Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. If none of them work then we know \y\ does not have an inverse. Modular arithmetic practice carnegie mellon university. This shows that the set of common factors of a and b is the same as the set of common factors of a and r. Given two integers a and m, find modular multiplicative inverse of a under modulo m. The prime factorizations would no longer be unique if 1 was prime.
Handbook of applied cryptography centre for applied. It is the study of numbers and various ways in which we can combine them, such as through addition and subtraction, multiplication and division. Modular arithmetic is a way of systematically ignoring di. Chapter 4 deals with the computation, to arbitrary precision, of functions. This printable worksheet and interactive quiz will provide an assessment of your proficiency at solving math problems using modular arithmetic. The latter operation is the most interesting one and creates a complicated structure on integer numbers. Faster modular exponentiation using double precision. Constructive algorithms, based on the extended euclidean algorithm, to compute modular inverses and solve the chinese remainder problem is presented. We say \a is congruent to b modulo m and write a b mod m if m ja b i. Read and learn for free about the following article.
Pdf a modularpositional computation technique for multiple. Number theory modular arithmetic and gcd misha lavrov arml practice 92220. That is, they are not limited by the computer wordsize of 32 or 64 bits, only by the memory and time available for the computation. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around when reaching a certain value, called the modulus.
The upshot is that when arithmetic is done modulo n, there are really only n different kinds of numbers to worry about, because there are only n possible remainders. Followup exercise read up on euclids algorithm for fnding the greatest common divisor of. The modular arithmetic properties allow me to solve this equation the way i would solve a linear equation, up to a point. Euclids algorithm can be extended to prove the following result, and indeed to construct the s and t the theorem claims exist. Modular arithmetic victor adamchik fall of 2005 plan 1. Providing support for extended precision modular arithmetic and polynomial operations can increase the performance of cryptosystems. Modular arithmetic and elementary algebra 1 euclids algorithm. Modular arithmetic practice joseph zoller september, 2015 practice problem solutions 1. Modular arithmetic in this week we will discuss integer numbers and standard operations on them. However, on some processors the traditional fpu uses an extended precision. In modular arithmetic, numbers wrap around upon reaching a given fixed quantity this given quantity is known as the modulus to leave a remainder. An integer b is divisible by an integer a, not zero, if there is an integer x. Basically, it is a kind of integer arithmetic that reduces all numbers to ones that belongs to a fixed set 0.
Modular arithmetic is a special type of arithmetic that involves only integers. A multiply unit includes an extended precision accumulator. This discussion extends the classical context of modular. Highperformance and embedded systems for cryptography. Jacobian elliptic functions and weierstrass ellipticmodular functions 72, sec. Further note that the implementation of class residue requires a mantissa precision according to the ieee standard for floatingpoint arithmetic ieee 754. Implementation of multipleprecision modular multiplication on gpu. Modular arithmetics pdf the expression a b mod n, pronounced a is congruent to b modulo n, means that mike holt pdf a b is a multiple of n. Chapter 2 is concerned with the fft and modular arithmetic, and their. Extended precision refers to floating point number formats that provide greater precision than the basic floating point formats. Of course, one can come up with homebrewed 10liner of extended euclidean algorithm, but why reinvent the wheel for example, javas biginteger has modinverse method. Applications of modular arithmetic are given to divisibility tests and to block.
In particular, we will need them to describe the rsa cryptosystem, the primality. In this article, i will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. Modular arithmetics pdf modular arithmetics pdf modular arithmetics pdf download. The algorithms that we present are mainly intended for arbitraryprecision arithmetic. Introduction to number theory and its applications lucia moura winter 2010 \mathematics is the queen of sciences and the theory of numbers is the queen of mathematics.
Modular arithmetic peter mcnamra bucknell university and trinity college dublin. The expression parm1 modulo parm2 is defined as follows, with the brackets indicating an integer part. Does some standard python module contain a function to compute modular multiplicative inverse of a number, i. In fact, this is the example we shall need for the rsa cryptosystem. In modular arithmetic, we select an integer, n, to be our \modulus. Cuda 1 4, which extends the c programming language. That is, the largest integer whose magnitude does not exceed the magnitude of parm1 parm2 is used. Introduction to modular arithmetic university of georgia. Faster modular exponentiation using double precision floating point arithmetic on the gpu niall emmart. If 1 is prime it breaks the fundamental theorem of arithmetic which says that all numbers 1 have a unique prime factorization.
Modular arithmetic continued lecture notes for access 2011 by erin chamberlain and nick korevaar number theory refresher here are some words which will occur in our discussion today. Modular arithmetic, cryptography, and randomness for hundreds of years, number theory was among the least practical of mathematical disciplines. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. In contrast to extended precision, arbitraryprecision arithmetic refers to implementations of much larger numeric. Hence, it is indispensable that the proper mantissa length is enforced before performing any arithmetic operations. Modular arithmetic and elementary algebra lecturer. Number theory modular arithmetic stanford university. Remainder arithmetic 265 according to whether their remainders on division by 3 are 0, 1, or 2.
In particular, the greatest element in the two sets must be the same. Fast integer multiplication using modular arithmetic. College of information and compute sciences, university of massachusetts, amherst, ma 01003, usa state key laboratory of information security, institute of information engineering, chinese academy of sciences, beijing, china. For almost all its history, the study of modular arithmetic has been driven purely by its inherent. Extendedprecision accumulation of multiplier output. Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. Fast modular arithmetic on the kalray mppa256 processor for an.
The swiss mathematician leonhard euler pioneered the modern approach to. Modular arithmetic for linear algebra computations in the real field. A study of high performance multiple precision arithmetic on. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i. Analysis of modular arithmetic 9 the set of solutions of the homogene ous system a x 0 is the z m module gen erate d from the ve ctors.
Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. Modular arithmetic this lecture introduces the concept of modular arithmetic and presents a fast algorithm to compute the power of a number modulo another number. Techniquesdescribedforperforming modular reduction for an arbitrary modulusm are the classical method. Michel goemans these notes cover basic notions in algebra which will be needed for discussing several topics of this course. For any algorithm based on computing in q, we must. Math circle thursday january 22, 2015 what is modular arithmetic. Karl friedrich gauss csi2101 discrete structures winter 2010.
The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. If nis an integer, two integers are equal mod nif they di. Amd radeon hd 8750 m, ati, computer science, extended precision, gpu cluster, linear algebra, modular arithmetic, mpi, opencl. Modular arithmetic, in its most elementary form, arithmetic done with a count that resets itself to zero every time a certain whole number n greater than one, known as the modulus mod, has been reached. Some cryptographic systems, such as those performing rsa cryptographic signature authentication, perform successive extended precision modular multiplications, accumulating the sum of the results. I multiply out the left side, then get the xs on one side. Highprecision arithmetic in mathematical physics mdpi. Microprocessor instructions are provided for manipulating portions of the extended precision accumulator including an instruction to move the contents of a portion of the extended accumulator to a generalpurpose register mflhxu and an instruction to move the contents of a generalpurpose register to a portion of the extended. Examples are a digital clock in the 24hour system, which resets itself to 0 at midnight n. Modular arithmetic is often tied to prime numbers, for instance, in wilsons theorem, lucass theorem, and hensels lemma, and generally appears in fields. Modular arithmetic, a fundamental system for operating on numbers within the limit of a modulus, is known to mathematicians since the days of euclid and finds applications in diverse areas ranging. Ceesxmod performs the mathematical function modular arithmetic by using the equation. The modern approach to modular arithmetic was developed by carl friedrich gauss in his book disquisitiones arithmeticae, published in 1801 a familiar use of modular arithmetic is in the 12hour clock, in which the day is divided into two 12.
1094 283 349 1430 84 933 538 622 880 156 189 1386 1343 1486 693 717 309 258 734 1315 1007 564 1369 35 547 1161 1217 1295 950 460 1331 249 128 966 1101 439 261 883 1274 166 743 807 1350 1414 387 149