Cryptography — Introduction Version: 2012/02/06 16:06:05
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4 Information Theory
Jay Daigle Occidental College Math 400: Cryptology 4 Information Theory Definition 4.1. A (symmetric) encryption system is composed of: • A set of possible messages M • A set of possible keys K • A set of possible ciphertexts C • An encryption function e : K × M ! C • A decryption function d : K × C ! M such that the decryption function is a partial inverse to the encryption function: that is d(k; e(k; m)) = m e(k; d(k; c)) = c: −1 We often write ek(m) = e(k; m) and dk(c) = d(k; c). Thus for each k 2 K, dk = ek . This implies that each ek is one-to-one. Of course, some encryption systems are terrible. To be good, we'd like our cryptosystem to have the following properties: 1. Given any k 2 K; m 2 M, it's easy to compute e(k; m). 2. Given any k 2 K; c 2 C, it's easy to compute d(k; c). 3. Given a set of ciphertexts ci 2 C, it's difficult to compute dk(ci) without knowing k. 4. Given a collection of pairs (mi; ci), it's difficult to decrypt a ciphertext whose plaintext is not already known. (\known-plaintext attack"). The first two principles make a cryptosystem practically usable; the third and fourth make it secure. The fourth property is by far the most difficult to achieve. You'll notice that all of the cryptosystems we've studied so far satisfy the first two properties, and several of them do at least okay on the third, none of them achieve the fourth at all. -
Amy Bell Abilene, TX December 2005
Compositional Cryptology Thesis Presented to the Honors Committee of McMurry University In partial fulfillment of the requirements for Undergraduate Honors in Math By Amy Bell Abilene, TX December 2005 i ii Acknowledgements I could not have completed this thesis without all the support of my professors, family, and friends. Dr. McCoun especially deserves many thanks for helping me to develop the idea of compositional cryptology and for all the countless hours spent discussing new ideas and ways to expand my thesis. Because of his persistence and dedication, I was able to learn and go deeper into the subject matter than I ever expected. My committee members, Dr. Rittenhouse and Dr. Thornburg were also extremely helpful in giving me great advice for presenting my thesis. I also want to thank my family for always supporting me through everything. Without their love and encouragement I would never have been able to complete my thesis. Thanks also should go to my wonderful roommates who helped to keep me motivated during the final stressful months of my thesis. I especially want to thank my fiancé, Gian Falco, who has always believed in me and given me so much love and support throughout my college career. There are many more professors, coaches, and friends that I want to thank not only for encouraging me with my thesis, but also for helping me through all my pursuits at school. Thank you to all of my McMurry family! iii Preface The goal of this research was to gain a deeper understanding of some existing cryptosystems, to implement these cryptosystems in a computer programming language of my choice, and to discover whether the composition of cryptosystems leads to greater security. -
Codebusters Coaches Institute Notes
BEING COVER AGENT FIXED DELAY, PILOT RIGHT PLANE, CATCH SMALL RADIO (CODEBUSTERS) This is the first year CodeBusters will be a National event. A few changes have been made since the North Carolina trial event last year. 1. The Atbash Cipher has been added. 2. The running key cipher has been removed. 3. K2 alphabets have been added in addition to K1 alphabets 4. Hill Cipher decryption has been added with a given decryption matrix. 5. The points scale has been doubled, but the timing bonus has been increased by only 50% in order to further balance the test. 1 TYPES OF PROBLEMS 1.1 ARISTOCRAT (EASY TO HARD DIFFICULTY) http://www.cryptograms.org/tutorial.php An Aristocrat is the typical Crypto-quote you see in the newspaper. Word spaces are preserved. No letter will stand for itself and the replacement table is given as a guide (but doesn’t need to be filled in by the team to get credit). FXP PGYAPYF FIKP ME JAKXPT AY FXP GTAYFMJTGF THE EASIEST TYPE OF CIPHER IS THE ARISTOCRAT A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Frequency 4 1 6 3 1 2 2 2 6 3 3 4 Replacement I F T A Y C P O E R H S 1.2 ARISTOCRATS WITH SPELLING AND/OR GRAMMAR ERRORS (MEDIUM TO VERY HARD DIFFICULTY) For these, either words will be misspelled or grammatical errors introduced. From a student perspective, it is what they might expect when someone finger fumbles a text message or has a bad voice transcription. -
The Mathemathics of Secrets.Pdf
THE MATHEMATICS OF SECRETS THE MATHEMATICS OF SECRETS CRYPTOGRAPHY FROM CAESAR CIPHERS TO DIGITAL ENCRYPTION JOSHUA HOLDEN PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu Jacket image courtesy of Shutterstock; design by Lorraine Betz Doneker All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Holden, Joshua, 1970– author. Title: The mathematics of secrets : cryptography from Caesar ciphers to digital encryption / Joshua Holden. Description: Princeton : Princeton University Press, [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016014840 | ISBN 9780691141756 (hardcover : alk. paper) Subjects: LCSH: Cryptography—Mathematics. | Ciphers. | Computer security. Classification: LCC Z103 .H664 2017 | DDC 005.8/2—dc23 LC record available at https://lccn.loc.gov/2016014840 British Library Cataloging-in-Publication Data is available This book has been composed in Linux Libertine Printed on acid-free paper. ∞ Printed in the United States of America 13579108642 To Lana and Richard for their love and support CONTENTS Preface xi Acknowledgments xiii Introduction to Ciphers and Substitution 1 1.1 Alice and Bob and Carl and Julius: Terminology and Caesar Cipher 1 1.2 The Key to the Matter: Generalizing the Caesar Cipher 4 1.3 Multiplicative Ciphers 6 -
Analysis of Stream Ciphers Based on Theoretic Approach
International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Impact Factor (2012): 3.358 Analysis of Stream Ciphers Based on Theoretic Approach Shisif Pokhrel1, Ahmed Abdul Kadhim Basheer2 1Jawaharlal Nehru Technological University Hyderabad, Andhra Pradesh, Hyderabad, Kukatpally - 500085 2Jawaharlal Nehru Technological University Hyderabad, Andhra Pradesh, Hyderabad, Kukatpally - 500085 Abstract: This paper is intended to determine the strength of modern security systems by theoretic approach. The design of existing security system is based on complexity of algorithm and secrecy of key. Besides, several parameters exists that may be useful to determine the strength of cryptosystem. Spurious keys gives text like text as decrypted output to a cryptogram and leads to confusion in proceeding towards unique solution during cryptanalysis. A system rich in spurious keys can be considered as more secured system. This paper presents analysis on stream ciphers and provides a construct model with rich spurious keys. The paper also shows the effect of implementation of Natural Language Model in security system Keywords: Cryptosystem, Cryptogram, Spurious Keys, Code-Points, Unicity Distance 1. Introduction Cryptography is a major tool concerned with security of information. The core of cryptography deals with key establishment and secured communication. The secret key is implemented for encryption and decryption. A basic convention exists in cryptography that the algorithm is public, cant be kept hidden to the attacker and the secrecy of the system solely depend on the secrecy of key and complexity of algorithm. However, there are different parameters defined in information theory of secrecy system like Entropy, Redundancy, Unicity Distance, Equivocation, which describe the strength of cryptosystem. -
Decrypt Cryptotexts: GBLVMUB JOGPSNBUJLZ VMNIR RPNBMZ EBMFLP OFABKEFT Decrypt: VHFUHW GH GHXA VHFUHW GH GLHX, VHFUHW GH WURLV VH
PROLOGUE - I. Decrypt cryptotexts: Part IV GBLVMUB JOGPSNBUJLZ Secret-key cryptosystems VMNIR RPNBMZ EBMFLP OFABKEFT prof. Jozef Gruska IV054 4. Secret-key cryptosystems 2/99 PROLOGUE - II. CHAPTER 4: SECRET-KEY (SYMMETRIC) CRYPTOGRAPHY Decrypt: In this chapter we deal with some of the very old, or quite old, classical (secret-key or symmetric) cryptosystems and their cryptanalysis that were primarily used in the pre-computer era. VHFUHW GH GHXA These cryptosystems are too weak nowadays, too easy to break, especially VHFUHW GH GLHX, with computers. However, these simple cryptosystems give a good illustration of several of the VHFUHW GH WURLV important ideas of the cryptography and cryptanalysis. Moreover, most of them can be very useful in combination with more modern VHFUHW GH WRXV. cryptosystem - to add a new level of security. prof. Jozef Gruska IV054 4. Secret-key cryptosystems 3/99 prof. Jozef Gruska IV054 4. Secret-key cryptosystems 4/99 BASICS CRYPTOLOGY - HISTORY + APPLICATIONS Cryptology (= cryptography + cryptanalysis) has more than four thousand years long history. Some historical observation People have always had fascination with keeping information away from others. Some people – rulers, diplomats, military people, businessmen – have always had needs to keep some information away from others. BASICS Importance of cryptography nowadays Applications: cryptography is the key tool to make modern information transmission secure, and to create secure information society. Foundations: cryptography gave rise to several new key concepts of the foundation of informatics: one-way functions, computationally perfect pseudorandom generators, zero-knowledge proofs, holographic proofs, program self-testing and self-correcting, . prof. Jozef Gruska IV054 4. Secret-key cryptosystems 5/99 prof. -
Substitution Ciphers
Foundations of Computer Security Lecture 40: Substitution Ciphers Dr. Bill Young Department of Computer Sciences University of Texas at Austin Lecture 40: 1 Substitution Ciphers Substitution Ciphers A substitution cipher is one in which each symbol of the plaintext is exchanged for another symbol. If this is done uniformly this is called a monoalphabetic cipher or simple substitution cipher. If different substitutions are made depending on where in the plaintext the symbol occurs, this is called a polyalphabetic substitution. Lecture 40: 2 Substitution Ciphers Simple Substitution A simple substitution cipher is an injection (1-1 mapping) of the alphabet into itself or another alphabet. What is the key? A simple substitution is breakable; we could try all k! mappings from the plaintext to ciphertext alphabets. That’s usually not necessary. Redundancies in the plaintext (letter frequencies, digrams, etc.) are reflected in the ciphertext. Not all substitution ciphers are simple substitution ciphers. Lecture 40: 3 Substitution Ciphers Caesar Cipher The Caesar Cipher is a monoalphabetic cipher in which each letter is replaced in the encryption by another letter a fixed “distance” away in the alphabet. For example, A is replaced by C, B by D, ..., Y by A, Z by B, etc. What is the key? What is the size of the keyspace? Is the algorithm strong? Lecture 40: 4 Substitution Ciphers Vigen`ere Cipher The Vigen`ere Cipher is an example of a polyalphabetic cipher, sometimes called a running key cipher because the key is another text. Start with a key string: “monitors to go to the bathroom” and a plaintext to encrypt: “four score and seven years ago.” Align the two texts, possibly removing spaces: plaintext: fours corea ndsev enyea rsago key: monit orsto gotot hebat hroom ciphertext: rcizl qfkxo trlso lrzet yjoua Then use the letter pairs to look up an encryption in a table (called a Vigen`ere Tableau or tabula recta). -
Introduction to Cryptography with Open-Source Software
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN INTRODUCTION TO CRYPTOGRAPHY WITH OPEN-SOURCE SOFTWARE Alasdair McAndrew Victoria University Melbourne, Victoria, Australia CRC Press Taylor &. Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK Contents Preface xv 1 Introduction to cryptography 1 1.1 Hiding information: Confidentiality 1 1.2 Some basic definitions 3 1.3 Attacks on a cryptosystem 5 1.4 Some cryptographic problems 7 1.5 Cryptographic protocols 8 1.6 Some simple ciphers 12 1.7 Cryptography and computer security 18 1.8 Glossary 19 Exercises 20 2 Basic number theory 23 2.1 Introduction 23 2.2 Some basic definitions 23 2.3 Some number theoretic calculations 27 2.4 Primality testing 44 2.5 Glossary 47 Exercises 48 3 Classical cryptosystems 55 3.1 Introduction 55 3.2 The Caesar cipher 56 3.3 Translation ciphers 57 3.4 Transposition ciphers 58 3.5 The Vigenere cipher 61 3.6 The one-time pad 65 3.7 Permutation ciphers 65 3.8 Matrix ciphers 66 3.9 Glossary 71 Exercises 71 4 Introduction to information theory 79 4.1 Entropy and uncertainty 79 ix X 4.2 Perfect secrecy 82 4.3 Estimating the entropy of English 84 4.4 Unicity distance 88 4.5 Glossary • 89 Exercises 89 5 Public-key cryptosystems based on factoring 93 5.1 Introduction 93 93 5.2 The RSA cryptosystem . 5.3 Attacks against RSA 99 5.4 RSA in Sage 101 5.5 Rabin's cryptosystem 104 5.6 Rabin's cryptosystem in Sage 109 5.7 Some notes on security HI 5.8 Factoring H2 5.9 -
ISA 562 Information Security Theory & Practice
ISA 562 Information Security Theory & Practice Introduction to Cryptography Agenda • Basics & Definitions • Classical Cryptography • Symmetric (Secret Key) Cryptography • DES (Data Encryption Standard) • Multiple Encryptions • Modes of Block Cipher Operations • Math Essential • Asymmetric (Public Key) Cryptography 2 1 Basics & Definitions Security Concepts (I) • Confidentiality – Prevent information from being exposed to unintended party – Ex: An employee should not come to know the salary of his manager • Integrity – Assure that the information has not been tempered – Ex: An employee should not be able to modify the employee's own salary • Identity – Assure that the party of concern is authentic - it is what it claims to be – Ex: An employee should be able to uniquely identify and authenticate himself/herself 4 2 Security Concepts (II) • Availability – Assure that unused service or resource is available to legitimate users – Ex: Paychecks should be printed on time as stipulated by law • Anonymity – Assure that the identity of some party is remain anonymous – Ex: The manager should not know who had a critical review of him • Non-Repudiation – Assure that authenticated party has indeed done something that cannot be denied – Ex: Once the employee has cashed his paycheck, he can’t deny it. 5 Cryptography • Crypt = secret • Graph = writing • Cryptography is the science / art of transforming meaningful information into unintelligible text Becoming a science that relies on mathematics (number theory, algebra) • Cryptanalysis is the science / art of breaking cryptographic codes • Cryptology is the science / art / study of both cryptography and cryptanalysis 6 3 Applications of Cryptography • Assuring document integrity • Assuring document confidentiality • Authenticating parties • Document signature • Non-repudiation • Secure transactions • Exchanging keys • Sharing Secrets • Digital cash • Preserving anonymity • Copyright protection • More . -
A Complete Bibliography of Publications in Cryptologia
A Complete Bibliography of Publications in Cryptologia Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 04 September 2021 Version 3.64 Title word cross-reference 10016-8810 [?, ?]. 1221 [?]. 125 [?]. 15.00/$23.60.0 [?]. 15th [?, ?]. 16th [?]. 17-18 [?]. 18 [?]. 180-4 [?]. 1812 [?]. 18th (t; m)[?]. (t; n)[?, ?]. $10.00 [?]. $12.00 [?, ?, ?, ?, ?]. 18th-Century [?]. 1930s [?]. [?]. 128 [?]. $139.99 [?]. $15.00 [?]. $16.95 1939 [?]. 1940 [?, ?]. 1940s [?]. 1941 [?]. [?]. $16.96 [?]. $18.95 [?]. $24.00 [?]. 1942 [?]. 1943 [?]. 1945 [?, ?, ?, ?, ?]. $24.00/$34 [?]. $24.95 [?, ?]. $26.95 [?]. 1946 [?, ?]. 1950s [?]. 1970s [?]. 1980s [?]. $29.95 [?]. $30.95 [?]. $39 [?]. $43.39 [?]. 1989 [?]. 19th [?, ?]. $45.00 [?]. $5.95 [?]. $54.00 [?]. $54.95 [?]. $54.99 [?]. $6.50 [?]. $6.95 [?]. $69.00 2 [?, ?]. 200/220 [?]. 2000 [?]. 2004 [?, ?]. [?]. $69.95 [?]. $75.00 [?]. $89.95 [?]. th 2008 [?]. 2009 [?]. 2011 [?]. 2013 [?, ?]. [?]. A [?]. A3 [?, ?]. χ [?]. H [?]. k [?, ?]. M 2014 [?]. 2017 [?]. 2019 [?]. 20755-6886 [?, ?]. M 3 [?]. n [?, ?, ?]. [?]. 209 [?, ?, ?, ?, ?, ?]. 20th [?]. 21 [?]. 22 [?]. 220 [?]. 24-Hour [?, ?, ?]. 25 [?, ?]. -Bit [?]. -out-of- [?, ?]. -tests [?]. 25.00/$39.30 [?]. 25.00/839.30 [?]. 25A1 [?]. 25B [?]. 26 [?, ?]. 28147 [?]. 28147-89 000 [?]. 01Q [?, ?]. [?]. 285 [?]. 294 [?]. 2in [?, ?]. 2nd [?, ?, ?, ?]. 1 [?, ?, ?, ?]. 1-4398-1763-4 [?]. 1/2in [?, ?]. 10 [?]. 100 [?]. 10011-4211 [?]. 3 [?, ?, ?, ?]. 3/4in [?, ?]. 30 [?]. 310 1 2 [?, ?, ?, ?, ?, ?, ?]. 312 [?]. 325 [?]. 3336 [?, ?, ?, ?, ?, ?]. affine [?]. [?]. 35 [?]. 36 [?]. 3rd [?]. Afluisterstation [?, ?]. After [?]. Aftermath [?]. Again [?, ?]. Against 4 [?]. 40 [?]. 44 [?]. 45 [?]. 45th [?]. 47 [?]. [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?]. Age 4in [?, ?]. [?, ?]. Agencies [?]. Agency [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?]. -
Decoding Running Key Ciphers
Decoding Running Key Ciphers Sravana Reddy∗ Kevin Knight Department of Computer Science Information Sciences Institute The University of Chicago University of Southern California 1100 E. 58th Street 4676 Admiralty Way Chicago, IL 60637, USA Marina del Rey, CA 90292, USA [email protected] [email protected] Abstract Since we know that the plaintext and running key are both drawn from natural language, our objective There has been recent interest in the problem function for the solution plaintext under some lan- of decoding letter substitution ciphers using techniques inspired by natural language pro- guage model is: cessing. We consider a different type of classi- ˆ cal encoding scheme known as the running key P = arg max log Pr(P ) Pr(RP,C ) (1) cipher, and propose a search solution using P Gibbs sampling with a word language model. where the running key RP,C is the key that corre- We evaluate our method on synthetic cipher- sponds to plaintext P and ciphertext C. texts of different lengths, and find that it out- Note that if RP,C is a perfectly random sequence performs previous work that employs Viterbi of letters, this scheme is effectively a ‘one-time pad’, decoding with character-based models. which is provably unbreakable (Shannon, 1949). The knowledge that both the plaintext and the key 1 Introduction are natural language strings is important in breaking The running key cipher is an encoding scheme that a running key cipher. uses a secret key R that is typically a string of words, The letter-frequency distribution of running key usually taken from a book or other text that is agreed ciphertexts is notably flatter than than the plaintext upon by the sender and receiver. -
Solving Substitution Ciphers
Solving Substitution Ciphers Sam Hasinoff Department of Computer Science, University of Toronto [email protected] Abstract We present QUIPSTER, an experimental system for the automatic solu- tion of short substitution ciphers (Cryptoquotes). The system operates using an -gram model of English characters and stochastic local search over the space of ¡£¢¥¤§¦©¨ possible keys. Experimental results show a median of 94% cipher letters correctly decoded, which is typi- cally good enough for an unskilled human to finish decoding the cipher with minimal additional effort. Extensions incorporating a dictionary with word frequencies and a database of word patterns are also discussed. 1 Introduction In Arthur Conan Doyle’s short story “The Adventure of the Dancing Men” (1903), the pro- tagonist Sherlock Holmes solves a murder mystery by realizing that a series of unusual stick figure drawings (Figure 1) are actually messages encoded using a substitution cipher [5]. The substitution cipher is a well-known classical cipher in which every plaintext character in all its occurrences in a message is replaced by a unique ciphertext character. Figure 1: Dancing men ciphertext from “The Adventure of the Dancing Men” (1903). Thus, each permutation of the 26 letters of the English alphabet (there are ¡¢¤¦¨ in total) gives a unique key for encrypting a message. If a particular permutation is used to encrypt a message, then the inverse of that permutation can be used to decrypt it. A worked out example of a substitution cipher is given in Figure 2. Decoding substitution ciphers is a popular activity among amateur cryptographers and peo- ple who enjoy word puzzles.