The prime factorizations would no longer be unique if 1 was prime. If none of them work then we know \y\ does not have an inverse. Modular arithmetic is a way of systematically ignoring di. Constructive algorithms, based on the extended euclidean algorithm, to compute modular inverses and solve the chinese remainder problem is presented. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. The algorithms that we present are mainly intended for arbitraryprecision arithmetic. Read and learn for free about the following article. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers.
Providing support for extended precision modular arithmetic and polynomial operations can increase the performance of cryptosystems. 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. 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. 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. This section introduces our modular method for limiting the precision needed in sum ming real inputs. 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. Extended precision refers to floating point number formats that provide greater precision than the basic floating point formats. Michel goemans these notes cover basic notions in algebra which will be needed for discussing several topics of this course. 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.
Modular arithmetic practice carnegie mellon university. Handbook of applied cryptography centre for applied. The modular arithmetic properties allow me to solve this equation the way i would solve a linear equation, up to a point. Chapter 4 deals with the computation, to arbitrary precision, of functions. Modular arithmetic for linear algebra computations in the real field. Remainder arithmetic 265 according to whether their remainders on division by 3 are 0, 1, or 2. If 1 is prime it breaks the fundamental theorem of arithmetic which says that all numbers 1 have a unique prime factorization. Amd radeon hd 8750 m, ati, computer science, extended precision, gpu cluster, linear algebra, modular arithmetic, mpi, opencl.
The latter operation is the most interesting one and creates a complicated structure on integer numbers. Pdf a modularpositional computation technique for multiple. Does some standard python module contain a function to compute modular multiplicative inverse of a number, i. 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. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. In contrast to extended precision, arbitraryprecision arithmetic refers to implementations of much larger numeric. The modular multiplicative inverse is an integer x such that. 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, 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. 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 and elementary algebra lecturer. In particular, we will need them to describe the rsa cryptosystem, the primality. Basically, it is a kind of integer arithmetic that reduces all numbers to ones that belongs to a fixed set 0. Euclids algorithm can be extended to prove the following result, and indeed to construct the s and t the theorem claims exist. The swiss mathematician leonhard euler pioneered the modern approach to. Techniquesdescribedforperforming modular reduction for an arbitrary modulusm are the classical method. However, on some processors the traditional fpu uses an extended precision. Extended precision modular arithmetic library that targets cuda. Extendedprecision accumulation of multiplier output. 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, 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. Given two integers a and m, find modular multiplicative inverse of a under modulo m.
Further note that the implementation of class residue requires a mantissa precision according to the ieee standard for floatingpoint arithmetic ieee 754. Faster modular exponentiation using double precision. Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. 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. Modular arithmetic, cryptography, and randomness for hundreds of years, number theory was among the least practical of mathematical disciplines. For almost all its history, the study of modular arithmetic has been driven purely by its inherent. 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. For any algorithm based on computing in q, we must.
This discussion extends the classical context of modular. In particular, the greatest element in the two sets must be the same. Math circle thursday january 22, 2015 what is modular arithmetic. Highperformance and embedded systems for cryptography. In this article, i will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. If nis an integer, two integers are equal mod nif they di. Fast integer multiplication using modular arithmetic. Applications of modular arithmetic are given to divisibility tests and to block. 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. Modular arithmetic is a special type of arithmetic that involves only integers. An integer b is divisible by an integer a, not zero, if there is an integer x. We say \a is congruent to b modulo m and write a b mod m if m ja b i. Examples are a digital clock in the 24hour system, which resets itself to 0 at midnight n.
Modular arithmetics pdf modular arithmetics pdf modular arithmetics pdf download. Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. Modular arithmetic practice joseph zoller september, 2015 practice problem solutions 1. 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. Followup exercise read up on euclids algorithm for fnding the greatest common divisor of. Modular arithmetic is a system of arithmetic for integers, which considers the remainder. A multiply unit includes an extended precision accumulator. Modular arithmetic victor adamchik fall of 2005 plan 1. Faster modular exponentiation using double precision floating point arithmetic on the gpu niall emmart.
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. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as. In fact, this is the example we shall need for the rsa cryptosystem. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. Highprecision arithmetic in mathematical physics mdpi. Number theory modular arithmetic and gcd misha lavrov arml practice 92220.
Introduction to modular arithmetic university of georgia. Number theory modular arithmetic stanford university. In modular arithmetic, numbers wrap around upon reaching a given fixed quantity this given quantity is known as the modulus to leave a remainder. Some cryptographic systems, such as those performing rsa cryptographic signature authentication, perform successive extended precision modular multiplications, accumulating the sum of the results. Modular arithmetic peter mcnamra bucknell university and trinity college dublin. This printable worksheet and interactive quiz will provide an assessment of your proficiency at solving math problems using modular arithmetic. The expression parm1 modulo parm2 is defined as follows, with the brackets indicating an integer part. Jacobian elliptic functions and weierstrass ellipticmodular functions 72, sec. Since modular arithmetic is such a broadly useful tool in number theory, we divide its explanations into several levels.
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. Karl friedrich gauss csi2101 discrete structures winter 2010. 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. Implementation of multipleprecision modular multiplication on gpu. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around when reaching a certain value, called the modulus.
A study of high performance multiple precision arithmetic on. I multiply out the left side, then get the xs on one side. In modular arithmetic, we select an integer, n, to be our \modulus. Modular arithmetic in this week we will discuss integer numbers and standard operations on them. Fast modular arithmetic on the kalray mppa256 processor for an. Chapter 2 is concerned with the fft and modular arithmetic, and their. Modular arithmetic and elementary algebra 1 euclids algorithm. Hence, it is indispensable that the proper mantissa length is enforced before performing any arithmetic operations. 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.
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