Uses Of Modular Arithmetic In Cryptography

We introduced Vigenere cipher, did an example, and began. In particular, modular multiplications are a critical. Modular arithmetic is a key ingredient of many public key crypto-systems. Modular arithmetic; 29. com MMU E-Learning Portal for Cryptography added a new photo. - mr-ma Mar 1 '15 at 11:44. average / arithmetic mean. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. Modular arithmetic is widely used in computer science and cryptography. Modular Arithmetic and Cryptography 12/28/09 Page 5 of 11 2. for cryptography), and boolean matrices (all of arbitrary size) very efficiently. Uses Of Modular Arithmetic In Cryptography. Maybe I give it a try on Crypto forum or edit my question with source code as you suggested. The methods we will cover in this paper can be easily picked up with some knowledge of basic arithmetic and some modular algebra. Fast Exponentiation algorithm. It is not only important for receiving transmissions from space probes, but much closer to. Principal wishing to participate in secure communication with others makes pair of keys, Ke and Kd, and keeps decryption key Kd secret. In some sense, modular arithmetic is easier than integer arithmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. In particular, we’ll get applications to divisibility tests, necessary conditions for solutions of various Diophantine equations (including non-solvability results), as well as an application to modern cryptography. Euclidean Algorithm provides GCD and MMI. These are some of the functions used in modular arithmetic and cryptography. First, our message M will be coded as an integer or block of integers. Show that casting out 3s is true for 73452 and 19548. 5 Knapsack Encipherment and Decipherment of ASCII-Plaintext. Use BigInteger. modular arithmetic rules pdf A. thanks anyway. Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a smaller range of numbers. Algorithms available for the use of most include MD5, SHA1, SHA2, SHA3, and others. Modular arithmetic. This means that every operation within the basic arithmetic works modulo a chosen prime number. Chapter 1: cryptography fundamentals; private key and public key cryptosystems; Chapter 2: Caesar. MODULAR ARITHMETIC, RSA ALGORITHM 59 (a) Units: elements with multiplicative inverse. The typical approach in finding is to use the extended Euclidean algorithm. It is often used in calculating checksums and check digits. modular arithmetic operations (they help evaluate numbers modulo m): addition: (a+ b)modm= amodm+ bmodm modm subtraction: (a b)modm= amodm bmodm modm multiplication: (ab)modm= amodmbmodm modm 12. The notation used for expressions involving modular arithmetic is: x = y (mod m) Which reads as "x is equivalent to y, modulo m". Back to Number Theory and Cryptography Primes, Modular Arithmetic, and Public Key Cryptography (April 15, 2004) Introduction. Let a;b;c 2Z with a 6= 0. The authors introduce the core principles of modern cryptography, with an emphasis on formal definitions, clear assumptions, and rigorous proofs. Cryptography deals with the actual securing of digital data. Both the sending party and the receiving party has a private and public key. Copy the code using the icon below and paste it into the item you are classifying only if you have fully classified the item. We have already discussed modular arithmetic and Diffie-Hellman key exchange algorithm. Intermediate Python Skills 6. In this section of Data Communication and Networking - Cryptography MCQ (Multiple Choice) Based Questions and Answers,it cover the below lists of topics. Questions tagged [modular-arithmetic] Ask Question Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value… the modulus. Modular reasoning. Shyamsunder, Ganesan Kaliyaperumal* TIFAC-CORE in Automotive Infotronics, VIT University, Vellore, Pincode-632014, India. A Math: Modular Arithmetic (and Cryptography) module has been added to AEA PD Online's Student Personalized Learning system which can be paired with teacher learning of the module Mathematics: Networks (Vertex-Edge Graphs) Cluster found in the Training System. Why is modular arithmetic used in cryptography? Cryptography: Cryptographical Techniques are those that help to maintain the security of systems. Use modular arithmetic to solve 13999 mod 10 2. [57] propose algorithms for incomplete modular arithmetic, namely modular addition, subtraction and multiplication using Montgomery arithmetic for 2. range of technologies. I'm using NTL for modular arithmetic and have double checked all values using online modulo calculators. And that's what we're going to do now. Several important cryptosystems make use of modular arithmetic. To encrypt our integer message M, pick a (large) modulus m and an encryption key e that is an integer that is relatively. Introduction history of encryption; 33. Material icons are delightful, beautifully crafted symbols for common actions and items. Use Cases of Public-Key Cryptography. 3 (primality testing) and 1. The chapter has several objectives: To review integer arithmetic, concentrating on divisibility and finding the greatest common divisor using the. the hours on a clock face. Bob, likewise,. example as used in keeping information secure through cryptography. for cryptography), and boolean matrices (all of arbitrary size) very efficiently. For instance, in cryptography the use of modular arithmetic permits some operations to be carried out more quickly and with fewer storage requirements, while other operations become more difficult. So you please anyone help me how to use Galois Field in MATLAB. Analogs to discrete logarithm and other one-way functions suitable for public key crypto exist in EC. Convention tends to use 0 to represent the even numbers and 1 to represent the odd ones. I am currently doing project on Cryptography for images. Introduction The purpose of this lecture is to introduce you to terminology used in microbiology. For the computations involving modular arithmetic one can use GAP. Modular arithmetic was systematized by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. , smaller than \$2^{62}\$) and one for the others. Show that casting out 3s is true for 73452 and 19548. Step (1) is our augmented matrix that we will use to find the inverse of A. Its use is purely in passing the type annotation into the function natVal so that the type level integer n representing the modulus can be extracted and used at value level to actually do the modulus operation. Using Cryptography hashing Show how the ID number 1111111111111 of a student with 13 data items can be hash checked for storage errors 3. Yanik et al. Diffie-Hellman continued; 30. Cryptography has been used in many technologies. Modular Arithmetic Operations • can perform arithmetic with residues • use a finite number of values, and loop back from either end Zn={0,1,,(n –1)} • modular arithmetic is doing addition and multiplication and modulo reduce answer • can do reduction at any point, i. The field of cryptography makes heavy use of number theory and congruencies Cryptography is the study of secret messages Uses of cryptography: Protecting medical records Storing and transmitting military secrets Secure web browsing … Congruencies are used in cryptosystems from antiquity, as well as in modern-day algorithms. The basic idea behind the autokey Vigenère cipher can, however, be used to produce a more secure cipher. Modular arithmetic has a variety of applications in cryptography. • 3+8 mod 9 • 21+20 mod 35 • 5 9 mod 11 • 5 5 mod 24 • 2 3 mod 6 • 7 3 5 mod 10 • 23 mod 5 • 36 mod 8 • (5+7)3 mod 10. While the term itself may seem intimidating Therefore, cryptography requires one-way hash functions. This site provides order information, updates, errata, supplementary information, chapter bibliographies, and other information for the Handbook of Applied Cryptography by Menezes, van Oorschot and Vanstone. Introduction to Polygraphic Ciphers 10. Modular Arithmetic • define modulo operator “a mod n” to be remainder when a is divided by n • use the term congruence for: a = b mod n – when divided by n, a & b have same remainder – eg. 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. Hard to reverse operation Scalar Multiplication Group Operations Point Addition, Point Doubling Modular Arithmetic Addition, Subtraction, Multiplication and Inversion. It refers to the design of mechanisms based on mathematical algorithms that provide fundamental information security services. It is not only important for receiving transmissions from space probes, but much closer to. First divide 4294967296 by 1073741827 and find the remainder. In general, Cryptography is a hidden writing. If you have any questions, please contact. Janet Beissinger and Vera Pless (AK Peters, 2006). Historically, cryptography was used to send important messages which were only meant for a select set of eyes. , (n - 1)} • modular arithmetic is when do addition & multiplication and modulo reduce answer • can do reduction at any point, ie. Affine Cipher 5. Modular arithmetic is the most fundamental operation in Elliptic Curve Cryptography. Mathematical Arrays Strings Dynamic Programming Hash Sorting Bit Magic Matrix Tree Searching Stack STL Greedy Graph Linked List Recursion Prime Number Java CPP Numbers Misc number-theory Binary Search Heap Binary Search Tree Queue Modular Arithmetic sieve Java-Collections DFS series sliding-window Combinatorial logical-thinking Map Fibonacci. For example, a number of techniques for doing modular multiplication and exponentiation are discussed in x14. In this paper we highlight the benefits of using genus-2 curves in public-key cryptography. The difference, (a - b), is 32 - 11 = 21. divided by some particular number. 122 relations. Caesar Cipher 4. 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. XOR mathematics. PETER MCNAMRA Bucknell University and Trinity College Dublin. Textbook Authors: Rosen, Kenneth, ISBN-10: 0073383090, ISBN-13: 978-0-07338-309-5, Publisher: McGraw-Hill Education. I'm using NTL for modular arithmetic and have double checked all values using online modulo calculators. (Of course one can also use 1110 - 1 = (11 - 1)(119 + 118 + gcd and the Fundamental Theorem of Arithmetic. and widely used example of public-key cryptography is the RSA system (Rivest, Shamir and Adleman, 1978). Or 13 mod 10 = 3 because 12=1×10+3. 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. back to readable. Since the public key can only be used to decrypt messages signed with Alice's private key, we can trust that Alice was the author of the original message. RSA encryption is widely used and is the de-facto encryption standard. Caesar and affine cipher later on). We benchmarked the library’s arithmetic operations over R. Blockchain,cryptography , and consensus free download Page 1. (MPE) is carried out via repeated use of modular polynomial multiplication (MPM) algorithm. The operation (to be performed between the two operands) is defined by an operator. Use security algorithms and protocols. 18), Cambridge University Press, November 2010, 236 pages. Modular arithmetic. Also, it can identify if the sequence is arithmetic or geometric. These Modular Arithmetic Properties are used in solving Modular Arithmetic related competitive questions. Multiplicative Ciphers 4. Let's begin with what is called a shift cipher. In particular, you are using a mod 27 system, since you are limited to the 27 integers between 0. Secondly, always use the most convenient residue (recall Definition 4. Selected Preprints and Reprints. If you have not taken Applied Finite Mathematics, then you will need to spend a considerable amount of time learning the material in the lesson. Most encryption techniques use rather simple mathematical formulas using functions such as modular arithmetic, but applied to very large numbers and in conjunction with other functions. PETER MCNAMRA Bucknell University and Trinity College Dublin. In competitive programming, Modular Arithmetic Properties are essential tools in solving big number problems. RSA uses arithmetic on integers at least 200 digits long. Modular Arithmetic and Cryptography Jeffhraim Balilla • Many real-world occurences operate in an unusually cyclical process. So, in a daily time basis, 15 hours added with 18 hours doesn't count 33 hours in. 4 (Cryptography specifically RSA). We're going to start off with Euler's. They report a speedup of around 10% in elliptic GF 1 p curve cryptography through incomplete modular arithmetic. CRYPTOGRAPHY We hope you understood the modular arithmetic lesson. Modular arithmetic. Modulo Challenge. But Cn+1 = Cn, and therefore the relationship holds. Hence why prime pmakes modular arithmetic. So called Fundamental theorem of arithmetic, known since Euclid, claims that factorization of an integer n into a power of. Elliptic-curve cryptography relies on modular arithmetic, meaning that the values of the numbers that figure into the computation are assigned a limit. 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. RSA Algorithm is used to encrypt and decrypt data in modern computer systems and other electronic devices. We recommend using the recipes layer whenever possible, and falling back to the hazmat layer only. Modular arithmetic; 29. Prerequisites. Modular Exponentiation. Cryptography deals with the actual securing of digital data. 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). Modular arithmetic was systematized by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Example: 7 MOD 4 = 3. Fast Exponentiation algorithm. We use this kind of arithmetic whenever we tell the time. , the greatest common divisor of modulus pand the number for which we seek an inverse must be one [8]. In the past the primary use of cryptography was for government and military intelligence, and this use continues to be important. Caesar cipher is a simple cryptography scheme, used by Julius Caesar to communicate with his generals. We’ll start with a bit of definition review and progress through some mathematical properties involving “mod”. 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. 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. What this means is that x and y leave the same remainder when divided by m. Textbook Authors: Rosen, Kenneth, ISBN-10: 0073383090, ISBN-13: 978-0-07338-309-5, Publisher: McGraw-Hill Education. 6 Basic Arithmetic Operations for Large Integers 14 - Addition and subtraction methods, and introduce applications in cryptography and various protocols. an Internet encryption and authentication system that uses an algorithm developed by Ron Rivest, Adi Shamir, and Leonard Adleman. For example 6 ≡ 1 (mod 5). ! Prerequisite Knowledge • Properties of real numbers • Long division Learning Goals Understand that: • Modular arithmetic is used to keep information secure on the Internet. Notes on Modular Arithmetic. Convention tends to use 0 to represent the even numbers and 1 to represent the odd ones. (In other words m (b− a)). ) • Some historical ciphers can be elegantly described with modular arithmetic (cf. useful non-cryptographic algorithms. CRYPTOGRAPHY We hope you understood the modular arithmetic lesson. It is often used in calculating checksums and check digits. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. N2 - finite fields have been used for numerous applications including error-control coding and cryptography. · Modular Arithmetic · But for the purposes of cryptography in this book, we’ll only be modding positive numbers. Context The activity requires addition, subtraction, multiplication and division. Playing with numbers has very important uses in cryptography. The people of these fields utilize a modular arithmetic calculator. Ma- jority of execution time of ECC is spent in performing modular operations on underlying elds. ECC operates on elliptic curves over finite fields. The sender has a message or plaintext. 3answers 343 views How to reverse in modular arithmetic. The RSA algorithm includes modular arithmetic as well as Montgomery modular exponentiation on the large numbers which have the key size of 1024 to 4096 bit typically. 1dw4l75al6qdqla aw1ts8c1b1osjlc zuium7oq8wlvl 7t4imyfu0x6jlj ewohcmswx6 zndwkdjb9zj ke7ka033dsivdx 7icbq5pgtt6. , 16 or 32 bits) For crypto, need to work with integers that are much longer (e. RSA keys are based on modular exponentiation modulo the product of two large primes. Using modular arithmetic with discrete logarithm produces the unlimited (or limited by our large numbers) possibilities of solutions when in fact only one of which is the one that has the capability of decrypting our ciphers. This volume contains the proceedings of the 12th conference on Arithmetic, Geometry, Cryptography and Coding Theory, held in Marseille, France from March 30 to April 3, 2009, as well as the first Geocrypt conference, held in Pointe-à-Pitre, Guadeloupe from April 27 to May 1, 2009, and the European Science Foundation exploratory workshop on Curves, Coding Theory, and Cryptography, held in. Affine Cipher 5. The applications of Modular arithmetic in cryptography are endless. InteropServices; The reference to InteropServices in the top of your class will allow ZeroMemory: this method will be used to remove the password from the memory, increasing the security of the encryption. This process is designed to use something called a Decryption Key. Almost any cipher from the Caesar Cipher to the RSA Cipher use it. (MPE) is carried out via repeated use of modular polynomial multiplication (MPM) algorithm. Theoretically speaking, modular arithmetic is not anyhow special. Encryption First, we need to specify our alphabet. In this section we will consider modular arithmetic and applications to cryptography and to generating “ random numbers ” by deterministic computers. Campbell R. modular arithmetic operations. Show that there are no integer solutions to w 6 + x6 + y 6 + z6 = 1012 + 7. range of technologies. I used the AJAX Javascript library for the rest of the codes. Modular Arithmetic. The two numbers can be literals Operators and Operands. 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). These are some of the functions used in modular arithmetic and cryptography. That can be a scary phrase but, for now, know that the encryption key consists of two integers and - expressed as - where is the exponent and is the modulus used in the encryption function. In ECC cryptography, elliptic curves over the finite fields are used, where the modulus p and the order n are very large integers (n is usually prime number), e. Hence why prime pmakes modular arithmetic. If you are interested in a Comprehensive Cryptography Course then check out this course! RSA Cryptosystem – Integer Factorization. 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. 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. This is an Open Source project, code licensed MIT. 18), Cambridge University Press, November 2010, 236 pages. The encryption and decryption operations. Public key cryptography; 28. 1 Modular Arithmetic \Modular arithmetic [] is a system of arithmetic for integers, where numbers \wrap around" after they reach a certain value - the modulus"2, here denoted 10 as N. In particular, we’ll get applications to divisibility tests, necessary conditions for solutions of various Diophantine equations (including non-solvability results), as well as an application to modern cryptography. We’ll use many ideas developed in Chapter 1about proof methods and proof strategy in our exploration of number theory. So called Fundamental theorem of arithmetic, known since Euclid, claims that factorization of an integer n into a power of. If n is a negative number then you. free C++ library for cryptography: includes ciphers, message authentication codes, one-way hash functions, public-key cryptosystems, key agreement schemes, and finite field arithmetics, including GF(p) and GF(2^n). Cryptography is ubiquitous and plays a key role in ensuring data secrecy and integrity as well as in securing computer systems more broadly. the hours on a clock face. Modular Arithmetic If you’ve ever wondered why you’re able to keep track of 12-hour clock easier then figuring out whether June has 30 or 31 days in it (it’s 30 by the way) its probably because they use modular arithmetic. The proposed processor supports dual-field computations, and employs various design techniques across the algorithm, architecture, and arithmetic circuit levels to minimize power and energy consumption. Da Vinci's cryptography invention is a tube with lettered dials. If you're using wireless (or a mobile phone), then all your data is being broadcast, and can be picked up by any wireless receiver in the vicinity. 4 Modular Arithmetic Modular Inverses Modular Exponentiation 5 Number-Theoretic Theorems Euler’s Totient Function Euler’s theorem 6 Number-Theoretic Algorithms Bezout’s identity Modular Multiplicative Inverses Modular Exponentiation Primality Testing 7 Discrete Logarithms 8 Summary Modes of Operations 7/84. The Euclidean Algorithm. Maybe I give it a try on Crypto forum or edit my question with source code as you suggested. Let a;b 2Z with a 6= 0. Applied Cryptography Updated: November, 2019 Page 3 Assignment 3 (9 points): Solve five modular arithmetic problems, and practice on the fundamental operations of the public cryptosystem. 3 + 9 = 12 ≡ 2 mod 10) Modular addition • Given: c = x + k mod n, with c, x, k ∈ Zn → if x + k < n : c = a + b → if x + k ≥ n : c = j, where x + k = i ⋅ n + j and j < n • Can be used. Modular Arithmetic Exponent Law 1 De nition 9 (Modular Arithmetic Exponent Law). It turns out that modular arithmetic is quite useful to represent those results. the modulo operation, consists of a simple conditional subtraction (addition) of the modulus. Real numbers can have an arbitrary number of digits, and computers have only a finite amount of memory. Compute the exponentiation [a. In this and other related lessons, we will briefly explain basic math operations. Aug 27, 2020 the arithmetic of elliptic curves graduate texts in mathematics. , the greatest common divisor of modulus pand the number for which we seek an inverse must be one [8]. Modular Exponentiation has two complex parts in it. Assorted Modular Arithmetic from CryptoClub book Applications of Modular Arithmetic Presentations: Calendar problems Chapter 8 Modular Arithmetic. 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. I'm using NTL for modular arithmetic and have double checked all values using online modulo calculators. The decrypted message Real-life encryption with public and private keys also uses modular arithmetic. It is often used in calculating checksums and check digits. A type of cryptography that cannot secure correspondence until after the two parties exchange keys. They can offer the same level of security for modular arithmetic operations over much smaller prime fields. But first let’s define a symbol to represent each group. Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm. The operation (to be performed between the two operands) is defined by an operator. Cryptography. It is not concerned with equality (which is represented by two lines, one on top of the other), but instead is One of the most useful types of cryptography developed is known as public-key cryptography, which is used in the present day for internet security. 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. PETER MCNAMRA Bucknell University and Trinity College Dublin. It is not only important for receiving transmissions from space probes, but much closer to. They report a speedup of around 10% in elliptic GF 1 p curve cryptography through incomplete modular arithmetic. In particular, you are using a mod 27 system, since you are limited to the 27 integers between 0. InteropServices; The reference to InteropServices in the top of your class will allow ZeroMemory: this method will be used to remove the password from the memory, increasing the security of the encryption. The applications of Modular arithmetic in cryptography are endless. Computing Computer science Cryptography Modular arithmetic. We’ll use many ideas developed in Chapter 1about proof methods and proof strategy in our exploration of number theory. Let's begin with what is called a shift cipher. As a first step towards finding such errors automatically, we will describe two modular arithmetic decision […]. But first let’s define a symbol to represent each group. ModularInverse [k, n] gives the number r such that the remainder of the division of r k by n is equal to 1. To use this plagiarism checker, please copy and paste your content in the box below, and then click on the big blue button that says "Check Plagiarism It is used and trusted by millions of people all around the world and can easily boast of being the single most sophisticated, feature-rich, user-friendly. I'm using NTL for modular arithmetic and have double checked all values using online modulo calculators. For decryption we can use d(x) = a-1(x-b) mod m. java that takes a command-line argument N and prints out an N-bit integer that is (probably) prime. Title: Modular Arithmetic 1 Modular Arithmetic. Modular arithmetic is useful in these settings, since it limits numbers to a prede-fined range f0;1;:::;N 1g, and wraps around whenever you try to leave this range — like the hand of a. This package provides a type for integers modulo some constant, usually written as ℤ/n. If we apply the exponent after simplifying, we may need to simplify again!. A Math: Modular Arithmetic (and Cryptography) module has been added to AEA PD Online's Student Personalized Learning system which can be paired with teacher learning of the module Mathematics: Networks (Vertex-Edge Graphs) Cluster found in the Training System. On this page, we collect resources about some basic cryptography, the mathematics of encrypting and decrypting (hopefully) secret messages, including some basic modular arithmetic. To calculate the value of n mod m, you take away as many multiples of m as possible until you are left with an answer between 0 and m. This is when the answer to a calculation is always in the range 0 m where m is the modulus. 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. One of the most interesting applications, to me, of modular arithmetic is in generating pseudo-random sequences. To see how it works, following along while Alice and Babette share a message. Aug 27, 2020 the arithmetic of elliptic curves graduate texts in mathematics. Discrete Mathematics and Its Applications, Seventh Edition answers to Chapter 4 - Section 4. The basic principle of RSA is the fact that it is practical to find three very large positive integers e,. This process is designed to use something called a Decryption Key. 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. Assignment 4 (9 points): Write a Python program to generate prime numbers and implement. Solving modular linear equations; Modular exponentiation; Chinese Remainder Theorem; Discrete Logarithm Problem; Quadratic Residues; Key Exchange. Several important cryptosystems make use of modular arithmetic. 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). Proprietary cryptography is currently deployed in many products which are used on a daily basis by the majority of people world-wide. An Introduction to Cryptography Exercises for Chapter 1 Section. We talked about modular arithmetic. CODING THEORY. For example, a number of techniques for doing modular multiplication and exponentiation are discussed in x14. It is embedded in the computational core of many wireless and contactless devices used in access control systems and vehicle immobilizers. - mr-ma Mar 1 '15 at 11:44. As an implied precondition, parameters must be integers unless otherwise noted. A generic approach to searching for Jacobians , Mathematics of Computation 78 (2009), 485-507. 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. Arithmetic Geometry, co-editor with Gary Cornell, Springer-Verlag, 1986. A Second View of Modular Arithmetic: \Z_{𝑛} and 𝑈_{𝑛} 73 86; 4. 2015 Talk RNS modular multiplication through reduced base extensions at ASAP, Jun. All these bits are stored in status or flag ALU used four general purpose register. Extension of Euclid's Algorithm. 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. The typical approach in finding is to use the extended Euclidean algorithm. 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. Elliptic Curve Cryptography, however, relies on arithmetic within a finite set of numbers. Cryptography is used to secure and protect data during communication. Private key cryptography. In computer science, modular arithmetic is often applied in bitwise. Similarly, no matter how many hours you count off on a traditional analog clock. Using Cryptography hashing Show how the ID number 1111111111111 of a student with 13 data items can be hash checked for storage errors 3. Talk RNS Modular Arithmetic: Introduction and Cryptographic Applications at Séminaire sécurité des systèmes électoniques embarqués, Sep. Many public-key algorithms heavily depend on modular arithmetic. That can be a scary phrase but, for now, know that the encryption key consists of two integers and – expressed as – where is the exponent and is the modulus used in the encryption function. We know that 11 mod 5. Affine Ciphers and Linear Congruences 66 79; 3. modular arithmetic operations (they help evaluate numbers modulo m): addition: (a+ b)modm= amodm+ bmodm modm subtraction: (a b)modm= amodm bmodm modm multiplication: (ab)modm= amodmbmodm modm 12. Introduction. Blockchain,cryptography , and In this paper we propose an efficient modular multiplication algorithm for primes of the form p= 2· 2 a 3 It uses the human vision to decrypt the encrypted images without any cryptographic computations. It turns out that modular arithmetic is quite useful to represent those results. Modular arithmetic: Goal: Computation in finite sets (used in about 95% of modern cryptography) Finite sets are often represented in circles, e. Proprietary cryptography is currently deployed in many products which are used on a daily basis by the majority of people world-wide. Modulo 2 addition is performed using an exclusive OR (xor) operation on the corresponding binary digits of each operand. Modular arithmetic is useful in these settings, since it limits numbers to a prede-fined range f0;1;:::;N 1g, and wraps around whenever you try to leave this range — like the hand of a. Bob, likewise,. 3 (primality testing) and 1. A flowchart can be helpful for both writing programs and explaining the program to others. Modular arithmetic is useful in many fields, such as cryptography, computer science, and music. 4 Modular Arithmetic We begin the chapter on number theory by introducing modular integer arithmetic. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms. Finally, in music theory, modulo 12 arithmetic is. Modular arithmetic was systematized by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. However, the laboratory exercise at the end of this chapter contains an interesting application to cryptography called Hill codes. So we need something much more fundamental in order to create a scheme that is really secure. Secret key cryptography uses one key across both the sending encryption and the receiver's decryption. As well as the analysis of letter frequencies, other patterns can also be detected that may help to decipher a piece of ciphertext. Students explore the concept of modular arithmetic and cryptography. These are some of the functions used in modular arithmetic and cryptography. De nition 5. Extension eld arithmetic are very important to improve the perfor-mance of pairing. What we did in class: We took up the Modular Arithmetic User's Manual Concept Check Sheet and talked about some subtleties of modular arithmetic. Web Cryptography API. Although modular arithmetic is often denoted using (mod n) after an equation, here we will use in Z N instead. 5 Knapsack Encipherment and Decipherment of ASCII-Plaintext. How cryptography works >IN PROGRESS< 1. Euclid’s Algorithm. Now the time has come to see this in details. Using Cryptography hashing Show how the ID number 1111111111111 of a student with 13 data items…. Modular Arithmetic and Cryptography 12/28/09 Page 5 of 11 2. Perform multifaceted calculations. If n is a negative number then you. Elliot Gorokhovsky. Cryptography. Performing calculations for Windows 98 / ME / 2000 / XP / Vista / 7. Geeks love to use technical words in regular contexts. So this is the same as to say that our number is equal to 3 x 10 to the power of 3 + 4 x 10 to the power of 2 + 7 x 10 + 5. Notice board. In addition to using fixtures in test functions, fixture functions can use other fixtures themselves. We teach the classic elements of programming, using an. Mel Frequency Cepstral Coefficents (MFCCs) are a feature widely used in automatic speech and speaker recognition. using System. com MMU E-Learning Portal for Cryptography added a new photo. Secondly, always use the most convenient residue (recall Definition 4. Computing Computer science Cryptography Modular arithmetic. You have also picked up a very important algorithm called the Euclidean Algorithm. 1 Introduction to Cryptography Since ancient times, people desiring to transmit messages privately have devised methods of encoding messages, so that no person but the intended recipient. , the greatest common divisor of modulus pand the number for which we seek an inverse must be one [8]. Today, the same concept is applied to computer graphics, as well as sculptures and modern paintings. 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. Chapter 1: cryptography fundamentals; private key and public key cryptosystems; Chapter 2: Caesar. Common Exponentiation algorithm. While the term itself may seem intimidating Therefore, cryptography requires one-way hash functions. It is helpful to prevent unauthorized person or group of users from accessing any Symmetric-key encryption are algorithms which use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext. Modular arithmetic is a "calculus of remainders". You have also picked up a very important algorithm called the Euclidean Algorithm. Then you will encrypt it picking a shift of the wheel,. It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra. (Modular Arithmetic) Compute the following numbers: 6 + 11 + 14 + 3 (mod 7) 7 * 5 * 13 (mod 12) 7 + (10 * 9) (mod 11) 3. For decryption we can use d(x) = a-1(x-b) mod m. In fact, I bet you use it all the time. average / arithmetic mean. Why is modular arithmetic used in cryptography? Cryptography: Cryptographical Techniques are those that help to maintain the security of systems. I'm using NTL for modular arithmetic and have double checked all values using online modulo calculators. Each of those parties has their own secret and public key. This contributes to a modular design of your fixtures and allows re-use of framework-specific fixtures across many projects. In number theory modular arithmetic occurs frequently. Modular arithmetic: Goal: Computation in finite sets (used in about 95% of modern cryptography) Finite sets are often represented in circles, e. Keyword Method. In general, Cryptography is a hidden writing. Web Cryptography API. Modular arithmetic is a foundational subject in number theory, but as we will see, it also has interesting practical applications—for example, it finds many uses in cryptography (the study and design of secret codes). Cryptography; using System. and widely used example of public-key cryptography is the RSA system (Rivest, Shamir and Adleman, 1978). The steps are summarized as follows: Using the Euclidean algorithm to compute , the greatest common divisor (GCD) of and. 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. now do arithmetic modulo m as follows: after adding or multiplying two integers,replacetheirsumorproduct,respectively,byitsresiduemodulo m. A necessary part of Hill ciphers is modular arithmetic. Please click here for more information on the CIS group. The use of elliptic curves for cryptography was suggested, independently, by Neal Koblitz and Victor Miller in 1985. Furthermore, a number only has a modular inverse if it is coprime to the modulus, i. 167 5 5 bronze badges. Including the principles of modular arithmetic and the Chinese remainder theorem, with efficient methods, is given in detail, together with their implementation for transformation from integer fixed-radix to modular and back again. 1985, which is the most rare algorithm used in public-key cryptography; serve as an efficient algorithm for modular multiplication and exponentiation operations [23]. Using both the scaled modulus and the inversion algorithm we define a cryptographic processor for Elliptic curves Cryptography (ECC). Modular arithmetic is often used to calculate checksums that are used within identifiers - International Bank Account Numbers (IBANs) for example make use of modulo 97 arithmetic to trap user input errors in bank account numbers. In some sense, modular arithmetic is easier than integer arithmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. Low system requirements. DecodeString(keyString) plaintext := []byte(stringToEncrypt). Eve will know these two numbers, and it won't matter! Alice chooses a number A, which we'll call her " secret exponent. This is an example of what is called modular arithmetic. 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. On a standard piano keyboard, one note has the same name as one 12 half-tones above it. Modular arithmetics is, generally speaking, an arithmetic system for integers, where numbers Modulo operations are used to calculate the checksums of serial numbers. MODULAR ARITHMETIC, GROUPS, FINITE FIELDS AND PROBABILITY. 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. So this is the same as to say that our number is equal to 3 x 10 to the power of 3 + 4 x 10 to the power of 2 + 7 x 10 + 5. Modulo 2 addition is performed using an exclusive OR (xor) operation on the corresponding binary digits of each operand. The arithmetic operations used in elliptic curves are different from the standard algebraic operations. Mod-arithmetic is the central mathematical concept in cryptography. Contactless access control cards are used in various security systems. Modulo Challenge. Talk Improving Modular Inversion in RNS using the Plus-Minus Method at CHES, Apr. Let a;b;c 2Z with a 6= 0. In this tutorial, you'll learn what kinds of mistakes can be made when you're rounding numbers and how you can best manage or avoid them. Kubernetes for DevOps. Compute the exponentiation [a. Discrete Mathematics and Its Applications, Seventh Edition answers to Chapter 4 - Section 4. Mathematical Arrays Strings Dynamic Programming Hash Sorting Bit Magic Matrix Tree Searching Stack STL Greedy Graph Linked List Recursion Prime Number Java CPP Numbers Misc number-theory Binary Search Heap Binary Search Tree Queue Modular Arithmetic sieve Java-Collections DFS series sliding-window Combinatorial logical-thinking Map Fibonacci. As well as the analysis of letter frequencies, other patterns can also be detected that may help to decipher a piece of ciphertext. Secrecy: Some hash algorithms are available for public use, but proprietary interests can make their own. It is necessary to asymmetric encryption algorithms. Patents may prevent use of some ECC algorithms. cryptography arithmetic modular-arithmetic gmp jni-wrapper modular-exponentiation verificatum primality-testing-routines verificatum-gmpmee Updated Sep 1, 2019 M4. In modular arithmetic we are only interested in the remainder after division. Properties of prime numbers. The prince. It uses modular arithmetic with. Modular arithmetic in the context of cryptography: I think it was really fortunate that we had a lesson in class on the beginnings of modular arithmetic. edu) May 7, 2020 12:21Noon c2020AvinashKak,PurdueUniversity Goals: • To review modular arithmetic • To present Euclid’s GCD algorithms • To present the prime finite field Z p. In modular arithmetic, numbers "wrap around" upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder. Show that casting out 3s is true for 73452 and 19548. the modulo operation, consists of a simple conditional subtraction (addition) of the modulus. Also, it can identify if the sequence is arithmetic or geometric. Try these exercises from the Modular Arithmetic lecture in Octave. 6, respectively. Faster implementations of public-key cryptography and in particular of RSA are of uttmost importance nowadays. 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. 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. RNS is also an appealing method in Elliptic Curve Cryptography (ECC) where the sizes range from 160 to 256 bits. If you have not taken Applied Finite Mathematics, then you will need to spend a considerable amount of time learning the material in the lesson. It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra. Material icons are delightful, beautifully crafted symbols for common actions and items. Using Cryptography hashing Show how the ID number 1111111111111 of a student with 13 data items…. Fast Exponentiation algorithm. Cryptography and Computer Programming 1. A generic approach to searching for Jacobians , Mathematics of Computation 78 (2009), 485-507. However, this work focuses on modular arithmetic, also known as GF(p) mathematics, in order to support backwards compatibility with RSA. easily implement ideal lattice cryptography. 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. Modular arithmetic. Moving to Jupyter! Introduction to modular arithmetic, including integer rings. (2) Polynomial arithmetic in which the arithmetic on the coefficients is performed modulo p; that is, the coefficients are in Zp. Modern cryptography applies theories from mathematics, computer science, and electrical engineering, to design and implement strong encryption methods which are important to the safety of computer systems today. Mel Frequency Cepstral Coefficents (MFCCs) are a feature widely used in automatic speech and speaker recognition. Modular Arithmetic 3/34 Modular addition Create the modular addition table for Z 7, x +y mod 7. average / arithmetic mean. Chapman & Hall/CRC (2nd ed. 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). Each of those parties has their own secret and public key. y 2 =x 3 +ax+b. FCE Reading and Use of English Practice Test 1. not true for division (division is not de ned for modular arithmetic. 5, 2016 (RSA Public Key Encryption) Week 4 (Primality Testing - Sections 3. Modular arithmetic is a key ingredient of many public key crypto-systems. Modular arithmetic is a system of arithmetic for integers, which considers the remainder. Arithmetic is the oldest and most elementary branch of mathematics. 3 + 9 = 12 ≡ 2 mod 10) Modular addition • Given: c = x + k mod n, with c, x, k ∈ Zn → if x + k < n : c = a + b → if x + k ≥ n : c = j, where x + k = i ⋅ n + j and j < n • Can be used. The elements of GF(pn) are polynomials over GF(p. 6, respectively. A First Pass at Modular Arithmetic 56 69; 3. Discrete Logarithm – Inverse of Modular Exponentiation Like factoring problem, discrete logarithm problem (DLP) is believed to be difficult. To encrypt our integer message M, pick a (large) modulus m and an encryption key e that is an integer that is relatively. Elliot Gorokhovsky. Eve will know these two numbers, and it won't matter! Alice chooses a number A, which we'll call her " secret exponent. How do we write modular arithmetic? Continuing the example above with modulus 5, we write: 2+1 = 3 (mod 5) = 3 2+2 = 4 (mod 5) = 4 2+3 = 5 (mod 5) = 0 2+4 = 6 (mod 5) = 1 Challenge question! What is 134 (mod 5)? It might help us to think about modular arithmetic as the remainder when we divide by the modulus. Much of modern cryptography is based on modular arithmetic, which we now briefly review. Any crypto-system that uses modular arithmetic can be constructed in an analogous way with a group having certain properties under associated group operations. Modular arithmetic is useful in many fields, and its applications include RSA algorithm in cryptography, primality testing in computer science, and 12-tone arithmetic in music. (modular arithmetic) and a little probability before. See full list on globalsoftwaresupport. … - Selection from Computer Security and Cryptography [Book]. First we must translate our message into our numerical alphabet. See full list on calculators. He's using TikZ anyway, in order to produce these figures; that's the point of his question. the group has the order of p. now do arithmetic modulo m as follows: after adding or multiplying two integers,replacetheirsumorproduct,respectively,byitsresiduemodulo m. Class exercise: Prove that 2=5=8=11 (mod 3) How to factor numbers? Modular arithmetic –working only with remainders. The pattern is continued by adding 3 to the last number each time, like this: In General we could write an arithmetic sequence like this. We have a kind of long arithmetic calculation here, and we are asked about the result modulo some number ( result % M in many languages). off original price! The coupon code you entered is expired or invalid, but the course is still available! Modular Arithmetic Available in days days after you enroll Start Modular arithmetic. Fundamental theorem of arithmetic. In this section of Data Communication and Networking - Cryptography MCQ (Multiple Choice) Based Questions and Answers,it cover the below lists of topics. Material icons are delightful, beautifully crafted symbols for common actions and items. We say that a≡ b(mod m) if the integers a and b differ by a multiple of m. It is also widely used in computer science. Prove that 1110 - 1 is divisible by 100. forming the basic arithmetic operations in this structure (addition, subtraction, multiplica-tion, division, and exponentiation). Modular arithmetic has lots of applications. Number Theory (Modular Arithmetic) and Modern Cryptography Cryptography: Derived from Greek words “crypto” (secret) and “graphy” (writing), cryptography is “the art and science of transforming information into an intermediate form which secures that information while in storage or in transit” Cryptography may be used to provide: confidentiality exchange secret (encrypted) messages. Furthermore, a number only has a modular inverse if it is coprime to the modulus, i. Algorithms available for the use of most include MD5, SHA1, SHA2, SHA3, and others. Multiplicative and Affine Ciphers. 5, 2016 (RSA Public Key Encryption) Week 4 (Primality Testing - Sections 3. Assorted Modular Arithmetic from CryptoClub book Applications of Modular Arithmetic Presentations: Calendar problems Chapter 8 Modular Arithmetic. Recall that for a monic polynomial g(x) of positive degree, we define a(x) mod g(x) to the unique polynomial a0(x) with dega0. Here's the demo for subtraction and division We won't need these operators any time soon, as web development has little use of them, but in some special areas, such as cryptography, they are useful. The following statement is a basic principle of modular arithmetic: a + kp = a (mod p). An exploration of modular arithmetic - adding, multiplying, powers, orders, residues. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour. Further modular reduction can be performed at any point, thus (a+b) mod n = ((a mod n) + (b mod n)) mod n. It is also the foundation of some of the most recent applications, which progress in Cryptography has introduced in everyday life, from ATM cards to the e-commerce. See Modular arithmetic for thorough explanations. Relating Modular Arithmetic to the Caesar Cipher 4. An Introduction to Cryptography Exercises for Chapter 1 Section. Extension eld arithmetic are very important to improve the perfor-mance of pairing. In this modular arithmetic and cryptography lesson, students use applets to explore modular arithmetic using a clock and Caesar Ciphers. One commonly used public-key cryptography method is the _____ algorithm. example, the modular inverse of 16 modulo 37 is 7 since 16 7 1 (mod 37) since 16 7 = 112 = 111 + 1 = 3(37) + 1. ) • Some historical ciphers can be elegantly described with modular arithmetic (cf. Modular Arithmetic is also. Cryptography. As an increasing amount of our personal. First we have to talk a bit about the problem of integer factorization. Modular arithmetic is a key ingredient of many public key crypto-systems. There are other groups which would work (i. CrypTool 1 (CT1) is an open-source Windows program for cryptography and cryptanalysis. In this article we are going discuss RSA cryptosystem. Chapter 4 - Number Theory and Cryptography 4. This line will intersect the curve at a third point, -R. 10 Modular Arithmetic and Cryptography 10. But first let’s define a symbol to represent each group. It is like hiding our key in a room full of fake keys with only one authentic key. A First Pass at Modular Arithmetic 56 69; 3. asked Aug 19 '18 at 22:24. Here’s the gist: You can think of modular arithmetic as a system of arithmetic for integers where the number line isn’t an infinitely long and straight line (as we’ve talked about in past discussions of integers), but is instead a line that curves around into a circle. Once a message was. Furthermore, a number only has a modular inverse if it is coprime to the modulus, i. Exponentiation is performed by a chain of these, and it is the fundamental building block of RSA, ElGamal, or elliptic curve cryptosystems or the Diffie-Hellman key exchange protocol [1]. The binary method is adapted to perform square and multiplication operations. It uses modular arithmetic with a modulus of 26. The latter operation is the most interesting one and creates a complicated structure on integer numbers. [57] propose algorithms for incomplete modular arithmetic, namely modular addition, subtraction and multiplication using Montgomery arithmetic for 2. 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. average / arithmetic mean. 6 Cryptanalysis of the Merkle–Hellman Knapsack System (Modular Mapping). - mr-ma Mar 1 '15 at 11:44. Multiplicative Ciphers 4. In this case we know that the variable B is being multiplied by the variable C since the multiplication operator * appears between them in the expression. In this section we will consider modular arithmetic and applications to cryptography and to generating “ random numbers ” by deterministic computers. RSA has been implemented many times in hardware, but if it is only used for key exchange, a software implementation is fast enough. It involves history, political science, technology, languages, ethics, mathematics and computer science. Introduction to Modern Cryptography provides a rigorous yet accessible treatment of this fascinating subject. Aug 28, 2020 the web of modularity arithmetic of the coefficients of modular forms and q series cbms regional conference series in mathematics Posted By Robin CookLtd TEXT ID f12919360 Online PDF Ebook Epub Library. Harvey 2018. The next article will introduce Elliptic Curves used in cryptography. Moving to Jupyter! Introduction to modular arithmetic, including integer rings. Large Integer Arithmetic in GPU forCryptography free download ABSTRACT Most computer. thanks anyway. And we showed how to use that to break Turing's code, version number two. Let me show you. 2020 by fimim. Abstract— HEORIESCryptography is a science and art that using modular arithmetic to secure messages by encrypting it into another unreadable text and decrypting it back. Although modular arithmetic is often denoted using (mod n) after an equation, here we will use in Z N instead. Modular arithmetic has several practical applications including: music, banking, book publishing, cryptography… and of course math. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms. 5 Residue Class Rings and Prime Fields 14 2. Apart from that I introduce everything needed from scratch. 1 thought on “ Cryptography: A modern use for modular arithmetic ” benbraun October 10, 2014 at 9:37 am. Let's begin with what is called a shift cipher. Most applications, including OpenSSL, use elliptic curves over coordinates that use modular arithmetic, where the modulus is a large prime number. A Second View of Modular Arithmetic: \Z_{𝑛} and 𝑈_{𝑛} 73 86; 4. How to use modular arithmetic in a sentence. Selected Preprints and Reprints. The elliptic curve cryptography (ECC) uses elliptic curves over the finite field 𝔽p (where p is prime and p > 3) or 𝔽2 m (where the fields size p = 2 m). Elliptic curve cryptography is used by the cryptocurrency Bitcoin. Private key cryptography. Fractions and the Notion of an Equivalence. Modular arithmetic was systematized by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Substitution and transposition ciphers are two categories of ciphers used in classical cryptography. Posted By Danielle SteelMedia TEXT ID d63014bb.