding to does not work well here.  I actually find it difficult to respond to you in detail at this point.  —Quondum 02:58, 20 January 2017 (UTC)


 * , if you check the box "Add pages and files I edit to my watchlist" in Preferences → Watchlist → Advanced options, any changes to this talk page will appear on your watchlist and there would be no need for you to get a special invite to look at this page. Also, interweaving responses in an editor's post is really frowned upon here. With your and Quondum's forgiveness, I hope, I have taken the liberty of restoring Quondom's post and providing indentation in your reply. --Bill Cherowitzo (talk) 03:35, 20 January 2017 (UTC)


 * Now, as to substance. There are many points of disagreement, but for the moment I shall limit myself to just two. First, this issue of categories. You say, "As I all ready said and explained. I think it is okay to do this.  Please respond, to my point above that discusses why I think it is important to use this type of link." The above point was, "I disagree with your thoughts on the category page discussed here. For one, category pages are a feature that not everyone knows about and they are not used as SOURCES they are used as LINKS.  Please understand the difference."  You seem to be saying that such a link should be used because people don't know about them. I fail to see any logic in this. Yes, this is a link, but you are using it as a source, namely the source of the statement that this subject is in the list of inverse problems. One of several difficulties here is that modular multiplicative inverses are not inverse problems! A quick look at the page Inverse problem should convince you of that. Modular multiplicative inverse is not correctly categorized (all wikis contain errors which, in particular, is why you can not use them as sources). Even if this was corrected (it's an inverse operation, not an inverse problem) there is no need to include this in the body of the article as it already appears in the footer with the other categories that the subject belongs to. The second point is that I am baffled by your insistence that comments made on this talk page, before you made any changes, form a consensus in agreement with your changes. I quote, " I see consensus on the talk page to support my changes" and "Again, I have consensus from the view of this talk page." Consensus is the agreement that has been achieved after discussion. There has been no discussion, ergo, there can be no consensus. You are only inferring that the people who made those comments would like what you have done, but that is your personal point of view, not consensus. --Bill Cherowitzo  (talk) 04:42, 20 January 2017 (UTC)


 * I agree with Bill on both points, though my point about categories is a matter of the style of Wikipedia. Categories were created as a way of grouping articles by type, not as a linking mechanism between articles; they appear in a separate box at the very bottom of each article and their display probably really belongs under the "Tools" part of the left bar, along with "What links here".  There is the main article space, and there is the article talk space (Talk:), user space (User: and User_talk:), Wikipedia meta spaces (Wikipedia:, Wikipedia_talk: and others such as Category:) and outside links.  In essence, we never link to any of the others from the main article space, with limited standardized exceptions: primarily links to the OEIS via a template, links to categorization of the article itself at the very end of the article, links to outside links in the References, External links and Notes sections.  No matter the rationale, to use category links inside an article, even in the See also section breaks the existing style, which should not be done lightly and would require a broader discussion than an article talk page.  And no matter how logical a change seems, this type of change is well-night impossible to obtain consensus on.  If you want to suggest a style change, feel free to start a discussion at Wikipedia talk:Manual of Style, or similar forums (starting with places like the teahouse linked on your talk page would be a gentler experience).  I think interacting with a greater spectrum of editors would be useful for gaining a feel for the interaction and consensus-building process on Wikipedia.  (Think of this as my response to your "...  point above that discusses why I think it is important to use this type of link."  You seem to have missed that I've already responded, so I'm trying to make it clearer that no matter how valid your rationale, there is a higher-level consideration that does not allow one at this point to consider the possibility of using category links as you did.)  —Quondum 16:03, 21 January 2017 (UTC)


 * Hi, please see if my changes address most of your issues. With respect to fixing the issue of using "category links" as links, (a) I found a page that discusses "inverse elements" as a group, and yes, upon reviewing the link "category:inverse problems" it was quite confusing. I study "inverse problems" and  I thought the "category:inverse problems" would be more general than what it actually is, but there is absolutely consensus on how the category inverse problems, is defined.  In any case, I suppose the best way to handle the linking of "category" pages is to find a page that actually talks about related concepts, but is not yet defined as a category on Wiki.  Also, please keep in mind that this is basic concept and I am trying to make it very accessible.
 * PinkAppleFlower (talk) 17:22, 21 January 2017 (UTC)


 * Hello again, I want to respond to Quondum's response to my point (3):


 * Previously Quondum said, This assumes a specific teaching context, not appropriate here. As a mathematical article, definition and membership of mathematical categories (rings, etc.) rank above examples and algorithms. The latter are more like addenda to the mathematical presentation. You'll get a feel for the style as you work with multiple articles.


 * Okay, I see your point because I looked up the definition of Encyclopedia, which says "a book or set of books giving  information on many subjects or on many aspects of one subject and typically arranged alphabetically".  In other words it is not pedagogical. Yet, what I am trying to say is that the Computation Section does not provide appropriate context. I think it should likely have an introductory paragraph. Also, having the Example section in the article, does seem to be questionable because your critique seems to apply here as well.  So, what I think may need to happen, is perhaps providing more coherent narrative and context on the algorithms maybe merge the example and computation sections, and perhaps we say, see external links for solved examples explicitly somewhere in the article? Perhaps we also add in some of the code snippets here (I can check them and host them on my site if they are not hosted elsewhere).  PinkAppleFlower (talk) 18:06, 21 January 2017 (UTC)


 * , your recent edit is certainly an improvement, but there are still concerns that I have. I consider the source that you found for the RSA algorithm to be far too technical and not user friendly. If the purpose of the citation is to support the statement that the modular inverse operation is important in this algorithm, there are far better references that readers can go to. This topic has become a standard example in modern number theory introductory texts and any of them would be a better source for our readers. You would already have to know the details of the RSA algorithm to get anything out of the technical specs for its internet application, defeating the purpose of the citation. Since an introduction is primarily an abstract of the contents of the article (see Lead), I usually do not put citations in the introduction, but save them for the body of the article where the material is actually discussed. Another issue, which I have raised before, is that I do not see the relevance of talking about general properties of binary operators in relation to this topic. Interjecting such extraneous material would only cause readers to shake their heads in befuddlement. Taking a modular inverse is a unary, not binary, operation. The binary operation relevant to this topic is modular multiplication and in a discussion of that talking about the associative and commutative laws would make sense, but not here. --Bill Cherowitzo  (talk) 18:53, 21 January 2017 (UTC)


 * , I am glad we are reaching agreement on some points. Compromise is good.
 * #List item I am talking about multiplication being a binary operator.
 * #List item I think it is necessary to provide a source for the claim that the modulo inverse is used in the RSA Algorithm. Yes, it is technical, but it a primary source, and again provides context. For those users who understand the technical reference, which is what brought me to this particular page, it is helpful.  For those who are not familiar with that technicality, it provides proof, outside of Wiki.  People do use the RSA algorithm today, now we have proof.
 * #List item To understand the importance of "inverse elements" and the significance of this concept to other areas of mathematics, the reference to binary operators is needed. I think it is more appropriate for you to add in the information about rings and groups if you would like to.  I was trying to provide the necessary background for your extensions.
 * #List item To understand things like "the natural numbers are closed under addition". Properties of operators, inverse elements, provide the key context as well as the extensions.


 * Please let me know what issues remain to be fixed in the introduction and the see also sections. Otherwise I am moving on to the subsequent sections. Thanks! PinkAppleFlower (talk) 19:20, 21 January 2017 (UTC)


 * I am not sure what agreements you are seeing, but ... again two points. First of all, with respect to your second item, we do not use primary sources unless there is no alternative. Reliable secondary sources are what we depend on. We are reporting on what is in the literature, that is, what authors are saying about the primary sources, not what we think the primary sources are saying. Also, Wikipedia does not prove anything, your statement that the fact that RSA is used today requires proof is bewildering. A simple statement that modular inverses are used in RSA is all that is needed in the introduction, actual citations and an expansion of this statement belong in the applications section (which needs to be rewritten in any event). Secondly, I don't think that you have fully grasped the nature of encyclopedic writing. If the most important thing about a modular inverse was that it is an inverse element, then this whole page should be a section of the inverse element page and modular multiplicative inverse would be a redirect to that section. I don't think that either of us would think that that is appropriate. Thus, this page needs to establish the importance of the subject for what it is and not for its being a type of something more general. In this case, the broader context of binary operations should be dealt with by the appropriate wikilinks and a discussion of that context needs to be limited to those aspects that are directly related to the topic at hand. I am far from finished with the introduction, I just started on this paragraph because I thought it would be the easiest to deal with. In particular, the first sentence needs to be redone and expanded. In your attempt to simplify, you replaced an unfamiliar but standard notation (≡) by an even more obscure and non-standard notation for modular multiplication. Notation is now inconsistent with the rest of the article. You may work on any part of the article that you like, you certainly don't need my or anyone else's approval for that. However, I would suggest that you not make blanket changes to notation without getting some concensus for it on this talk page.--Bill Cherowitzo  (talk) 22:35, 21 January 2017 (UTC)

Excellent article
I don't know if this section has any validity here (and I'll understand if it gets deleted), as I am not suggesting any improvements, but I just wanted to say that, as a beginner in math, this is one of the very few articles that I almost fully understood and, for the biggest part, is accessible to people unfamiliar with these concepts already. For the biggest part, it is clear and it has a structure that makes sense, so again thanks to everyone involved for this article. 2A02:587:4526:EA00:4940:D912:B4:B87C (talk) 02:42, 16 August 2017 (UTC)

Special / easier algorithm for "inverse modulo 2n"?
I remember vaguely that there is an easier algorithm for "multiplicative inverse modulo 2n", especially when n is the word size of the processor (e.g. 32 or 64). It would be nice if this algorithm is also described here (or at least mentioned, if it has its own name/article). --RokerHRO (talk) 10:08, 16 August 2018 (UTC)