Introduction to modern cryptography pdf free download. Chapter 8 number cryptography network theory security. Elementary number theory primes, congruences, and secrets. May 07, 2018 how are prime numbers used in cryptography. Cryptography is the science of using mathematics to hide data behind encryption. Dec 09, 2012 cryptography and number theory over 300 years ago, a mathematician named fermat discovered a subtle property about prime numbers. If you are able to factorize the public key and find these prime numbers, you will then be able to find the private key. Or, 150 as a product of 15 and 10, which can be further broken down and written as the product of 3, 5, 2 and 5 all prime numbers. In this post, im going to explain public key cryptography. Cryptography and modern cryptography the setting of privatekey encryption historical ciphers and their cryptanalysis the basic principles of modern cryptography 1.
Many encryption systems relied on the secret key that 2 or more parties had used in decrypting information which is encrypted by the typically agreed method. Cryptography, or cryptology, is the practice and study of hiding information. Public key crypto algorithms utilize prime numbers exten sively. New findings about prime number distribution almost certainly. Prime numbers prime numbers only have divisors of 1 and self they cannot be written as a product of other numbers note. A natural number greater than 1 that is not prime is called a composite number. Nov 30, 2011 pupils have to find the prime factorisation of large numbers, the prime factors relate to letters that unjumble to make a very bad maths joke. The book features the source code to several ciphers and hacking programs for these ciphers. The prime number theorem journey into cryptography. Can be adapted to make other words using the code given on the sheet.
Public key cryptography seems magical to everyone, even those who understand it. I should also mention that many other cryptosystems do not use prime numbers. A simple explanation of how prime numbers are used in public key cryptography from abc1 science program catalyst. There are probably more prime numbers in the 2 2048 600 digits base 10 space than there are atoms in the entire known universe. A computational introduction to number theory and algebra. Modern cryptography is a mix of mathematics, computer science, and electrical engineering.
The objective of this work was the study of rsa encryption in which an algorithm was developed to analyze a possible security attack through the euler function to try to find the prime factors of. Feb 05, 2019 public key cryptography simply explained. Encryption and decryption are carried out using two di. Number of keys required to use public key cryptography, each individual requires two keys one public key and one private key. The systematic study of number theory was initiated around 300b. This is because the numbers, which are only divisible by themselves and one, can be multiplied together. Why are primes important for encryption cryptography. Prime numbers and computer methods for factorization hans. Example 2 the numbers 4, 6, and 9 are relatively prime but not pairwise relatively prime. The two keys in such a key pair are referred to as the public key and the private key. These keywords were added by machine and not by the.
In rsa, the function used is based on factorization of prime numbers however it is not the only option elliptic curve is another one for example. See 1, 4, 5 for many historical comments on public key cryptography. The prime numbers have a lot of useful properties which were going to study, and then use in the cryptographic algorithms. Elliptic curve cryptography, or ecc, builds upon the complexity of the elliptic curve discrete logarithm problem to provide strong security that is not dependent upon the factorization of prime numbers. Prime number encryption algorithm punctuation mark integer representation small prime factor. In other words, p is prime if its only factors in the natural numbers are itself and 1, and these factors are different. More specifically, some important cryptographic algorithms such as rsa critically depend on the fact that prime factorization of large numbers takes a long. Pdf on the one hand, the study of numbers and especially of prime numbers has fascinated mathematicians since ancient. For example, a prime that ends in 1 is less likely to be followed by another ending in. In this video, learn how cryptographers make use of these two algorithms.
For n individuals to communicate, number of keys required 2 x n 2n keys. Fermats little theorem states that if p is a prime number, then for any integer a, the number a p. General speaking, it is because many cryptosystems use prime numbers in the process. The elements of gfpn are polynomials over gfp which is the same as the set of residues z p.
This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. However, it is also considered neither prime nor composite. Historically, most encryption systems depended on a secret key that two or more parties used to decrypt information encrypted by a commonly agreed method. I understand the part where phi of the product of 2 primes is derived by doing a p1q1, and the main part is why does it work. Number theory and cryptography free computer books. These numbers are popular in encryption and cryptography. Prime numbers and cryptography a lessandro l anguasco, a lberto p erelli on the one hand, the study of numbers and es pecially of prime nu mbers has. Especially if all you have available to do it, is digital hardware and deterministic software.
Rsa public key encryption algorithm 1 rsa public key encryption algorithm. The first article below describes how a public key encryption scheme works, and the second explains the mathematics behind it. The rsa cryptosystem has made buying and selling on the internet practical and safe. Quantum computing attempts to use quantum mechanics for the same purpose. It is very easy, even trivial, to find a factor of it, but not all factors.
Both are designed to behave predictably, each time, every time. Numbers that are equally large but have more smaller factors are very much easier to factor, and hence are unsuitable for cryptography purposes for practical purposes, the difficulty of factoring a number varies with the size of its smallest prime factor, regardless of how big the number is, and so your prime factors need to be as large as. Public key cryptography simply explained brandons blog. Notes on primality testing and public key cryptography part 1. According to cal the largest known prime as of march 2007 is the 44th known mersenne prime p 232582657 1. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. A prime number or a prime is a natural number greater than 1 that is not a product of two smaller natural numbers. How many prime numbers are there in a 600 digit number. The exponents n which give mersenne primes are 2, 3, 5, 7, 17, 19, 31.
In other words, there is no prime number of the form 22m 2i 2j 1, where 1 i 7. Cryptography uses prime numbers, and needs to test if a certain number is prime. While cryptography is also used in the science of securing data, cryptanalysis. Basic facts about numbers in this section, we shall take a look at some of the most basic properties of z, the set of integers. Cryptography is used in atm bank cards, computer passwords, and shopping on the. This process of reducing a composite number to a product of prime numbers is known as prime factorization. Also, i understand the mod function, but why cryptography relies on it so much. In mathematics, a mersenne prime is a prime number that is one less than a power of two.
New findings about prime number distribution almost. If you had the technology to store 1 prime number not just 1 bit, an entire 600 digit prime using a single atom then youd still need more matter than exists in the entire universe to store them. Download number theory and cryptography download free online book chm pdf. However, cryptography is a totally new research field due to the modern time compu. And the examples of prime numbers are on the slide starting from 2, 3, 5, 7, 11, and so on. This is a textbook about classical elementary number theory and elliptic curves.
Learning and experiencing cryptography with cryptool and sagemath background reading for cryptool the free elearning crypto program cryptography, mathematics, and more 12th edition draft version 01. The second edition of this book is available under the title cracking codes with python. Asymmetric encryption algorithms the famous asymmetric encryption algorithms are rsa algorithm. This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. Finite fields of the form gf2n theoretical underpinnings of modern cryptography. Factoring that number is a nontrivial operation, and that fact is the source of a lot of cryptographic algorithms. Luiz felipe martins history one of the first encryption systems was the caesar cipher.
Prime numbers are whole numbers greater than 1 that are not divisible by any whole number other than 1 and itself. Prime numbers and computer methods for factorization. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number the number 3. Infinitude, density and substance, the prime number theorem and. Mar 23, 2016 prime numbers near to each other tend not to want to echo each others last digits, the mathematicians say. In this article, we will briefly outline the development of the theory of prime numbers.
We look at properties related to parity even, odd, prime factorization, irrationality of square roots, and modular arithmetic. The prime numbers cryptography public key cryptography standard security has been established on mathematical complexity of getting 2 prime factors that are larger numbers. Prime numbers building blocks for cryptography coursera. The use of publickey cryptography is pervasive in the information protection and. The author provides explicit algorithms and computer programs, and has attempted to discuss as many of the classically important results as possible, as well as. D used for hundreds of years until it was discovered how to break the code by using frequency analysis. Multiplying 2, a digit prime, and another digit prime, will give a number that is still very difficult to factor. It involves storing secret information with a key that people must have in order to access the raw data. Prime factorization or integer factorization is a commonly used mathematical problem often used to secure publickey encryption systems. Pupils have to find the prime factorisation of large numbers, the prime factors relate to letters that unjumble to make a very bad maths joke. So, basically you need two prime numbers for generating a rsa key pair. Gimps, founded in 1996, has discovered the last 17 mersenne primes. Hartley university of manchester, england, and national university of singapore the word cryptography derives from greek and means secret writing.
Or, a larger number such as 126, 356, which is composed of larger prime numbers 2,2,31 and 1019. There are simple steps to solve problems on the rsa algorithm. Last year a canadian college student found the biggest known prime. The fact that 1 is not counted as being prime is a convention, but is.
Indeed it is destined to become a definitive text on prime numbers and factoring. Since ancient times, cryptographic methods have been in use in diplomatic and military contexts for the transfer of secret information. If you multiply two large prime numbers, you get a huge nonprime number with only two large prime factors. It grew out of undergraduate courses that the author taught at harvard, uc san diego, and the university of washington. Cryptography and number theory over 300 years ago, a mathematician named fermat discovered a subtle property about prime numbers. Theorem 1 for all m 1, there is no prime number whose binary representation has precisely 2m signi cant bits, exactly two of which are zero bits. Steiger, and endre szemeredi, infinite sets of primes with fast primality tests and quick generation of large primes, math. A common practice is to use very large semiprimes that is, the result of the multiplication of two prime numbers as the number securing the encryption. Without cracking the cipher, its impossible to know what the original is. Mersenne primes were named for the french monk marin mersenne, who studied these numbers more than 350 years ago.
Multiplication of prime numbers ifp discrete logarithms mod p dlp. The recipient uses the relative prime numbers to convert the public key into the private key and, then, generates the subset s by solving the private key with respect to the number n. Thus 126,356 can be factored into 2 x 2 x 31 x 1,019, where 2, 31, and 1,019 are all prime. It is sometimes called code, but this is not really a correct name. Prime numbers in cryptology leon bykov, lisa stelmarski, daniel arraj advisor. Cryptography simple english wikipedia, the free encyclopedia. In general, the bigger the prime numbers you find, the safer your secret will be. This number is used to lock data and can be shared freely. The official proof of a prime is through its primality certificate. In the 1970s, three mathematicians at mit showed that his discovery could be used to formulate a remarkably powerful method for. In this specialization, students will learn basic security issues in computer communications, classical cryptographic algorithms, symmetrickey cryptography, publickey cryptography, authentication, and digital signatures.
Breaking the primenumber cipher, one proof at a time. Mathematical foundations for cryptography coursera. The message, the public key and the number n are transmitted to a recipient, who knows the value of the two relative prime numbers. The security of the standard public key cryptography pkc algorithm rsa is founded on the mathematical difficulty of finding two prime factors of a very large number. A natural number p is said to be prime if p 1 and, whenever p ab holds for some natural numbers a and b, we have either a p, b 1, or a 1, b p. The new prime is only the 51st known mersenne prime ever discovered. For example, if a 2 and p 7, then 2 7 128, and 128. Introduction and terminology cryptology is defined as the science of making communication incomprehensible to all people except those who have a right to read and understand it. They are named after marin mersenne, a french minim friar, who studied them in the early 17th century. Therefore, hardware and software designers, trying to find unpredictability, have to look outside of their normal operating environment to find it. Rsa algorithm is an asymmetric cryptography algorithm which means, there should be two keys involve while communicating, i. It is astonishing that nding prime numbers is easy even though factoring integers is. Determine if there is a number between 2 and n, the number to test that divides n, without a rest. In the modern age of almost universal computer usage, practically every individual in a technologically developed society has routine access to the most uptodate cryptographic technology that exists, the socalled rsa publickey cryptosystem.
And the definition of prime numbers is the following. Actually, any number divides zero, because zero times a equal to zero. Public key cryptography is based on asymmetric cryptography, so first let us talk about symmetric cryptography. In the 1970s, three mathematicians at mit showed that his discovery could be used to formulate a remarkably powerful method for encrypting information to be sent online. Do not use the new prime number for rsa encryption. Publickey cryptography algorithms like rsa get their security from the difficulty of factoring large composite numbers that are the product of two prime numbers. Ppt rsa public key encryption algorithm powerpoint. It is the science used to try to keep information secret and safe. If you multiply two primes together, you get a public key. Prime numbers in public key cryptography an introduction jerry crow abstract the use of public key cryptography is pervasive in the information protection and privacy arenas. Us8958547b2 generation of relative prime numbers for use in. A primality test is a method or algorithm to find out if a certain number is a prime number.