The Index of Coincidence and Its Applications In
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Introduction to Public Key Cryptography and Clock Arithmetic Lecture Notes for Access 2010, by Erin Chamberlain and Nick Korevaar
1 Introduction to Public Key Cryptography and Clock Arithmetic Lecture notes for Access 2010, by Erin Chamberlain and Nick Korevaar We’ve discussed Caesar Shifts and other mono-alphabetic substitution ciphers, and we’ve seen how easy it can be to break these ciphers by using frequency analysis. If Mary Queen of Scots had known this, perhaps she would not have been executed. It was a long time from Mary Queen of Scots and substitution ciphers until the end of the 1900’s. Cryptography underwent the evolutionary and revolutionary changes which Si- mon Singh chronicles in The Code Book. If you are so inclined and have appropriate leisure time, you might enjoy reading Chapters 2-5, to learn some of these historical cryptography highlights: People came up with more complicated substitution ciphers, for example the Vigen`ere square. This Great Cipher of France baffled people for quite a while but a de- termined cryptographer Etienne Bazeries had a Eureka moment after three years of work, cracked the code, and possibly found the true identity of the Man in the Iron Mask, one of the great mysteries of the seventeeth century. Edgar Allan Poe and Sir Arthur Conan Doyle even dabbled in cryptanalysis. Secrecy was still a problem though because the key needed to be sent, and with the key anyone could encrypt and decrypt the messages. Frequency analysis was used to break all of these codes. In 1918 Scherbius invented his Enigma machine, but Alan Turing’s machine (the first computer?) helped in figuring out that supposedly unbreakable code, and hastened the end of the second World War. -
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. -
Historical Ciphers • A
ECE 646 - Lecture 6 Required Reading • W. Stallings, Cryptography and Network Security, Chapter 2, Classical Encryption Techniques Historical Ciphers • A. Menezes et al., Handbook of Applied Cryptography, Chapter 7.3 Classical ciphers and historical development Why (not) to study historical ciphers? Secret Writing AGAINST FOR Steganography Cryptography (hidden messages) (encrypted messages) Not similar to Basic components became modern ciphers a part of modern ciphers Under special circumstances modern ciphers can be Substitution Transposition Long abandoned Ciphers reduced to historical ciphers Transformations (change the order Influence on world events of letters) Codes Substitution The only ciphers you Ciphers can break! (replace words) (replace letters) Selected world events affected by cryptology Mary, Queen of Scots 1586 - trial of Mary Queen of Scots - substitution cipher • Scottish Queen, a cousin of Elisabeth I of England • Forced to flee Scotland by uprising against 1917 - Zimmermann telegram, America enters World War I her and her husband • Treated as a candidate to the throne of England by many British Catholics unhappy about 1939-1945 Battle of England, Battle of Atlantic, D-day - a reign of Elisabeth I, a Protestant ENIGMA machine cipher • Imprisoned by Elisabeth for 19 years • Involved in several plots to assassinate Elisabeth 1944 – world’s first computer, Colossus - • Put on trial for treason by a court of about German Lorenz machine cipher 40 noblemen, including Catholics, after being implicated in the Babington Plot by her own 1950s – operation Venona – breaking ciphers of soviet spies letters sent from prison to her co-conspirators stealing secrets of the U.S. atomic bomb in the encrypted form – one-time pad 1 Mary, Queen of Scots – cont. -
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 -
Index-Of-Coincidence.Pdf
The Index of Coincidence William F. Friedman in the 1930s developed the index of coincidence. For a given text X, where X is the sequence of letters x1x2…xn, the index of coincidence IC(X) is defined to be the probability that two randomly selected letters in the ciphertext represent, the same plaintext symbol. For a given ciphertext of length n, let n0, n1, …, n25 be the respective letter counts of A, B, C, . , Z in the ciphertext. Then, the index of coincidence can be computed as 25 ni (ni −1) IC = ∑ i=0 n(n −1) We can also calculate this index for any language source. For some source of letters, let p be the probability of occurrence of the letter a, p be the probability of occurrence of a € b the letter b, and so on. Then the index of coincidence for this source is 25 2 Isource = pa pa + pb pb +…+ pz pz = ∑ pi i=0 We can interpret the index of coincidence as the probability of randomly selecting two identical letters from the source. To see why the index of coincidence gives us useful information, first€ note that the empirical probability of randomly selecting two identical letters from a large English plaintext is approximately 0.065. This implies that an (English) ciphertext having an index of coincidence I of approximately 0.065 is probably associated with a mono-alphabetic substitution cipher, since this statistic will not change if the letters are simply relabeled (which is the effect of encrypting with a simple substitution). The longer and more random a Vigenere cipher keyword is, the more evenly the letters are distributed throughout the ciphertext. -
A Hybrid Cryptosystem Based on Vigenère Cipher and Columnar Transposition Cipher
International Journal of Advanced Technology & Engineering Research (IJATER) www.ijater.com A HYBRID CRYPTOSYSTEM BASED ON VIGENÈRE CIPHER AND COLUMNAR TRANSPOSITION CIPHER Quist-Aphetsi Kester, MIEEE, Lecturer Faculty of Informatics, Ghana Technology University College, PMB 100 Accra North, Ghana Phone Contact +233 209822141 Email: [email protected] / [email protected] graphy that use the same cryptographic keys for both en- Abstract cryption of plaintext and decryption of cipher text. The keys may be identical or there may be a simple transformation to Privacy is one of the key issues addressed by information go between the two keys. The keys, in practice, represent a Security. Through cryptographic encryption methods, one shared secret between two or more parties that can be used can prevent a third party from understanding transmitted raw to maintain a private information link [5]. This requirement data over unsecured channel during signal transmission. The that both parties have access to the secret key is one of the cryptographic methods for enhancing the security of digital main drawbacks of symmetric key encryption, in compari- contents have gained high significance in the current era. son to public-key encryption. Typical examples symmetric Breach of security and misuse of confidential information algorithms are Advanced Encryption Standard (AES), Blow- that has been intercepted by unauthorized parties are key fish, Tripple Data Encryption Standard (3DES) and Serpent problems that information security tries to solve. [6]. This paper sets out to contribute to the general body of Asymmetric or Public key encryption on the other hand is an knowledge in the area of classical cryptography by develop- encryption method where a message encrypted with a reci- ing a new hybrid way of encryption of plaintext. -
The Enigma History and Mathematics
The Enigma History and Mathematics by Stephanie Faint A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Mathematics m Pure Mathematics Waterloo, Ontario, Canada, 1999 @Stephanie Faint 1999 I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by pho tocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. 11 The University of Waterloo requires the signatures of all persons using or pho tocopying this thesis. Please sign below, and give address and date. ill Abstract In this thesis we look at 'the solution to the German code machine, the Enigma machine. This solution was originally found by Polish cryptologists. We look at the solution from a historical perspective, but most importantly, from a mathematical point of view. Although there are no complete records of the Polish solution, we try to reconstruct what was done, sometimes filling in blanks, and sometimes finding a more mathematical way than was originally found. We also look at whether the solution would have been possible without the help of information obtained from a German spy. IV Acknowledgements I would like to thank all of the people who helped me write this thesis, and who encouraged me to keep going with it. In particular, I would like to thank my friends and fellow grad students for their support, especially Nico Spronk and Philippe Larocque for their help with latex. -
Shift Cipher Substitution Cipher Vigenère Cipher Hill Cipher
Lecture 2 Classical Cryptosystems Shift cipher Substitution cipher Vigenère cipher Hill cipher 1 Shift Cipher • A Substitution Cipher • The Key Space: – [0 … 25] • Encryption given a key K: – each letter in the plaintext P is replaced with the K’th letter following the corresponding number ( shift right ) • Decryption given K: – shift left • History: K = 3, Caesar’s cipher 2 Shift Cipher • Formally: • Let P=C= K=Z 26 For 0≤K≤25 ek(x) = x+K mod 26 and dk(y) = y-K mod 26 ʚͬ, ͭ ∈ ͔ͦͪ ʛ 3 Shift Cipher: An Example ABCDEFGHIJKLMNOPQRSTUVWXYZ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 • P = CRYPTOGRAPHYISFUN Note that punctuation is often • K = 11 eliminated • C = NCJAVZRCLASJTDQFY • C → 2; 2+11 mod 26 = 13 → N • R → 17; 17+11 mod 26 = 2 → C • … • N → 13; 13+11 mod 26 = 24 → Y 4 Shift Cipher: Cryptanalysis • Can an attacker find K? – YES: exhaustive search, key space is small (<= 26 possible keys). – Once K is found, very easy to decrypt Exercise 1: decrypt the following ciphertext hphtwwxppelextoytrse Exercise 2: decrypt the following ciphertext jbcrclqrwcrvnbjenbwrwn VERY useful MATLAB functions can be found here: http://www2.math.umd.edu/~lcw/MatlabCode/ 5 General Mono-alphabetical Substitution Cipher • The key space: all possible permutations of Σ = {A, B, C, …, Z} • Encryption, given a key (permutation) π: – each letter X in the plaintext P is replaced with π(X) • Decryption, given a key π: – each letter Y in the ciphertext C is replaced with π-1(Y) • Example ABCDEFGHIJKLMNOPQRSTUVWXYZ πBADCZHWYGOQXSVTRNMSKJI PEFU • BECAUSE AZDBJSZ 6 Strength of the General Substitution Cipher • Exhaustive search is now infeasible – key space size is 26! ≈ 4*10 26 • Dominates the art of secret writing throughout the first millennium A.D. -
Encryption Is Futile: Delay Attacks on High-Precision Clock Synchronization 1
R. ANNESSI, J. FABINI, F.IGLESIAS, AND T. ZSEBY: ENCRYPTION IS FUTILE: DELAY ATTACKS ON HIGH-PRECISION CLOCK SYNCHRONIZATION 1 Encryption is Futile: Delay Attacks on High-Precision Clock Synchronization Robert Annessi, Joachim Fabini, Felix Iglesias, and Tanja Zseby Institute of Telecommunications TU Wien, Austria Email: fi[email protected] Abstract—Clock synchronization has become essential to mod- endanger control decisions, and adversely affect the overall ern societies since many critical infrastructures depend on a functionality of a wide range of (critical) services that depend precise notion of time. This paper analyzes security aspects on accurate time. of high-precision clock synchronization protocols, particularly their alleged protection against delay attacks when clock syn- In recent years, security of clock synchronization received chronization traffic is encrypted using standard network security increased attention as various attacks on clock synchronization protocols such as IPsec, MACsec, or TLS. We use the Precision protocols (and countermeasures) were proposed. For this Time Protocol (PTP), the most widely used protocol for high- reason, clock synchronization protocols need to be secured precision clock synchronization, to demonstrate that statistical whenever used outside of fully trusted network environments. traffic analysis can identify properties that support selective message delay attacks even for encrypted traffic. We furthermore Clock synchronization protocols are specifically susceptible to identify a fundamental conflict in secure clock synchronization delay attacks since the times when messages are sent and between the need of deterministic traffic to improve precision and received have an actual effect on the receiver’s notion of the need to obfuscate traffic in order to mitigate delay attacks. -
Classic Crypto
Classic Crypto Classic Crypto 1 Overview We briefly consider the following classic (pen and paper) ciphers o Transposition ciphers o Substitution ciphers o One-time pad o Codebook These were all chosen for a reason o We see same principles in modern ciphers Classic Crypto 2 Transposition Ciphers In transposition ciphers, we transpose (scramble) the plaintext letters o The scrambled text is the ciphertext o The transposition is the key Corresponds to Shannon’s principle of diffusion (more about this later) o This idea is widely used in modern ciphers Classic Crypto 3 Scytale Spartans, circa 500 BC Wind strip of leather around a rod Write message across the rod T H E T I M E H A S C O M E T H E W A L R U S S A I D T O T A L K O F M A N Y T H I N G S When unwrapped, letters are scrambled TSATAHCLONEORTYTMUATIESLHMTS… Classic Crypto 4 Scytale Suppose Alice and Bob use Scytale to encrypt a message o What is the key? o How hard is it for Trudy to break without key? Suppose many different rod diameters are available to Alice and Bob… o How hard is it for Trudy to break a message? o Can Trudy attack messages automatically—without manually examining each putative decrypt? Classic Crypto 5 Columnar Transposition Put plaintext into rows of matrix then read ciphertext out of columns For example, suppose matrix is 3 x 4 o Plaintext: SEETHELIGHT o Ciphertext: SHGEEHELTTIX Same effect as Scytale o What is the key? Classic Crypto 6 Keyword Columnar Transposition For example o Plaintext: CRYPTOISFUN o Matrix 3 x 4 and keyword MATH o Ciphertext: -
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. -
Towards Low Energy Stream Ciphers
IACR Transactions on Symmetric Cryptology ISSN 2519-173X, Vol. 2018, No. 2, pp. 1–19. DOI:10.13154/tosc.v2018.i2.1-19 Towards Low Energy Stream Ciphers Subhadeep Banik1, Vasily Mikhalev2, Frederik Armknecht2, Takanori Isobe3, Willi Meier4, Andrey Bogdanov5, Yuhei Watanabe6 and Francesco Regazzoni7 1 Security and Cryptography Laboratory (LASEC), École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland [email protected] 2 University of Mannheim, Mannheim, Germany {mikhalev,armknecht}@uni-mannheim.de 3 University of Hyogo, Hyogo, Japan [email protected] 4 University of Applied Sciences and Arts Northwestern Switzerland (FHNW), Windisch, Switzerland [email protected] 5 Department of Applied Mathematics and Computer Science (DTU Compute), Technical University of Denmark, Kongens Lyngby, Denmark [email protected] 6 National Institute of Advanced Industrial Science and Technology, Osaka, Japan [email protected] 7 Università della Svizzera italiana (USI), Lugano, Switzerland [email protected] Abstract. Energy optimization is an important design aspect of lightweight cryptog- raphy. Since low energy ciphers drain less battery, they are invaluable components of devices that operate on a tight energy budget such as handheld devices or RFID tags. At Asiacrypt 2015, Banik et al. presented the block cipher family Midori which was designed to optimize the energy consumed per encryption and which reduces the energy consumption by more than 30% compared to previous block ciphers. However, if one has to encrypt/decrypt longer streams of data, i.e. for bulk data encryption/decryption, it is expected that a stream cipher should perform even better than block ciphers in terms of energy required to encrypt.