A Comparative Study of Classical Substitution Ciphers
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COS433/Math 473: Cryptography Mark Zhandry Princeton University Spring 2017 Cryptography Is Everywhere a Long & Rich History
COS433/Math 473: Cryptography Mark Zhandry Princeton University Spring 2017 Cryptography Is Everywhere A Long & Rich History Examples: • ~50 B.C. – Caesar Cipher • 1587 – Babington Plot • WWI – Zimmermann Telegram • WWII – Enigma • 1976/77 – Public Key Cryptography • 1990’s – Widespread adoption on the Internet Increasingly Important COS 433 Practice Theory Inherent to the study of crypto • Working knowledge of fundamentals is crucial • Cannot discern security by experimentation • Proofs, reductions, probability are necessary COS 433 What you should expect to learn: • Foundations and principles of modern cryptography • Core building blocks • Applications Bonus: • Debunking some Hollywood crypto • Better understanding of crypto news COS 433 What you will not learn: • Hacking • Crypto implementations • How to design secure systems • Viruses, worms, buffer overflows, etc Administrivia Course Information Instructor: Mark Zhandry (mzhandry@p) TA: Fermi Ma (fermima1@g) Lectures: MW 1:30-2:50pm Webpage: cs.princeton.edu/~mzhandry/2017-Spring-COS433/ Office Hours: please fill out Doodle poll Piazza piaZZa.com/princeton/spring2017/cos433mat473_s2017 Main channel of communication • Course announcements • Discuss homework problems with other students • Find study groups • Ask content questions to instructors, other students Prerequisites • Ability to read and write mathematical proofs • Familiarity with algorithms, analyZing running time, proving correctness, O notation • Basic probability (random variables, expectation) Helpful: • Familiarity with NP-Completeness, reductions • Basic number theory (modular arithmetic, etc) Reading No required text Computer Science/Mathematics Chapman & Hall/CRC If you want a text to follow along with: Second CRYPTOGRAPHY AND NETWORK SECURITY Cryptography is ubiquitous and plays a key role in ensuring data secrecy and Edition integrity as well as in securing computer systems more broadly. -
Battle Management Language: History, Employment and NATO Technical Activities
Battle Management Language: History, Employment and NATO Technical Activities Mr. Kevin Galvin Quintec Mountbatten House, Basing View, Basingstoke Hampshire, RG21 4HJ UNITED KINGDOM [email protected] ABSTRACT This paper is one of a coordinated set prepared for a NATO Modelling and Simulation Group Lecture Series in Command and Control – Simulation Interoperability (C2SIM). This paper provides an introduction to the concept and historical use and employment of Battle Management Language as they have developed, and the technical activities that were started to achieve interoperability between digitised command and control and simulation systems. 1.0 INTRODUCTION This paper provides a background to the historical employment and implementation of Battle Management Languages (BML) and the challenges that face the military forces today as they deploy digitised C2 systems and have increasingly used simulation tools to both stimulate the training of commanders and their staffs at all echelons of command. The specific areas covered within this section include the following: • The current problem space. • Historical background to the development and employment of Battle Management Languages (BML) as technology evolved to communicate within military organisations. • The challenges that NATO and nations face in C2SIM interoperation. • Strategy and Policy Statements on interoperability between C2 and simulation systems. • NATO technical activities that have been instigated to examine C2Sim interoperation. 2.0 CURRENT PROBLEM SPACE “Linking sensors, decision makers and weapon systems so that information can be translated into synchronised and overwhelming military effect at optimum tempo” (Lt Gen Sir Robert Fulton, Deputy Chief of Defence Staff, 29th May 2002) Although General Fulton made that statement in 2002 at a time when the concept of network enabled operations was being formulated by the UK and within other nations, the requirement remains extant. -
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. -
Affine Cipher Project 1 Introduction
Affine Cipher Project 141KECBZ0H5CRK1HUZK1CGPCR.5PUGUZU1WCU.CM1CUBHUCAK.6.Z5WCP1RK1UCH5 WC0EPU1KECU.C141KEC.UB1KXC,,RBHKV1PCWGRQ15P7CHCUHV1C.6CU9.CRGUG1P Directions: • Answer all numbered questions completely. • Show non-trivial work, and put your final answer in the box provided. • Questions without boxes should be answered in complete sentences in the space provided. 1 Introduction Cryptography is the study of secret codes, or the secure transmission of information that nobody except the desired recipient can read. By the end of this project, you will be able to decipher the quote printed above. The mathematical study of ciphers will lead us through a world in which the number line is a closed curve, and fractions do not exist. This project is designed to help you to: • read and understand definitions and notation • observe patterns and generalize • think logically, analytically, and abstractly • express problems and solutions precisely • follow examples • combine ideas to solve problems and create applications 1 2 Caesar Cipher A cipher is a function or algorithm for translating plaintext into encrypted ciphertext. Throughout history, governments and merchants have used ciphers to safely transmit sensitive information. Julius Caesar is said to have use a simple system of substituting each letter with the letter 3 spots over, wrapping around the alphabet if necessary. Here is the mapping: 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 # # # # # # # # # # # # # # # # # # # # # # # # # # D E F G H I J K L M N O P Q R S T U V W X Y Z A B C Space and punctuation are discarded. -
Elements of Cryptography
Elements Of Cryptography The discussion of computer security issues and threats in the previous chap- ters makes it clear that cryptography provides a solution to many security problems. Without cryptography, the main task of a hacker would be to break into a computer, locate sensitive data, and copy it. Alternatively, the hacker may intercept data sent between computers, analyze it, and help himself to any important or useful “nuggets.” Encrypting sensitive data com- plicates these tasks, because in addition to obtaining the data, the wrongdoer also has to decrypt it. Cryptography is therefore a very useful tool in the hands of security workers, but is not a panacea. Even the strongest cryp- tographic methods cannot prevent a virus from damaging data or deleting files. Similarly, DoS attacks are possible even in environments where all data is encrypted. Because of the importance of cryptography, this chapter provides an introduction to the principles and concepts behind the many encryption al- gorithms used by modern cryptography. More historical and background ma- terial, descriptions of algorithms, and examples, can be found in [Salomon 03] and in the many other texts on cryptography, code breaking, and data hiding that are currently available in libraries, bookstores, and the Internet. Cryptography is the art and science of making data impossible to read. The task of the various encryption methods is to start with plain, readable data (the plaintext) and scramble it so it becomes an unreadable ciphertext. Each encryption method must also specify how the ciphertext can be de- crypted back into the plaintext it came from, and Figure 1 illustrates the relation between plaintext, ciphertext, encryption, and decryption. -
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 -
Short History Polybius's Square History – Ancient Greece
CRYPTOLOGY : CRYPTOGRAPHY + CRYPTANALYSIS Polybius’s square Polybius, Ancient Greece : communication with torches Cryptology = science of secrecy. How : 12345 encipher a plaintext into a ciphertext to protect its secrecy. 1 abcde The recipient deciphers the ciphertext to recover the plaintext. 2 f g h ij k A cryptanalyst shouldn’t complete a successful cryptanalysis. 3 lmnop 4 qrstu Attacks [6] : 5 vwxyz known ciphertext : access only to the ciphertext • known plaintexts/ciphertexts : known pairs TEXT changed in 44,15,53,44. Characteristics • (plaintext,ciphertext) ; search for the key encoding letters by numbers chosen plaintext : known cipher, chosen cleartexts ; • shorten the alphabet’s size • search for the key encode• a character x over alphabet A in y finite word over B. Polybius square : a,...,z 1,...,5 2. { } ! { } Short history History – ancient Greece J. Stern [8] : 3 ages : 500 BC : scytale of Sparta’s generals craft age : hieroglyph, bible, ..., renaissance, WW2 ! • technical age : complex cipher machines • paradoxical age : PKC • Evolves through maths’ history, computing and cryptanalysis : manual • electro-mechanical • by computer Secret key : diameter of the stick • History – Caesar Goals of cryptology Increasing number of goals : secrecy : an enemy shouldn’t gain access to information • authentication : provides evidence that the message • comes from its claimed sender signature : same as auth but for a third party • minimality : encipher only what is needed. • Change each char by a char 3 positions farther A becomes d, B becomes e... The plaintext TOUTE LA GAULE becomes wrxwh od jdxoh. Why enciphering ? The tools Yesterday : • I for strategic purposes (the enemy shouldn’t be able to read messages) Information Theory : perfect cipher I by the church • Complexity : most of the ciphers just ensure computational I diplomacy • security Computer science : all make use of algorithms • Mathematics : number theory, probability, statistics, Today, with our numerical environment • algebra, algebraic geometry,.. -
With Answers
Module 10.4: Exploring Some Historical Ciphers Gregory V. Bard January 30, 2020 • This is a practice workbook the Affine Cipher, touching on some shift ciphers (such as the Caesar cipher, the ROT-13 cipher), the Atbash cipher, and the Vigen`ereCipher. • The Affine Cipher was introduced in Module 10-2: The Basics of Modular Arithmetic, and some skills from Module 10-3: Modular Inverses are needed. • There is a with-answers version, and a without-answers version. • In the with-answers version of this workbook, the black ink represents the question, and the blue ink represents the answer. Question 10-4-1 Suppose Boris normally communicates with his handlers in Moscow using the affine cipher, and the encryption function c = fB(p) = 7p + 20 mod 26, while Natasha normally uses c = fN (p) = 11p + 8 mod 26. They have a very secret message to send back to Moscow, so they're going to encrypt the message twice, for added security. As you can probably guess, this is equivalent to using the affine cipher only once, but with a different function. • If Boris encrypts the plaintext first, followed by Natasha second, then what would BAT encrypt to? (B, A, T) becomes (1, 0, 19) and encrypts to (fB(1); fB(0); fB(19)) ≡ (1; 20; 23). We encrypt again to (fN (1); fN (20); fN (23)) ≡ (19; 20; 1), which becomes (T, U, B) or TUB. • If Natasha encrypts the plaintext first, followed by Boris second, then what would BAT encrypt to? (B, A, T) becomes (1, 0, 19) and encrypts to (fN (1); fN (0); fN (19)) ≡ (19; 8; 9). -
MAT 302: LECTURE SUMMARY Last Class We Discussed Two Classical Ciphers, Both of Which Turned out to Be Rather Insecure (As Evide
MAT 302: LECTURE SUMMARY Last class we discussed two classical ciphers, both of which turned out to be rather insecure (as evidenced by your cracking them manually during lecture): • The Scytale cipher • The Caesar (aka shift) cipher We began today’s lecture by writing the latter down in mathematical notation. To this end, it is convenient to use the set A = f0; 1; 2;:::; 25g to represent the English alphabet, i.e. 0 represents A, 1 represents B, etc. The Caesar Cipher. The Caesar cipher (or shift cipher) consists of a key k 2 Z, an encryption function ek : A −! A x 7−! x + k (mod 26) and a decryption function dk : A −! A x 7−! x − k (mod 26): The Caesar cipher is insecure because the space of all possible keys is rather small (how many are there?), so it is a trivial matter for Oscar to check all possible keys. We next discussed a variant of this, whose key space is somewhat larger: the affine cipher. The Affine Cipher. The affine cipher consists of a key (a; b) 2 Z2, an encryption function e(a;b) : A −! A x 7−! ax + b (mod 26) and a decryption function d(a;b) : A −! A x 7−! a−1(x − b) (mod 26) Actually, as stated this isn’t well-defined: a−1 doesn’t always exist (mod 26). Recall that a−1 is the element of A satisfying a−1a ≡ 1 (mod 26): For example, 0 has no (multiplicative) inverse (mod 26). Less trivially, 2 has no multiplicative inverse. -
FROM JULIUS CAESAR to the BLOCKCHAIN: a BRIEF HISTORY of CRYPTOGRAPHY by Côme JEAN JARRY & Romain ROUPHAEL, Cofounders of BELEM
Corporate identity in the digital era #9 JANUARY 2017 FROM JULIUS CAESAR TO THE BLOCKCHAIN: A BRIEF HISTORY OF CRYPTOGRAPHY By Côme JEAN JARRY & Romain ROUPHAEL, cofounders of BELEM 22 The world’s most important asset is information. Now message was inscribed lengthwise. Once the parchment more than ever. With computer theft and hacking was unrolled, the letters of the message were mixed becoming a common threat, protecting information up and the message meaningless. The receiver would is crucial to ensure a trusted global economy. need an identical stick to decipher the text. The E-commerce, online banking, social networking or scytale transposition cipher relied on changing the emailing, online medical results checking, all our order of the letters, rather than the letters themselves. transactions made across digital networks and This cryptographic technique still prevails today. insecure channels of communication, such as the Internet, mobile phones or ATMs, are subjected to vulnerabilities. Our best answer is cryptography. And THE ART OF SUBSTITUTION it has always been. As a science and as an art, it is Julius Caesar was also known to use encryption to an essential way to protect communication. convey messages to his army generals posted in the Cryptography goes back to older times, as far back as war front. The Caesar cipher is a simple substitution the Ancient World. cipher in which each letter of the plaintext is rotated left or right by some number of positions down the alphabet. The receiver of the message would then Early cryptography was solely concerned with shift the letters back by the same number of positions concealing and protecting messages. -
Tap Code 1 2 3 4 5 1 a B C D E 2 F G H I J 3 L M N O P 4 Q R S T U 5
Solution to Kryptos 2: Challenge 1 The audio files indicate that the prisoners are communicating with each other by “tapping”. There are two distinct tapping sounds which indicate the two prisoners are exchanging some sort of information back and forth. Furthermore, unlike Morse Code, each audio file is made up of pairs of taps which leads one to believe that each pair of taps corresponds to a word or letter. The longest tapping sequence is five, so if one assumes that each tap-pair corresponds to a pair of integers between 1 and 5, then this code could encipher 25 items – most likely letters. At this point, one could try a frequency analysis to help identify which pairs of integers correspond to which letters. Alternatively, a little research might yield the “tap code” (see e.g. Wikipedia) which has been used by prisoners of war in various conflicts throughout history, most recently Vietnam. It is based on the Polybius square, some of which have slightly different arrangements of the 26 letters placed into 25 cells. Challenge 1 was based on the following key found on Wikipedia: Tap code 1 2 3 4 5 1 A B C D E 2 F G H I J 3 L M N O P 4 Q R S T U 5 V W X Y Z The tap code table This has all letters except “K”, so “C” is used instead. The letter “X” is used to separate sentences. The first audio file begins with the following tap-pairs: (2,4), (4,3), (1,3), (3,4), (5,1), (1,5), (4,2),… which yields the following plaintext: i, s, c, o, v, e, r, … By adding in the likely punctuation, the complete conversation can be decoded: PLAINTEXT Prisoner 1: Is cover intact? Prisoner 2: Yes. -
An Introduction to Cryptography
abstract examples cryptanalysis harder cryptography schemes An Introduction to Cryptography Joe Fields http://www.southernct.edu/~fields/ Joe Fields Cryptography abstract examples Terminology cryptanalysis harder cryptography schemes Cryptography is the study of "secret writing." This is the only branch of mathematics to be designated by the U.S. government as export-controlled. Cryptographic knowledge is considered to be "war materials!" While we won't head off into TOP SECRET territory we will have a bit of fun working out how to make (and to break) good secret codes. Joe Fields Cryptography abstract examples Terminology cryptanalysis harder cryptography schemes the enigma Joe Fields Cryptography abstract examples Terminology cryptanalysis harder cryptography schemes WACs Joe Fields Cryptography Cryptology means \the study of secrets" practically speaking, they are synonyms... abstract examples Terminology cryptanalysis harder cryptography schemes Cryptography or Cryptology? Cryptography means \secret writing" Joe Fields Cryptography practically speaking, they are synonyms... abstract examples Terminology cryptanalysis harder cryptography schemes Cryptography or Cryptology? Cryptography means \secret writing" Cryptology means \the study of secrets" Joe Fields Cryptography abstract examples Terminology cryptanalysis harder cryptography schemes Cryptography or Cryptology? Cryptography means \secret writing" Cryptology means \the study of secrets" practically speaking, they are synonyms... Joe Fields Cryptography Bob (the recipient) but they are afraid that. Eve (the eavesdropper) will snoop on them and learn their secrets. abstract examples Terminology cryptanalysis harder cryptography schemes Cast of Characters Alice (the sender) wants to send a message to. Joe Fields Cryptography Eve (the eavesdropper) will snoop on them and learn their secrets. abstract examples Terminology cryptanalysis harder cryptography schemes Cast of Characters Alice (the sender) wants to send a message to.