![]() Partitions 9.1 Generating functions 9.2 Partitions 9.3 Pentagonal Number Theorem Representations 8.1 Sums of squares 8.2 Pell's equation 8.3 Binary quadratic forms 8.4 Finite continued fractions 8.5 In®nite continued fractions 8.6 p-Adic analysis ![]() Modular arithmetic 5.1 Congruence 5.2 Divisibility criteria 5.3 Euler's phi-function 5.4 Conditional linear congruences 5.5 Miscellaneous exercisesĬongruences of higher degree 6.1 Polynomial congruences 6.2 Quadratic congruences 6.3 Primitive roots 6.4 Miscellaneous exercisesĬryptology 7.1 Monoalphabetic ciphers 7.2 Polyalphabetic ciphers 7.3 Knapsack and block ciphers 7.4 Exponential ciphers Perfect and amicable numbers 4.1 Perfect numbers 4.2 Fermat numbersĤ.3 Amicable numbers 4.4 Perfect-type numbers Prime numbers 3.1 Euclid on primes 3.2 Number theoretic functions 3.3 Multiplicative functions 3.4 Factoring 3.5 The greatest integer function 3.6 Primes revisited 3.7 Miscellaneous exercises The intriguing natural numbers 1.1 Polygonal numbers 1.2 Sequences of natural numbers 1.3 The principle of mathematical induction 1.4 Miscellaneous exercisesĭivisibility 2.1 The division algorithm 2.2 The greatest common divisor 2.3 The Euclidean algorithm 2.4 Pythagorean triples 2.5 Miscellaneous exercises ISBN-13 978-3-3 hardback ISBN-10 3-1 hardback ISBN-13 978-1-6 paperback ISBN-10 1-7 paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. First published in print format 1999 ISBN-13 ISBN-10ĩ78-3-5 eBook (NetLibrary) 3-3 eBook (NetLibrary) Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States by Cambridge University Press, New York Information on this title: © Cambridge University Press 1999 This book is in copyright. Thus, odd periods are more secure than even against this form of cryptanalysis, because it would require more text to find a statistical anomaly in trigram plaintext statistics than bigram plaintext statistics.Elementary number theory in nine chapters For even periods, p, ciphertext lettersĪt a distance of p/2 are influenced by two plaintext letters, but for odd periods, p, ciphertext letters at distances of p/2 (rounded either up or down) are influenced by three plaintext letters. One way to detect the period uses bigram statistics on ciphertext letters separated by half the period. Longer messages are first broken up into blocks of fixed length, called the period, and the aboveĮncryption procedure is applied to each block. To decrypt, the procedure is simply reversed. In this way, each ciphertext character depends on two plaintext characters, so the bifid is a digraphic cipher, like the Playfair cipher. Then divided up into pairs again, and the pairs turned back into letters using the square: The message is converted to its coordinates in the usual manner, but they are written vertically beneath: It was invented around 1901 by Felix Delastelle.įirst, a mixed alphabet Polybius square is drawn up, where the I and the J share their position: In classical cryptography, the bifid cipher is a cipher which combines the Polybius square with transposition, and uses fractionation to achieve diffusion. JSTOR ( September 2014) ( Learn how and when to remove this template message).Unsourced material may be challenged and removed. Please help improve this article by adding citations to reliable sources. This article needs additional citations for verification.
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