Java Applets for a Cryptology Course. Ulrich A. Hoensch Rocky Mountain College Billings, Montana

Transcription

1 Java Applets for a Cryptology Course Ulrich A. Hoensch Rocky Mountain College Billings, Montana San Antonio, Texas January

2 Plaintext + Can be any keyboard-entered or cut-and-paste text. + Internally, all whitespace, and all non-alphabetic characters are discarded; alphabetic characters are all converted to lowerspace. + Character set for plaintext: P = {a,b,...,z}. Encryption Problem Given a plain text P[N] =(P 1,P 2,...,P N ) consisting of N characters P i P, generate a cipher text C[M] =(C 1,C 2,...,C M ),withc i C,where C is the set of ciphertext characters. Assumptions: + Character set for ciphertext: C = {A,B,...,Z}. + M = N (plaintext and ciphertext have equal number of characters). A one-to-one map χ : P[N] C[N] is called a cipher for the plaintext P[N]. 2

3 Monoalphabetic Encryption A bijective map κ : P Cis called the key for the monoalphabetic encryption cipher χ :(P 1,P 2,...,P N ) (κ(p 1 ),κ(p 2 ),...,κ(p N )). Monoalphabetic Encryption Applet + Generates random keys; + allows user-defined keys; + option: can use affine ciphers; + option: can use (monoalphabetic) keyword ciphers. Affine Ciphers Using the numerical coding a 0 A, b 1 B,..., z 25 Z, we can suppose that P = C = Z 26. An affine cipher is then given by the key κ a,b : Z 26 Z 26, X a X + b, wheregcd(a, 26) = 1, anda, b Z 26. 3

4 Statistical Analysis Applet + Can be used for cryptanalysis of monoalphabetic substitution ciphers; + displays histogram for single character frequencies; + displays histogram for digraph frequencies; + allows single character decryption. If the plaintext is English, the expected single character and digraph frequencies are: % abcdefghijklmnopqrstuvwxyz Frequency distribution of characters in the English language 4

5 per th er on an re he in ed nd ha at en es of or Distribution of most common digraphs in the English language 5

6 The Vigenère Cipher The Vigenère Cipher is a periodic, polyalphabetic keyword cipher. Using Z 26 to code plaintext, ciphertext, and keyword, the Vigenère Cipher (using the standard tableau) can be written as (C 1,C 2,...,C N )=χ(p 1,P 2,...,P N ), C i =(P i + K[(i 1)%p]) %26, where K =(K[0],K[1],...,K[p 1]) (Z 26 ) p is the keyword, and i i%n is reduction modulo n. Vigenère Cipher Applet + Encrypts using the standard Vigenère tableau; + decrypts using the standard Vigenère tableau; + possible future extension: support other Vigenère tableaux. 6

7 The standard Vigenère tableau 7

8 Cryptanalysis of Vigenère Ciphers I. Determining Keyword Length: Kasiski Test Applet + Counts trigraph repetitions, records the distances in the ciphertext at which each trigraph repeats, and computes the greatest common divisor (GCD) of these distances for each trigraph; + trigraphs that repeat in the plaintext at a multiple of the keyword length will also do so in the ciphertext; + also have random (non-periodic) repetitions of trigraphs in the ciphertext; + histogram shows the frequencies of the GCD; note frequency peak at the keyword length, and the period-echo. 8

9 Cryptanalysis of Vigenère Ciphers II. Finding the Keyword: Polyalphabetic Analysis Applet + Enter (probable) keyword length; + displays histograms of sections of the ciphertext; +ifavigenère Cipher (using the standard tableau) with keyword length p has been used, the ith section, i =1, 2,...,p, was encrypted using a shift cipher X X + K[i 1]; + thus the keyword can be found by detecting the position corresponding to the plaintext character a. 9

10 Hill Ciphers Hill Ciphers are block ciphers where blocks X of a fixed length n are encrypted using a fixed affine cipher X A X + B, where A GL (n, Z 26 ),andb (Z 26 ) n. Hill Cipher Applet + Allows encryption using Hill ciphers of dimension n =2to n =5; + can be used for decryption, provided the inverse matrix A 1 has been computed in GL (n, Z 26 ); + does not check whether A GL (n, Z 26 ); + for cryptanalysis, the Kasiski Test Applet can be used to indicate the dimension used for encryption. 10

11 Jefferson s Wheel Cipher Jefferson s Wheel Cipher is a keyword-based block cipher. It physically consists of 26 wheels, the arrangement of which is given by the keyword used. I. Student Project: Build a working physical model of Jefferson s Wheel Cipher. Jefferson s Wheel Cipher, by Magon Jones, 2005 II. Student Project: Write a Java program implementing Jefferson s Wheel Cipher (by Randy Senn, 2005). java CipherGenerator generates a virtual JW Cipher; java JeffersonWheelCipher calls the encryption / decryption program. 11

12 Conclusion + A cryptology class is a great way to get students interested in modular arithmetic, linear and abstract algebra, probability theory, and number theory; + the applets presented facilitate encryption, decryption, and cryptanalysis for some standard classical ciphers; + the applets give students with a weaker background in math access to cryptological tools they can be used in a cryptological methods course, designed as a general education course in liberal arts colleges. For more information, please contact: Ulrich Hoensch hoenschu@rocky.edu 12