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Published: 09.05.2021  Many complex cryptographic algorithms are actually based on fairly simple modular arithmetic. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division.

## An Introduction to Modular Arithmetic

Modular arithmetic has been a major concern of mathematicians for at least years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity.

Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic.

To add 3 and 5, you start at 0, count 3 to the right, and then a further 5 to the right, ending on 8. To multiply 3 by 5, you start at 0 and count 3 to the right 5 times ending up at These sorts of operations should be familiar from elementary school.

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. It can be any number, not necessarily To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1. As we mentioned above, there is nothing special about 7.

Our usual clocks can be used to do arithmetic modulo This may seem a rather trite variant on our usual arithmetic, and the reader could legitimately wonder if it is more than a curiosity. I hope this article will convince her that it is. An important observation is that any arithmetic equality that is true in normal arithmetic is also true in modular arithmetic modulo any whole number you like. This easily results from the observation that one can wind the usual number line around the modular clock face, turning usual arithmetic into modular arithmetic.

To illustrate a major reason why mathematicians care about modular arithmetic, let me start with one of the oldest questions in mathematics : Find Pythagorean triples , i.

The 3,year-old Babylonian tablet Plimpton lists Pythagorean triples. The second column of the tablet lists values for X and the third column the corresponding value of Z ; the value of Y is not listed. In modern notation, the solutions listed on Plimpton are as follows:. Could it have been trial and error, or did the Babylonians know an algorithm? What is certain is that 1, years later the Greeks knew the algorithm to generate all whole number solutions to this equation.

We know this because Euclid explained the method in Book X of his famous Elements. But what if we change the problem slightly? The answer comes from modular arithmetic.

We can arrange that no whole number bigger than 1 divides all of X , Y , and Z. If it did, simply divide each of X , Y , and Z by this common factor, and they still form a solution to the same equation. If need be, we repeat this process. Note that as the numbers X , Y , and Z get smaller in absolute value each time, but remain whole numbers, this procedure must eventually stop. Then there would be a solution to the same equation in arithmetic modulo 3.

As we have reached a contradiction, the only possibility is that our initial assumption was flawed, i. This sort of argument works not only for this particular equation. This is actually a very practical criterion. It may appear that one needs to check for solutions to our equation in arithmetic modulo m for infinitely many m.

However, for higher degree equations, the corresponding theorem can fail. Nevertheless, when studying the whole number solutions to any polynomial equation, the study of solutions modulo m is often a key tool. This theorem gives a very efficient algorithm that reduces the study of the solutions to a polynomial equation in arithmetic modulo a whole number m , to the study of the same equation in arithmetic modulo the factors of m of the form p a , where p is a prime number and a is a positive whole number.

In fact, it turns out that the key case to consider is when m is a prime number. Thus, for the rest of this article, we will only consider arithmetic modulo a prime number p. Recall that a prime number is a whole number greater than 1, which is only divisible by 1 and by itself. Examples are 2, 3, 5, 7, 11, 13, 17, and 19, but not for instance 15, which is divisible by 3 and 5. Every positive whole number can be written uniquely as a product of prime numbers up to order.

In some way, prime numbers are a bit like the atoms of which all other whole numbers are composed. It states that. Thirty years after I first learnt how to prove this theorem, it still seems miraculous to me. For example, one could ask how many square roots 3 has in arithmetic modulo , which is a prime number. You could, in theory, check all the possibilities and determine the answer, but without a computer this would take a very long time.

One could ask for a similar method that given any number of polynomials in any number of variables helps one to determine the number of solutions to those equations in arithmetic modulo a variable prime number p. Such reciprocity laws are often referred to as non-abelian. This conjecture completely changed the development of number theory.

Meanwhile, in the mids, Robert Langlands Professor Emeritus, School of Mathematics had the extraordinary insight that the ideas of Eichler, Taniyama, and Shimura were a small part of a much bigger picture.

He was able to conjecture the ultimate reciprocity law, an enormous generalization of what had gone before, which applies to any number of equations, of any degree in any number of variables. One striking feature of all the non-abelian reciprocity laws is that the formula for the number of solutions is given in terms of symmetries of certain curved spaces—an extraordinary connection between solving algebraic equations and geometric symmetry.

I will conclude by discussing one further question about modular arithmetic which has seen recent progress. Instead of asking for a rule to predict how many solutions an equation will have in arithmetic modulo a varying prime p , one can ask about the statistical behavior of the number of solutions as the prime varies.

For other equations, the correct answer may be harder to guess. He proved this for all cubic equations in two variables. These celebrated conjectures led to a revolution in arithmetic algebraic geometry. Rather the natural question is to consider the normalized error term. There should, of course, be density theorems for any number of equations in any number of variables of any degree, but these remain very much conjectural.

The story is continuing. Richard Taylor, who became a Professor in the School of Mathematics in January, is a leader in number theory who, with his collaborators, has developed powerful new techniques that they have used to solve important long-standing problems. With Michael Harris, he proved the local Langlands conjecture. More recently, Taylor established the Sato-Tate Conjecture, another longstanding problem in the theory of elliptic curves.

Email Share Tweet. Modulo 7. MODULO 7 To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. It states that if p is a prime number, then the number of square roots of an integer n in arithmetic modulo p depends only on p modulo 4 n. Published in The Institute Letter Summer Abelian Reciprocity Law. Algebraic Geometry. Carl Friedrich Gauss.

Chinese Remainder Theorem. Eichler Reciprocity Law. Euclid's Elements. Ferdinand Frobenius. Fermat's Last Theorem. Langlands Reciprocity Conjecture. Law of Quadratic Reciprocity. Modular Arithmetic. Nicholas Shepherd-Barron. Non-Abelian Reciprocity Laws. Polynomial Equations. Pythagora's Theorem.

Pythagorean Triples. Sato-Tate Density Theorem. Shimura-Taniyama Reciprocity Law. Shimura-Taniyama-Weil Conjecture. Pierre Deligne. Kelly Devine Thomas. ## Modular arithmetic: you may not know it but you use it every day

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Computing Computer science Cryptography Modular arithmetic. What is modular arithmetic? Practice: Modulo operator. Practice: Congruence relation.

Modular arithmetic has been a major concern of mathematicians for at least years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. To add 3 and 5, you start at 0, count 3 to the right, and then a further 5 to the right, ending on 8. To multiply 3 by 5, you start at 0 and count 3 to the right 5 times ending up at

You may never have heard of modular arithmetic, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders. When reckoning hours, we count up to 12 and start again from one. Numbers that differ by a multiple of the modulus 12 are said to be congruent modulo A similar situation arises for the days of the week, which are computed modulo seven. Suppose today is Thursday. What weekday will it be 1, days from today? ## An Introduction to Modular Arithmetic

Modular arithmetic , sometimes also called clock arithmetic , is a way of doing arithmetic with integers. Much like hours on a clock , which repeat every twelve hours, once the numbers reach a certain value, called the modulus , they go back to zero. People talked about modular arithmetic in many ancient cultures. For instance, the Chinese remainder theorem is many centuries old.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs and how to get involved. Comments: fixed some typos and references Subjects: Symbolic Computation cs.

You may never have heard of modular arithmetic, but you use it every day without the slightest difficulty. In this system, numbers wrap around when they reach a certain size called the modulus; it is the arithmetic of remainders. When reckoning hours, we count up to 12 and start again from one.

Он хотел крикнуть, но в легких не было воздуха, с губ срывалось лишь невнятное мычание. - Нет! - закашлявшись, исторгнул он из груди. Но звук так и не сорвался с его губ. Беккер понимал, что, как только дверь за Меган закроется, она исчезнет навсегда. Он снова попробовал ее позвать, но язык отказывался ему подчиняться. Девушка почти уже добралась до двери. Беккер поднялся на ноги, пытаясь выровнять дыхание.

Чатрукьян посмотрел на телефонный аппарат и подумал, не позвонить ли этому парню: в лаборатории действовало неписаное правило, по которому сотрудники должны прикрывать друг друга. В шифровалке они считались людьми второго сорта и не очень-то ладили с местной элитой. Ни для кого не было секретом, что всем в этом многомиллиардном курятнике управляли шифровальщики. Сотрудников же лаборатории безопасности им приходилось терпеть, потому что те обеспечивали бесперебойную работу их игрушек.

Мидж покачала головой: - Настолько сложной, что она длится уже восемнадцать часов? - Она выдержала паузу.  - Маловероятно. Помимо всего прочего, в списке очередности указано, что это посторонний файл. Надо звонить Стратмору.

Испанская церковь гордится тем, что ей принадлежат его останки. Испанская церковь. Беккер отлично знал, что в Испании только одна церковь - римско-католическая.

Он был зашифрован с помощью некоего нового алгоритма, с которым фильтры еще не сталкивались. Джаббе потребовалось почти шесть часов, чтобы их настроить. Бринкерхофф выглядел растерянным. - Стратмор был вне. Он заставил Джаббу вмонтировать в ТРАНСТЕКСТ переключатель системы Сквозь строй, чтобы отключить фильтры в случае, если такое повторится.

Неужели она узнала. Этого не может. Стратмор был уверен, что предусмотрел. - Сьюзан, - сказал он, подходя ближе.  - В чем .

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4 Response
1. Martin A.

In mathematics , modular arithmetic is a system of arithmetic for integers , where numbers "wrap around" when reaching a certain value, called the modulus.

2. CleofГЎs U.

This is classical arithmetic, and it turns up in countless applications in our everyday lives. The reader is also likely familiar with another kind of arithmetic, even if.

3. Michael N.

4. 