1: Shift Ciphers and Modular Arithmetic Practice HW from Barr Textbook (not to hand in) p. Modular arithmetic Introduction and goals. A special form of a modulus and a modified Barrett reduction method are used to perform modular arithmetic in a cryptographic system. Modular Arithmetic-In this week we will discuss integer numbers and standard operations on them: addition, subtraction, multiplication and division. First we have to talk a bit about the problem of integer factorization. However, here's where I'm getting stuck. Adopted or used LibreTexts for your course? We want to hear from you. Let's see this with an example by taking. In this talk you will learn about the core components of cryptography used in. In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, as well as providing finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including AES, IDEA, and RC4. You have also picked up a very important algorithm called the Euclidean Algorithm. It is also used to protect classified government information. Among these RSA en-cryption and digital signature schemes, discrete logarithm problem (DLP) based schemes such as the Diﬃe-Helman key agreement [4] and El-Gamal encryption. and R is 4294967296. Playing with numbers has very important uses in cryptography. We have already discussed modular arithmetic and Diffie-Hellman key exchange algorithm. News; Computing Computer science Journey into cryptography Modular arithmetic. By studying different modular arithmetic approaches on these curves, we present a range of genus-2 implementations. Modular arithmetic also plays an important role in abstract algebra, cryptography, computer science, and the visual and musical arts. Among these, RSA encryption and digital signature schemes, discrete logarithm problem (DLP) based schemes such as the Di e-Helman key agreement [DH76] and El-Gamal encryption and signature. Here 6 means 6 o’clock and 9 means 9 hours later it will be 3:00. Anybody can master modular arithmetic with a little bit of practice. Let's begin with what is called a shift cipher. The first chapter of DPV Algorithms text, specifically section 1. Modular arithmetic can be tricky, so in case you didn't follow what just happened, we are going to walk through it now. Featured on Meta What posts should be escalated to staff using [status-review], and how do I…. However, here's where I'm getting stuck. Mathematicians have long considered number theory to be pure mathematics, but it has important applications to computer science and cryptography studied in Sections 4. The Caesar cipher shifted each letter 3 letters forward in the alphabet (cyclically, sending xyz to abc respectively): Decipher the message: JRRG OXFN LQ WKH PLGWHUP!. for use in certain applications. This comes to light in the compilation programming languages such as C; where for example arithmetic operations on "int" integers are. \ifnum and \advance to perform modular arithmetic by hand with TeX, rather than \pgfmathmod as. We will discuss division with a remainder and introduce an arithmetic on the remainders. A New Arithmetic. Equivalently, for some integer k, a = b +k ·n. Modular arithmetic is important in number theory and can be used to simplify calculations of very large numbers, because many numbers are equivalent in a specified MOD. Step (1) is our augmented matrix that we will use to find the inverse of A. Vercauteren Public-key cryptography. 1: Shift Ciphers and Modular Arithmetic Practice HW from Barr Textbook (not to hand in) p. Encryption by the sender takes the plaintext and a key and produces ciphertext. Large-Integer Arithmetic. Modular arithmetic is, indeed, used in cryptography, but the question is strangely mixing two different levels of the conceptual hierarchy. Public Key Cryptography is based on asymmetric cryptography, so first let us talk about symmetric cryptography. We spend two weeks on elliptic curve cryptography. Cryptography (ECC) and HyperElliptic Curve Cryptography (HECC) of genus 2 over GFð2nÞ. We say that “ a is congruent to b modulo n,” or a ⌘ b (mod n) if a and b have the same remainder when divided by n. Typically the time on some internal clock is used • Compute x n+1 = ax Private Key Cryptography. Public key cryptography; 28. When we add to , we arrive back at. Modular Arithmetic (Clock Arithmetic) Modular arithmetic is a system of arithmetic for integers, where values reset to zero and begin to increase again, after reaching a certain predefined value, called the modulus (modulo). GetInt32(Int32, Int32) Generates a random integer between a specified inclusive lower bound and a specified exclusive upper bound using a cryptographically strong random number generator. The original character in the message is shifted by that number of places; if it hits the last letter in the alphabet, it simply loops back around to the beginning again (in simple modular arithmetic). 1 Modular Arithmetic Primer One way to do arithmetic calculations is to perform them inside a finite field over a prime number, or F p. Named after its creators Whitfield Diffie and Martin Hellman, it's simply called the Diffie-Hellman key exchange. Public Key Cryptography RSA public key cryptography uses modular arithmetic and inverses to encrypt/decrypt information, but does so in way that makes it infeasible for an outsider to compute the inverse of e, even though the outsider knows e and the modulus m. Cryptography: Pages 304: 2 c, 4 c. Section 3 explains different implementation techniques for modular. Secrecy: Some hash algorithms are available for public use, but proprietary interests can make their own. Time-keeping on this clock uses arithmetic modulo 12. Modular arithmetic can be tricky, so in case you didn't follow what just happened, we are going to walk through it now. Video created by University of Colorado System for the course "Basic Cryptography and Programming with Crypto API". READE Even the purest of pure mathematics can have a crucial influence on practical problems. What works in favor of modular arithmetic is the implementation. Gopinath Ganapathy and K. … - Selection from Computer Security and Cryptography [Book]. You are currently browsing the tag archive for the ‘modular arithmetic’ tag. Public Key Cryptography Simply Explained. $\begingroup$ Of course, he wants modular arithmetic to facilitate producing figures with certain features. The numbers are constructed using prime numbers and modular arithmetic. Almost any cipher from the Caesar Cipher to the RSA Cipher use it. Cryptography (ECC) and HyperElliptic Curve Cryptography (HECC) of genus 2 over GFð2nÞ. Online calculator. This page gives a fairly detailed introduction. Functions using modular arithmetic tend to perform erratically, which in turn sometimes makes them one-way functions. Khan Academy > Algebra I > Sequences (intro) Extend arithmetic sequences Use arithmetic sequence formulas Recursive formulas for arithmetic sequences Explicit formulas for arithmetic sequences Converting recursive & explicit forms of arithmetic sequences Extend geometric sequences. Modular arithmetic. We will discuss division with a remainder and introduce an arithmetic on the remainders. 6) of a number modulo $$n\text{. We begin by defining how to perform basic arithmetic modulo \(n$$, where $$n$$ is a positive integer. We can find these by looking through a multiplication table, or by a simple extension of the Euclidean Algorithm. Modular arithmetic is extremely important in the field of cryptography , which encodes information using modulo operations that have a very large modulus. In Step (2), we multiplied the top row by 15 because, looking at our table , we see that 15 is the multiplicative inverse of 2, our current pivot. You can visualize the usual arithmetic as operating on points strung out along the “number line. Back to Number Theory and Cryptography Primes, Modular Arithmetic, and Public Key Cryptography (April 15, 2004) Introduction. One of its uses is in the en-cryption of secret messages. Modular arithmetic and cryptography. IA Numbers and Sets. So for example 9 mod 4 = 1 because 9=2×4+1. Introduction Montgomery-friendly primes were introduced in [27, 14, 15] for cryptographic applications on elliptic or hyperelliptic curves. An exploration of modular arithmetic - adding, multiplying, powers, orders, residues. Note: The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. Substitution ciphers. students and teacher. In fact, I bet you use it all the time. This project implements the cryptographic primitives used in the Bitcoin system, especially elliptic curve operations and hash functions. Modular arithmetic is sometimes called clock arithmetic, since analog clocks wrap around times past 12, meaning they work on a modulus of 12. This morning I got a bit of a surprise when he asked me what modular arithmetic has to do with cryptography. •Modular exponentiation is an expensive computation •Typical usage: use public-key cryptography to establish a shared secret, then switch to symmetric crypto -We'll see this in IPSec and SSL!Keys are longer •1024 bits (RSA) rather than 128 bits (AES)!Relies on unproven number-theoretic assumptions •What if factoring is easy?. So if k = 3, we have an example of the following table to find the ciphertext. Besides these fields, you can find a job in the biotech industry or in teaching math. , certain elliptic curves). Encryption history as a military technology; legislative attempts to control encryption keys; 34. Modular reasoning. Elliptic-curve cryptography relies on modular arithmetic, meaning that the values of the numbers that figure into the computation are assigned a limit. If you continue browsing the site, you agree to the use of cookies on this website. Resources 1 Question Sheet per student. Modular arithmetic. Modular Arithmetic, Prime, Relatively Prime, Euler's Cryptography and Network security in Hindi. Modular multiplication is a core operation in virtually all public-key cryptosystems in use today. Analogously, in any group G, powers b k can be defined for all integers k, and the discrete logarithm log b a is an integer k such that b k = a. In this talk you will learn about the core components of cryptography used in. From the Caesar cipher, we will present the public-key cryptography. Cryptography - Lesson 2 Modular Arithmetic is a fundamental component of cryptography. com Modular arithmetic is a system of arithmetic for integers where numbers wrap around upon reaching a certain value – the modulus. MD5 and other internet security systems like RSA also use modular arithmetic to disguise, simplify, and verify information. After the activity, they investigate the Diffie-Hellman method using prime numbers, exponents, and modular. Modular exponentiation; Turn (unit) Notes. algebraic expressions with exact coefficients, exact powers, etc). Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time. It starts with some occurrences of patterns in powers of numbers and in clock arithmetic. Learning with errors (LWE) is essentially linear regression with modular arithmetic. First we must translate our message into our numerical alphabet. By studying different modular arithmetic approaches on these curves, we present a range of genus-2 implementations. If you are interested in a Comprehensive Cryptography Course then check out this course! RSA Cryptosystem – Integer Factorization. Modular Arithmetic In Example 1 sometime integers are greater than 25 and they were replaced by their remainders after divided by 26. Named after its creators Whitfield Diffie and Martin Hellman, it's simply called the Diffie-Hellman key exchange. Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and ﬁnd-ing the greatest common divisor using the Euclidean algorithm. Also on the HP-50g in Numeric/Approx mode: ' 79^10 MOD 85' will also give the erroneous answer of 45. Fandom Apps Take your favorite fandoms with you and never miss a beat. Cryptography The Making and Breaking of Secret Codes. Polking, Rice University 2 Need for Cryptography • Uses modular arithmetic and several polynomial mappings • Has a 128 bit key - Or 192 or 256 • Won on the basis of security, performance,. On this page, we collect resources about some basic cryptography, the mathematics of encrypting and decrypting (hopefully) secret messages, including some basic modular arithmetic. These high speed designs take advantage of the transistor bounty provided by Moore's law and the continuously diminishing average cost of a transistor. Modular arithmetic can be tricky, so in case you didn't follow what just happened, we are going to walk through it now. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the. Khan Academy > Algebra I > Sequences (intro) Extend arithmetic sequences Use arithmetic sequence formulas Recursive formulas for arithmetic sequences Explicit formulas for arithmetic sequences Converting recursive & explicit forms of arithmetic sequences Extend geometric sequences. 1 Encryption and Decryption Encryption is used to send messages secretly. Cryptography - Lesson 2 Modular Arithmetic is a fundamental component of cryptography. 6) Slides Axioms for Integers For Friday, Read 1. In this talk you will learn about the core components of cryptography used in. This page gives a fairly detailed introduction. Modular arithmetic; Modular exponentiation; Greatest Common Divisor (GCD) Extended Euclidean algorithm; Modular multiplicative inverse; 1. The implementation will use the constant-time Montgomery ladder method described in RFC 7748 in order to prevent side channel attacks. However, here's where I'm getting stuck. MD5 and other internet security systems like RSA also use modular arithmetic to disguise, simplify, and verify information. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the. 5: Public Key Cryptography Last updated; Save as PDF Page ID 34279; Contributed by David Lippman; Professor (Mathematics) at Pierce College; Publisher: The OpenTextBookStore. 5: Public Key Cryptography. Pages 3-25. You are familiar with this set and the corresponding operations, but they are reviewed here to create a background for modular arithmetic. Modular Arithmetic, Groups, Finite Fields and Probability. The sender has a message or plaintext. We will be working with modular arithmetic, so please review the section on modular arithmetic from the first. Uses modular arithmetic and elementary number theories to perform computations using two large prime numbers. Modular Arithmetic Freeware These freeware command line programs will let you do modular arithmetic calculations with large integers. On this page, we collect resources about some basic cryptography, the mathematics of encrypting and decrypting (hopefully) secret messages, including some basic modular arithmetic. In the arithmetic modulo, integers “wrap around” upon reaching a given fixed quantity (this given quantity is said to be as the modulus) to leave a remainder. By studying different modular arithmetic approaches on these curves, we present a range of genus-2 implementations. 10 Modular Arithmetic and Cryptography 10. In 1992, an exception was formally added in the USML for non-encryption use of cryptography (and satellite TV descramblers) and a deal between NSA and the Software Publishers Association made 40-bit RC2 and RC4 encryption easily exportable using a Commodity Jurisdiction with special "7-day" and "15-day" review processes (which transferred. Syllabus Errata for the textbook Thursday, June 2: Introduction to Cryptography, Substitution Ciphers (1. steiner1745. 1 Arithmetic Primitives 1. The heart of SSL { as well. Powers in Modular Arithmetic, and RSA Public Key Cryptography Lecture notes for Access 2009, by Erin Chamberlain and Nick Korevaar. Modular arithmetic. equation solving using variables), and exact math (e. Substitution ciphers. Plain English. Learning with errors (LWE) is essentially linear regression with modular arithmetic. Also on the HP-50g in Numeric/Approx mode: ' 79^10 MOD 85' will also give the erroneous answer of 45. Adopted or used LibreTexts for your course? We want to hear from you. In particular, it can be used to obtain information about the solutions (or lack thereof) of a specific equation. Problem: We have a number of things, but we do not know exactly how many. Asymmetric key cryptography A cryptographic technique that uses 2 mathematically related keys- one key to encrypt data & another key to decrypt data. Chapter 3 Modular Arithmetic February 15, 2010 3 In computers information is represented digitally, and nearly always in binary. Some researchers have neglected the diﬀerences between common arithmetic and modular arith-metic, and falsely claimed that FHE enables arbitrary computations on encrypted data. W e are going to study mo dular arithmetic. What works in favor of modular arithmetic is the implementation. Modular Arithmetic, Groups, Finite Fields and Probability. Cryptography Shinichi Tokoro December 12, 2001 In modular arithmetic, one is given a positive integer m, called the modulus, and any two integers whose diﬀerence is an integer multiple of the modulus are regarded to be "equal" or "equivalent" with respect to the modulus. Elliptic-curve cryptography relies on modular arithmetic, meaning that the values of the numbers that figure into the computation are assigned a limit. I will discuss a simple method of enciphering and deciphering a message using matrix transformations and modular arithmetic, and show how elementary row operations can sometimes be used to break an opponent's code. Here are two generic practical rules for using modular arithmetic. Playing with numbers has very important uses in cryptography. In the latter case, we see that using modular arithmetic modulo 8, is not a field. Modular arithmetic is extremely important in the field of cryptography , which encodes information using modulo operations that have a very large modulus. The numbers n and e are used for encrypting. Modular arithmetic in cryptography? As part of some lecture notes I'm trying to calculate the following: x = (1/N) (Mod R) where N = 1073741827. Modular Arithmetic Part-1 Explained with Examples in Hindi Cryptography lecture series. For example: 7 mod 3 is 1 because 7 = 2 * 3 + 1. This is the same as modular arithmetic, modulo a prime number p. Mani, Maximization of Speed in Elliptic Curve Cryptography Using Fuzzy Modular Arithmetic over a Micro-controller based Environment, Proceedings of the World Congress on Engineering and Computer Science, vol. The Caesar cipher shifted each letter 3 letters forward in the alphabet (cyclically, sending xyz to abc respectively): Decipher the message: JRRG OXFN LQ WKH PLGWHUP!. Asymmetric crypto uses modular exponentiation as encryption/decryption. Herein, SIMD and multithreaded parallel algorithms and approaches are proposed for embedded systems, based on the aforementioned techniques, which can be exploited to enhance large-operands modular arithmetic, used in crypto- graphic applications. Congruence modulo. Hi Joe, 😉 nice example, the HP-35s gets it right by accident, not design. Equivalently, for some integer k, a = b +k ·n. The \s" at the end stands for \secure" meaning that a process called SSL is being used to encode the contents of your inbox and prevent people from hacking your account. , “bignum” = array of unsigned chars (bytes) Useful to also maintain a variable indicating the length of the array. Answer Save. }\) Example 4. If you are looking for a reviewer in datacom, topic in Electronics Systems and Technologies (Communications Engineering) this will definitely help you before taking the Board Exam. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This shows the two numbers are congruent because 21 = 3 * 7, and 7 is a factor of 21. The most commonly used one is of course the normal clock. Denoted a ≡ b (modN). Public Key Cryptography Simply Explained. 4 Divisibility Topics discussed in this section:. Most ciphers use modular arithmetic during some steps of the encryption and decryption process. Perl usually uses arithmetic modulo operator that is machine-independent. But it is from 1996 and in some aspects outdated (f. By using modular arithmetic, Diffie-Hellman allowed two or more parties to exchange secret messages without sharing a secret key in advance. The sender has a message or plaintext. It is often used in calculating checksums and check digits. Before we discuss the widely used public key. If you would like to learn more about modular arithmetic, click here for a short tutorial. It generates a much larger number of points under the same curve equation and the same prime p. 1 Arithmetic Primitives 1. Modular arithmetic has a variety of applications in cryptography. Modular Arithmetic, Groups, Finite Fields and Probability. Modular arithmetic is a system of arithmetic for integers where numbers wrap around upon reaching a certain value - the modulus. Public Key Cryptography Simply Explained. The talk is "Modular Arithmetic based Circuits and Systems for Emerging Technologies and Applications: Deep Neural Networks and Cryptography". Pages 3-25. Modular Arithmetic and Cryptography Jeffhraim Balilla • Many real-world occurences operate in an unusually cyclical process. However, here's where I'm getting stuck. READE Even the purest of pure mathematics can have a crucial influence on practical problems. The heart of SSL { as well. With the present paper we attempt to ﬁll this gap. Modular arithmetic in cryptography; 31. pdf from SCIT CT at Jomo Kenyatta University of Agriculture and Technology. The sender has a message or plaintext. Browse other questions tagged matrices modular-arithmetic inverse cryptography or ask your own question. divided by some particular number. If you continue browsing the site, you agree to the use of cookies on this website. Modular arithmetic is still defined as the remainder of division, however it can also be defined (and is more commonly defined) as a clock. I will discuss a simple method of enciphering and deciphering a message using matrix transformations and modular arithmetic, and show how elementary row operations can sometimes be used to break an opponent's code. Modular Arithmetic Article Continue reading “Modular Arithmetic Library” Author dprogrammer Posted on November 1, 2019 November 2, 2019 Categories C++ , Tutorial Tags c++ , congruence , library , math , modular Leave a comment on Modular Arithmetic Library. It is necessary to asymmetric encryption algorithms. And for speed it is necessary that the arithmetic be realized at the hardware (chip) level. In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, and provides finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including Advanced Encryption Standard (AES), International Data Encryption Algorithm (IDEA), and RC4. … - Selection from Computer Security and Cryptography [Book]. Euler’s Totient Function. 3) •Solving Congruences(4. Modular Arithmetic, Groups, Finite Fields and Probability. An eleven proof also exists. We present in particular how works the famous R. Back to Number Theory and Cryptography Primes, Modular Arithmetic, and Public Key Cryptography (April 15, 2004) Introduction. Course Description. Modular arithmetic is arithmetic for integers, where numbers wrap around when reaching a given value called modulus. Modular exponentiation; Turn (unit) Notes. In number theory, the more commonly used term is index: we can write x = ind r a (mod m) (read the index of a to the base r modulo. Because you already know how to do modular arithmetic even if you've never heard of it before. So, modular division is a complex and necessary operation at the same time, and also it is. The fundamental arithmetic operations performed by most computers are actually modular arithmetic, where the modulus is 2 b (b being the number of bits of the values being operated on). Today, everyone uses cryptography on a daily basis: Examples include unlocking a car or garage door with a remote-control device, connecting to a wireless LAN, buying goods with a credit or debit card in a brick and mortar store or on the Inter-net, installing a software update, making a phone call via voice-over-IP, or paying. Now we proceed to see how modular arithmetic can be used to deal with other familiar algebraic constructions such as functions and equations, and how it can help us to better understand primes and composite numbers. In this module we will learn the modular arithmetic, the Euler Totient Theorm, the RSA Asymmetric Crypto Algorithm, use OpenSSL. GetInt32(Int32, Int32) Generates a random integer between a specified inclusive lower bound and a specified exclusive upper bound using a cryptographically strong random number generator. Learn Math Tutorials Recommended for you. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the. Many public-key algorithms heavily depend on modular arithmetic. Cryptographic applications for computing asymmetric keys may use the present implementations to advantage. The most commonly used one is of course the normal clock. This project for my Linear Algebra class is about cryptography. numbers along with an auxiliary value. This means that every operation within the basic arithmetic works modulo a chosen prime number. This morning I got a bit of a surprise when he asked me what modular arithmetic has to do with cryptography. This new results allow. Home Conferences GLSVLSI Proceedings GLSVLSI '08 Reconfigurable solutions for very-long arithmetic with applications in cryptography. They identi-ﬁed common sub-string and computed modular multipli-cation only once instead of repeating another time. Important! Selecting a language below will dynamically change the complete page content to that language. Modular Multiplication Modular Arithmetic is even more useful when we are dealing with multiplication. Learn Math Tutorials Recommended for you. So if k = 3, we have an example of the following table to find the ciphertext. 4 Modular Arithmetic We begin the chapter on number theory by introducing modular integer arithmetic. Johann Carl Friedrich Gauss is usually attributed with the invention/discovery of modular arithmetic. Modular Arithmetic, Groups, Finite Fields and Probability. Throughout the semester, we will post additional reference material here: Examples of Proofs in Cryptography. Modular arithmetic basics Arithmetic over a finite domain (a. Bitcoin cryptography library. To calculate the value of n mod m, you take. The solution - hashing. Modular arithmetic is a system of arithmetic for integers where numbers wrap around upon reaching a certain value - the modulus. , 16 or 32 bits) For crypto, need to work with integers that are much longer (e. Starting at noon, the hour hand points in order to the following: This is the way in which we count in modulo 12. The methods for performing a Monoalphabetic Shift Cipher are:. Cours 1 : Application of the elementary Number Theory to cryptography (Alain TOGBE, Westville, USA) This course will use the basics of the elementary number theory to introduce cryptography. Learn Math Tutorials Recommended for you. Multiprecision modular arithmetic is needed in cryptography, for example, to implement RSA encipherment. We use trapdoor-functions: functions that are easy to compute in one direction, yet difficult to compute in the opposite direction (finding its inverse). Modular arithmetic can be tricky, so in case you didn't follow what just happened, we are going to walk through it now. We now have a good definition for division: $$x$$ divided by $$y$$ is $$x$$ multiplied by $$y^{-1}$$ if the inverse of $$y$$ exists, otherwise the answer is. A special form of a modulus and a modified Barrett reduction method are used to perform modular arithmetic in a cryptographic system. Modular arithmetic Much of modern number theory, and many practical problems (including problems in cryptography and computer science), are concerned with modular arithmetic. The most commonly used one is of course the normal clock. It is often used in calculating checksums and check digits. Applications of modular arithmetic Hashing, pseudo-random numbers, ciphers. Each letter of the alphabet will be represented by a number from 0 to 25. Deals only in integer numbers and the process can be done in a very fast manner. Elliptic Curve Cryptography ECC is based on the algebra of elliptic curves (like y2 = x3 + ax + b) over finite fields instead of being based on simple finite groups like modular arithmetic. Because of its signiﬁcance we need a brief lesson on some of the ideas. Bit::Vector is a (stand-alone) C library and an object-oriented Perl module (with overloaded operators) which allows you to handle bit vectors, sets (of integers), "big integer arithmetic" (e. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications. Gopinath Ganapathy and K. First we have to talk a bit about the problem of integer factorization. Named after its creators Whitfield Diffie and Martin Hellman, it's simply called the Diffie-Hellman key exchange. ECC operates on elliptic curves over finite fields. Modern cryptosystems, used in numerous applications that require secrecy or privacy - electronic mail, financial transactions, medical-record keeping, government affairs, social media etc. 5 Modular arithmetic. Modular arithmetic can be tricky, so in case you didn't follow what just happened, we are going to walk through it now. a computer arithmetic). To create their general-purpose elliptic-curve chip, the researchers decomposed the cryptographic computation into its constituent parts. 15 hours after 10:00 isn't 25:00. • For example, 2 7 (mod 5). It uses recursive approach that uses various number theory algorithm to perform basic operation. Donate or volunteer today! Site Navigation. and R is 4294967296. Section 2: The Real World of Modular Arithmetic: Check Digits and Check Sums UPCs, ISBNs, and bank accounts numbers are all examples of modular arithmetic in the real world. Get this from a library! Cryptography arithmetic : algorithms and hardware architectures. We have a kind of long arithmetic calculation here, and we are asked about the result modulo some number ( result % M in many languages). First we have to talk a bit about the problem of integer factorization. Cryptography is the process of hiding a message by encoding it in a reverseable (decodable) way. Modular Arithmetic and RSA Encryption Stuart Reges Principal Lecturer University of Washington Some basic terminology Alice wants to send a secret message to Bob Eve is eavesdropping Cryptographers tell Alice and Bob how to encode their messages Cryptanalysts help Eve to break the code Historic battle between the cryptographers and the cryptanalysts that continues today Public Key Encryption. In our examples so far we've been assigning numbers to each letter of a plaintext and then using modular arithmetic to construct a cipher, number by number (or letter by letter). Submit a legible hard copy of your answers in the classroom by 4 pm on the date specified. divided by some particular number. What is Modular Arithmetic - Introduction to Modular Arithmetic - Cryptography - Lesson 2 - Duration: 4:48. For example, because of the rotational orbit which the earth makes around the sun which is supposed to take place in 24 hours, our clock system cycles at 24-hour period. We present in particular how works the famous R. Arithmetic and modular operations: • Arithmetic addition, subtraction, multiplication, and comparison • Modular addition, subtraction, and reduction & inversion • Montgomery multiplication Thanks to these operations, the PKA supports many standard Pubic Key algorithms: Modular Exponentiation, CRT exponentiation, RSA cryptography, Elliptic. Number theory, in turn, is foundational for cryptography, the science of keeping one's communications and data secure from eavesdropping third parties. Many public-key algorithms heavily depend on modular arithmetic. Submit a legible hard copy of your answers in the classroom by 4 pm on the date specified. So the theory of congruences will be gently used to introduce cryptography. 3, 2016 (Fast Powering) Oct. 5: Public Key Cryptography. If the time is 7:00 now, then 8 hours later it will be 3:00. For examples and exceptions, see the Perl documentation on. This system is based in the notion of congruence and residue classes, which we’ll brie y address later (the interesting details of which are beyond the scope of this paper). Pages 3-25. It is commonly used in cryptographic schemes that are based on the hardness of the Ring Learning With Errors (R-LWE) problem to e ciently implement modular polynomial multiplica-tion. In our examples so far we've been assigning numbers to each letter of a plaintext and then using modular arithmetic to construct a cipher, number by number (or letter by letter). • Write a b (mod m). Modular Arithmetic Part-1 Explained with Examples in Hindi Cryptography lecture series. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as-. Modular Exponentiation. Modular arithmetic is useful in many fields, such as cryptography, computer science, and music. However, here's where I'm getting stuck. The talk is "Modular Arithmetic based Circuits and Systems for Emerging Technologies and Applications: Deep Neural Networks and Cryptography". Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme. Become a member and unlock all Study Answers. Pages 3-25. In this course, we study jointly fundamental notions of arithmetic and cryptography. , 16 or 32 bits) For crypto, need to work with integers that are much longer (e. the corresponding chaotic maps in the next section 6. Mashable CMS. : Image Encryption and Decryption Using Chaotic Maps and Modular Arithmetic. Typically the time on some internal clock is used • Compute x n+1 = ax Private Key Cryptography. If we apply the exponent after simplifying, we may need to simplify again!. Cryptography is the study of codes - writing codes, solving codes, and manipulating codes. Information about the audience, dates and location. Introduction. Description. A special form of a modulus and a modified Barrett reduction method are used to perform modular arithmetic in a cryptographic system. RNS Modular Arithmetic: Introduction and Cryptographic Applications Karim Bigou CNRS { IRISA { CAIRN May 29, 2015 Karim Bigou RNS for Asymmetric Cryptography May 29, 2015 1 / 40. We will then study RSA and Diffie-Hellman. Modular arithmetic is a branch of number theory that is useful in its own right and as a tool in such disciplines as integer factorization, calendrical and astronomical calculations, and cryptography. Primality test. In our examples so far we've been assigning numbers to each letter of a plaintext and then using modular arithmetic to construct a cipher, number by number (or letter by letter). Shyamsunder et al. Elliptic-curve cryptography relies on modular arithmetic, meaning that the values of the numbers that figure into the computation are assigned a limit. In this chapter, Stephenson expands on the fundamental mathematics behind the Enigma machine: modular arithmetic. Description¶. --- Sure he can use e. Biology; Cryptography Expand/collapse global location 16. This is a field of order 7 using modular arithmetic modulo 7. It was used during the Civil War in the 1860's and even thousands of years ago during Caesar's Roman Empire. Modular Arithmetic from CryptoClub book Applications of Modular Arithmetic Presentations: Calendar problems Chapter 8 Modular Arithmetic BIG IDEA: modular arithmetic Multiplicative cracking 65 Modular Arithmetic: Reducing mod m Affine Ciphers 67 Modular Arithmetic: Rules 69 Modular Arithmetic: More Congruence Equations. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications. Mani, Maximization of Speed in Elliptic Curve Cryptography Using Fuzzy Modular Arithmetic over a Micro-controller based Environment, Proceedings of the World Congress on Engineering and Computer Science, vol. There are five videos, each between 4 and 8 minutes in duration. Cryptography involves very large numbers! Standard (unsigned) integers in C are small, fixed length (e. Encryption by the sender takes the plaintext and a key and produces ciphertext. , 16 or 32 bits) For crypto, need to work with integers that are much longer (e. This is, in fact how HTTPS works - it uses public key cryptography to establish a secret key between two parties. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This popular course is designed for for mathematical enrichment for students in grades 4-6. This kind of mathematics is used in defending the United States and other countries during wars and other times when there is information that has to be kept confidential. It was also, in its deep relationships to group and ﬁeld theory, one of the most beautiful. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the. This inversion algorithm is an improvement on the available Euclidean algorithm, incorporating the use of the scaled modulus and proving to be of high performance and efficiency for hardware implementation. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications. This is particularly interesting for modular arithmetic used in cryptography. 3 Integer Division 2. Modular Arithmetic, Groups, Finite Fields and Probability. the modular world. Substitution ciphers. We will need this algorithm to fix our problems with division. Many public-key algorithms heavily depend on modular arithmetic. A type of cryptography that cannot secure correspondence until after the two parties exchange keys. In the ﬁrst section, we established the basics of modular arith-metic. Learn Math Tutorials Recommended for you. 3, 2016 (Euclidean Algorithm) Oct. the modular world. Here 6 means 6 o’clock and 9 means 9 hours later it will be 3:00. In the questions below, if it specifies an alphabetic cipher, then the original map used letters only: ABCDEFGHIJKLMNOPQRSTUVWXYZ. Modular arithmetic occurs on the Galois Field of size n, which means that all operations use operands and produce results in the range 0. We will use the standard 26­letter English alphabet, with. MEETING 6 - MODULAR ARITHMETIC AND INTRODUCTORY CRYPTOGRAPHY 3 Use the Euclidean algorithm to ﬁnd the multiplicative inverse of 23 (mod 17) and use it to ﬁnd all integers xthat satisfy 23x≡ 337 (mod 17). Modular reasoning. In this course, we will study the developments of code writing from ancient times to our modern era, and learn some of the classical ciphers, a modern cryptosystem, and certain aspects. For many years, artists have been using mathematical shapes based on formulas to create designs. Modular arithmetic is, indeed, used in cryptography, but the question is strangely mixing two different levels of the conceptual hierarchy. I have used the mod command and know what the results mean, but I don't understand the theory behind it and what is actually happening. Reference Material. RSA is a public key encryption algorithm based on exponentiation using modular arithmetic to use the scheme, first generate keys: Key-Generation by each user consists of: selecting two large primes at random (~100 digit), p, q calculating the system modulus R=p. The difference, (a - b), is 32 - 11 = 21. Bitcoin cryptography library. public-key cryptosystems. Modular arithmetic (sometimes referred to as clock arithmetic) is much easier to work with than the \standard" arithmetic we generally use. What is Modular Arithmetic - Introduction to Modular Arithmetic - Cryptography - Lesson 2 - Duration: 4:48. Integer Arithmetic Arithmetic in Finite Fields Arithmetic of Elliptic Curves GCD Modular Exponentiation Primality Testing Integer Arithmetic In cryptography, we deal with very large integers with full. Example 1 is an apparatus for providing modular exponentiation optimization for cryptographic systems comprising 1) a first register to store a plurality of components of cryptographic message; and 2) a microprocessor coupled to the first register to determine a plurality of components for a second power of the cryptographic message using the plurality of components of the cryptographic message, wherein the determining of the plurality of components for the second power of the cryptographic. Modular arithmetic properties Congruence, addition, multiplication, proofs. Modular Exponentiation • Problems of the form c = be modm given base b, exponent e, and modulus m • If b, e, and mare non-negative and b< m, then a unique solution cexists and has the property 0 ≤ c< m • For example, 12 = 52mod 13 • Modular exponentiation problems are easy to solve, even for very large numbers. Named after its creators Whitfield Diffie and Martin Hellman, it's simply called the Diffie-Hellman key exchange. Playing with numbers has very important uses in cryptography. In this work, a hardware implementation of modular arithmetic and group operations over the prime field for an Elliptic Curve Cryptography Processor (ECP) for an efficient security system is proposed. Modular exponentiation; Turn (unit) Notes. It involves encryption, in which a message, called plaintext, is converted into a form, called ciphertext, that may be sent over channels possibly open to view by outside parties. 2 Binary Operations 2. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 5, 2016 (Introduction to Modern Cryptography and RSA) Oct. The elliptic curve arithmetic is basically the same but works on complex numbers. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. The length of the circular number line in modular arithmetic is called the modulus. For example, an RSA private key operation using a 2048-bit key requires modular exponentiation of 1024-bit integers, assuming the use of the Chinese Remainder Algorithm. com Modular arithmetic is a system of arithmetic for integers where numbers wrap around upon reaching a certain value – the modulus. D&D Beyond. But it is from 1996 and in some aspects outdated (f. Uses modular arithmetic and elementary number theories to perform computations using two large prime numbers. We will need this algorithm to fix our problems with division. In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db), modular arithmetic is used in the consideration of the twelve tone equally tempered scale. A new approach is used to implement elliptic curve cryptography (ECC) over prime finite fields. A passage that I found especially intriguing in Neal Stephenson’s Cryptonomicon, is when Waterhouse and Turning are on a bike ride in the English countryside. plications in computer arithmetic, e. That is, 7 mod 3 equals 1 because 1 is what remains after dividing 3 into 7. Modular Arithmetic, Groups, Finite Fields and Probability. Modular arithmetic is useful in these settings, since it limits numbers to a prede-. In non-modular arithmetic the number always equals 15, but it does not work that with modular arithmetic. Modular arithmetic is quite a useful tool in number theory. Modulo (disambiguation) and modulo (jargon) – many uses of the word modulo, all of which grew out of Carl F. So for example 9 mod 4 = 1 because 9=2×4+1. 2 Modular Arithmetic Modular arithmetic has a variety of applications in cryptography. Modular arithmetic is a fundamental, elem. 15 hours after 10:00 isn’t 25:00. Learn Math Tutorials Recommended for you. By using modular arithmetic, Diffie-Hellman allowed two or more parties to exchange secret messages without sharing a secret key in advance. Math3024 Elementary Cryptography and Protocols Semester 1 Exercises and Solutions for Week 10 2004 Modular Arithmetic Reduction modulo a polynomial g(x) or modulo an integer m plays a central role in the mathematics of cryptography. ) Elliptic Curve k Q modular multipl. For security reasons, the bit length of the modulus n is typically in the range 5122048 bits or even more, and thus efficient long integer modular arithmetic is required for achieving high throughput rates at these bit precisions. The RSA cipher (named after its creators, Rivest, Shamir,. An eleven proof also exists. We can make a more “mathy” version by introducing some facts about modular arithmetic: Modular arithmetic finds the remainder of a division problem. The key idea used in fuzzy modular arithmetic is not to compute the result exactly as in the traditional modular arithmetic because the traditional Maximization of Speed in Elliptic Curve Cryptography Using Fuzzy Modular Arithmetic over a Microcontroller based Environment. Learning with errors (LWE) is essentially linear regression with modular arithmetic. The remainder of the paper is organized as follows. Around the time when RSA was invented, another public-key cryptography algorithm was born. Use CAS for symbolic math (e. Modular arithmetic is a system of arithmetic for integers, where values reset to zero and begin to increase again, after reaching a certain predefined value, called the modulus (modulo). Euclidean Algorithm provides GCD and MMI. MEETING 6 - MODULAR ARITHMETIC AND INTRODUCTORY CRYPTOGRAPHY 3 Use the Euclidean algorithm to ﬁnd the multiplicative inverse of 23 (mod 17) and use it to ﬁnd all integers xthat satisfy 23x≡ 337 (mod 17). An answer to the symmetric Key Distribution problem Based on Public Keys and Private Key pairs Conﬁdentiality. divisibility, and modular arithmetic arguments; Perform. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. , 16 or 32 bits) For crypto, need to work with integers that are much longer (e. 3 Modular Exponentiation Modular arithmetic is used in cryptography. This method of working with remainders is at the core of mathematics called modular arithmetic. 3 2The next article will give you an indication of how amazingly di cult this is 3Or SSL. Modular exponentiation; Turn (unit) Notes. Modulo (disambiguation) and modulo (jargon) – many uses of the word modulo, all of which grew out of Carl F. Often, the two parties who want to communicate in secret need to share a common piece of information, called a key Since messages are often encoded as numbers, the key is typically an integer. In the questions below, if it specifies an alphabetic cipher, then the original map used letters only: ABCDEFGHIJKLMNOPQRSTUVWXYZ. It was used during the Civil War in the 1860's and even thousands of years ago during Caesar's Roman Empire. Modular arithmetic is much more efficient than non-modular arithmetic as well; you can easily get a feel for this yourself, what's the last digit of $2459 \times 9823$ or $9823^4$? Finally, we don't need to have an asymmetric primitive such as RSA to handle large plaintext/ciphertext; we can use symmetric primitives for that, creating a hybrid. You now know one of the most important topics in number theory modular arithmetic. Apart from that I introduce everything needed from scratch. Applications of congruences: Page 292: 2 c & d, 6 (use seed 4). So for example 9 mod 4 = 1 because 9=2×4+1. Biology; Cryptography Expand/collapse global location 16. The main advantage of elliptic curves is their efficiency. Learning with errors (LWE) is essentially linear regression with modular arithmetic. After the activity, they investigate the Diffie-Hellman method using prime numbers, exponents, and modular. Repeated use of a key makes it easier to crack Key Distribution. 3 Modular Exponentiation Modular arithmetic is used in cryptography. modular arithmetic operations such as division and multiplication over finite fields GF(p) and GF(2n) are heavily used in several public-key cryptographic algorithms that are used to provide security services in many applications. 3 Integer Division 2. We’ll use many ideas developed in Chapter 1about proof methods and proof strategy in our exploration of number theory. Modular arithmetic also plays an important role in abstract algebra, cryptography, computer science, and the visual and musical arts. They comprise the. For example, because of the rotational orbit which the earth makes around the sun which is supposed to take place in 24 hours, our clock system cycles at 24-hour period. Modular Arithmetic has a wide usage in the field of Computer Science as well as Cryptography. Syllabus Errata for the textbook Thursday, June 2: Introduction to Cryptography, Substitution Ciphers (1. Donate or volunteer today! Site Navigation. "From the exciting history of its development in ancient times to the present day, Introduction to Cryptography with Mathematical Foundations and Computer Implementations provides a focused tour of the central concepts of cryptography. In this modular arithmetic and cryptography lesson, students use applets to explore modular arithmetic using a clock and Caesar Ciphers. Cliff McCullough. By using modular arithmetic, Diffie-Hellman allowed two or more parties to exchange secret messages without sharing a secret key in advance. Questions tagged [modular-arithmetic] Are the elements of a modular ring simply the set of all the numbers from 1 to p−1? cryptography modular-arithmetic. Encryption by the sender takes the plaintext and a key and produces ciphertext. We further present a new record for modular multiplications per second on a single desktop computer as well as a new record for the ECM factoring algorithm. Perl usually uses arithmetic modulo operator that is machine-independent. Almost any cipher from the Caesar Cipher (Julius Caesar) to the RSA Cipher use modular arithmetic. Modular arithmetic is useful in these settings, since it limits numbers to a prede-. In other words, modular arithmetic is a method for doing addition, subtraction, multiplication, and division with integers where the numbers curve around the number line cyclically instead of continuing on forever. In the case of AES encryption, when it is implemented in hardware, perhaps 10 more more encrypters operating in parallel, reducing the gate count is important. The latter operation is the most interesting one and creates a complicated structure on integer numbers. The following statement is a basic principle of modular arithmetic: a + kp = a (mod p). Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. Imagine a finite range of numbers, for example, 1 to 12. 3 A MODULAR ARITHMETIC KNAPSACK PROBLEM The first example of a public-key cryptosystem used a variant of the knapsack problem that results when integer arithmetic is replaced by modular arithmetic. equation solving using variables), and exact math (e. In Step (2), we multiplied the top row by 15 because, looking at our table , we see that 15 is the multiplicative inverse of 2, our current pivot. Pages 3-25. Many public-key algorithms heavily depend on modular arithmetic. MULTIPLICATIVE INVERSES in CRYPTOGRAPHY Modular arithmetic and inverses can be used to encrypt/decrypt messages, digital signatures, et c, as follows. Secondly, always use the most convenient residue (recall Definition 4. Modular arithmetic is a system of arithmetic for integers, where values reset to 0 after reaching a certain predefined value. Substitution ciphers. We can make a more “mathy” version by introducing some facts about modular arithmetic: Modular arithmetic finds the remainder of a division problem. - are based on sophisticated mathematics and algorithms that in implementation involve much computer arithmetic. Definition 6 Two integers a and b are said to be Congruent Modulo N if and only if N|(a−b), i. For many years, artists have been using mathematical shapes based on formulas to create designs. This cipher is a direct application of modular arithmetic using the equation. • For example, 2 7 (mod 5). Explains simple encoding and decoding of messages for student learning of modular arithmetic. Modular arithmetic. Since messages are often encoded as numbers, the key is typically an integer. ized modular arithmetic on contemporary embedded platforms, including cycle count performance and analysis of the instruction set. Effective cryptography is an essential tool in a network environment for addressing these concerns. What this means is that x and y leave the same remainder when divided by m. it is sometimes called the nine proof. modular arithmetic. Description¶. Modular Arithmetic in Cryptography Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Modular arithmetic. Today, the same concept is applied to computer graphics, as well as sculptures and modern paintings. Nov 15, 2017 Cryptography 101: Mathematical basis (I) - Modular arithmetic. Let's begin with what is called a shift cipher. I will discuss a simple method of enciphering and deciphering a message using matrix transformations and modular arithmetic, and show how elementary row operations can sometimes be used to break an opponent's code. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. numbers along with an auxiliary value. Let me show you. This method uses a discard-and-retry strategy to avoid the low value bias that a simple modular arithmetic operation would produce. The Multiplicative Cipher. Though modular arithmetic is best known for its use in telling time, we also use it in many other circumstances. 3 2The next article will give you an indication of how amazingly di cult this is 3Or SSL. Browse other questions tagged matrices modular-arithmetic inverse cryptography or ask your own question. Modular Arithmetic Congruences and Cryptography Cryptology is the study of secret messages. We can find these by looking through a multiplication table, or by a simple extension of the Euclidean Algorithm. Is an internet encryption and authentication system. We use an infinite system while he uses finite systems. I will discuss a simple method of enciphering and deciphering a message using matrix transformations and modular arithmetic, and show how elementary row operations can sometimes be used to break an opponent's code. , 1024{4096 in prime elds p prime exponentiation (Di e-Hellman, DSA, etc. Perl usually uses arithmetic modulo operator that is machine-independent. , rely on the efficient realization of arithmetic over finite fields of the form GF(2n), where n € Z and n ≥ 1,or the form GF(p), where p a prime. Substitution ciphers. Modular exponentiation; Turn (unit) Notes. 1000 times faster than modular exponentiation Diffie-Hellman protocol (by itself) does not provide. 3 A MODULAR ARITHMETIC KNAPSACK PROBLEM The first example of a public-key cryptosystem used a variant of the knapsack problem that results when integer arithmetic is replaced by modular arithmetic. Modern Cryptography Number Theory Modular Examples 123 3 (mod 10) 287569832 2 (mod 10) 5 1 (mod 4) 33 1 (mod 4) 1 3 (mod 4) 8 0 (mod 4) 4 0 (mod 4) Modern Cryptography Number Theory Modular Inverses Standard arithmetic operations + ; Problems with division: 2x 1 (mod 7) vs 1 (mod 6) When do inverses exist? Modern Cryptography Number Theory. The computational demands of modern cipher systems center around ordinary arithmetic of large integers. Modular arithmetic is a quick way to calculate various intervals between pitches or pitch classess. Hill who introduced them in two papers. Johann Carl Friedrich Gauss is usually attributed with the invention/discovery of modular arithmetic. They are also used in several integer factorization algorithms based on elliptic curves that have applications in cryptography, such as Lenstra elliptic-curve factorization. x-1 = 1 ! Fermat’s Little Theorem ! For any prime p that does not divide an integer ‘a’ a(p-1) = 1 mod p Public-key Cryptography: Background Mathematics Modular arithmetic and multiplicative inverse ". 1000 times faster than modular exponentiation Diffie-Hellman protocol (by itself) does not provide. A significant number of applications including cryptography, error correction coding, computer algebra, DSP, etc. The numbers n and e are used for encrypting. Mod Arithmetic Modular Arithmetic is a system of arithmetic for integers where numbers "wrap around" after they general we use prime numbers a lot in Cryptography. A special form of a modulus and a modified Barrett reduction method are used to perform modular arithmetic in a cryptographic system. A 12-hour clock provides the most common example of the wrap-around nature of modular arithmetic. How do I solve this? The answer is given as 1789569707. Modular arithmetic is quite a useful tool in number theory. I will discuss a simple method of enciphering and deciphering a message using matrix transformations and modular arithmetic, and show how elementary row operations can sometimes be used to break an opponent's code. public-key cryptosystems. Modular exponentiation is a type of exponentiation performed over a modulus. 3 A MODULAR ARITHMETIC KNAPSACK PROBLEM The first example of a public-key cryptosystem used a variant of the knapsack problem that results when integer arithmetic is replaced by modular arithmetic. Numerous public key cryptographic algorithms are available in the literature which comprise modular arithmetic modules such as modular addition, multiplication, inversion and exponentiation.