Code Breaking. Kurt Langfeld, David McMullan School of Mathematics & Statistics, University of Plymouth Plymouth, PL4 8AA, UK.

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1 Code Breaking Kurt Langfeld, David McMullan School of Mathematics & Statistics, University of Plymouth Plymouth, PL4 8AA, UK July 7, Encryption and Code Breaking 1.1 Importance In cryptography codes and ciphers are used to encrypt a message and to make it unreadable unless the recipient knows the secret to decrypt it. Encryption has been used for many thousands of years, and it has continuously gained importance for our everyday life. In the beginning, simple substitution cipher were used. These ciphers just swap pairs of letters of the alphabet. Already in 2,000 BC the Atbash code was in use: there, the letters of the alphabet were reversed. In other words, all as are replaced with zs, all bs are replaced with ys, and so on. Rather than our Latin alphabet, the Hebrew alphabet was the basis for the Atbash cipher. Another cipher of great historical importance is the Caesar Figure 1: The Atbash cipher (2000 BC), Hebrew alphabet. cipher. This cipher was unbroken for centuries and will be discussed in more detail below. The Caesar cipher and more sophisticated derivatives of this cipher were used throughout the middle ages. Also widely used were e.g. the Vigenère and the Beaufort cipher. Ciphers have been of great military importance at all times. In the Second World War, the cipher machine Enigma was used to transfer secret messages e.g. between the German headquarter and the German submarine fleet. The exact influence of the breaking of the 1

2 Enigma code on ending the war is under debate. A typical belief is that the European war was shortened by two years. In our modern computing age, cryptography has become an integral part of our everyday life: ciphers are used in e-commerce, online banking and for encrypting a conversation between two mobile phones. Most of the modern applications use Public Key Cryptography and the RSA standard. The general belief is that, for sufficiently large keys, these ciphers are safe. Decrypting is the process to generate plain text from an encrypted message while the secret of the cipher is known. In the case of code breaking, the secret of the cipher is not known, and modern mathematics is used to reconstruct the plain text. In this project, you will be confronted with a message which is encrypted using a yet unknown substitution cipher. You will learn the techniques to recover the plain text in this case. 1.2 Caesar cipher and friends The Caesar cipher is a simple substitution cipher where each letter of the plain text is replaced by a unique other letter of the alphabet. The original Caesar cipher uses a cyclic permutation of the alphabet. The alphabet is shifted by n places to the right where n = specifies the particular type of the Caesar cipher (see figure 2 for an illustration). The number n serves as a key or sort of password. There are only 25 plain text cypher key 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 v w x y z a b c d e f g h i j k l m n o p q r s t u shift 5 Figure 2: The Caesar cipher for n = 5. possibilities for the key in the Caesar cipher. The Caesar cipher is therefore not very safe: once it is known that the message was encrypted using the Caesar cipher, one can break the code by testing by brut force all the 25 possibilities. A more sophisticated version of the Caesar cipher makes use of a real password. In step (i), you write the password ( tuesday in the example below) and you cross out all the letters from the alphabet which appear in the password: 2

3 ab c d e f g h i j k l m n o p q r s t u v w x y z t u e s d a y In step (ii), you fill in the remaining letters, i.e., the letters which are not crossed out, starting with the letter after the last crossed-out letter: ab c d e f g h i j k l m n o p q r s t u v w x y z t u e s d ay z b c f g h i j k l m n o p q r v w x This procedure generates the following substitution cipher: ab c d e f g h i j k l m n o p q r s t u v w x y z encryption decryption t u e s d ay z b c f g h i j k l m n o p q r v w x In order to make this procedure work you must either use a password in which every letter of the alphabet only appears once or you cross out all redundant letters in your password. The advantage of the modified Caesar cipher is that there is a large variety of possible cipher (in fact there are roughly possibilities to encrypt a message). Secondly, the cipher can be easily communicated by simply exchanging the password. Knowing the password, the recipient can easily construct the substitution table. 1.3 Breaking a substitution cypher Substitution ciphers such as the Caesar cipher or even more sophisticated versions can be broken using frequency analysis. This method is based upon the fact the letters in an English text appear with a certain abundance. Figure 3 shows the abundance of the letters for an English text. If we sort the letters according to their abundance, we find E, T, A, O, I, N, S, R, H, L, D, C, U, M, F, P, G, W, Y, B, V, K, X, J, Q, Z Hence, the letter E is the most frequent one in an English text followed by T and A. In order to break a substitution cipher, you firstly count how often the letters appear in the encrypted text. The most frequent letter is most likely to be identified with an E, and the second frequent letter might be a T, etc. Note, however, that, depending on 3

4 Figure 3: Abundance of the letters in an English text [source: wikipedia]. the amount of text you have at your disposal, only the most frequent letter are identified correctly while the order of less frequent letters might be statistically insignificant. Besides solely relying on the abundance of letters, you also can look on the abundance of pairs of letters. Most frequent pairs in an English text are th, er, on, an, re, he, in, ed, nd. Most frequent two letters words also can help to reconstruct the substitution table. Those are (sorted according to abundance) of, to, in, it, is, be, as, at, so, we, he, be, by, or. 4

5 2 Project 2.1 Storyline The Plymouth Headquarter has intercepted two secret messages: a lengthy one and a subsequent short one. There are indications that the second short one contains a precious piece of information. Plymouth Intelligence found out: the second message is encrypted using a password based Caesar cipher; password: unknown. the first message contains a paragraph of a novel; name of the novel: unknown. the surname of the author of this novel is the password for the second short message; surname: unknown. Unfortunately, the second message is too short to allow for a direct frequency analysis. This implies that we must break the code of the first message and find the surname of the novel s author. We have four hours to find the plain text of the second message. We rely on you! 5

6 2.2 Message one: The first message intercepted by Plymouth Intelligence is: pg eqj pz gno cqggoy vqij mb joagorloy, qzv ymutn tqcoj nqv jog pz epgn owfoagpmzqc dpmcozfo. qcc vqi gno epzv nqv jfyoqrov qzv gno yqpz nqv loqgoz qtqpzjg gno epzvmej, jm gnqg odoz noyo pz gno noqyg mb tyoqg, nqzv-rqvo cmzvmz eo eoyo bmyfov gm yqpjo muy rpzvj bmy gno pzjgqzg bymr gno ymugpzo mb cpbo qzv gm yofmtzpjo gno ayojozfo mb gnmjo tyoqg ocorozgqc bmyfoj enpfn jnypos qg rqzspzv gnymutn gno lqyj mb npj fpdpcpjqgpmz, cpso uzgqrov loqjgj pz q fqto. qj odozpzt vyoe pz, gno jgmyr tyoe nptnoy qzv cmuvoy, qzv gno epzv fypov qzv jmllov cpso q fnpcv pz gno fnprzoi. jnoycmfs nmcroj jqg rmmvpci qg mzo jpvo mb gno bpyoacqfo fymjj-pzvowpzt npj yofmyvj mb fypro, enpco p qg gno mgnoy eqj vooa pz mzo mb fcqys yujjoccj bpzo joq-jgmypoj uzgpc gno nmec mb gno tqco bymr epgnmug joorov gm lcozv epgn gno gowg, qzv gno jacqjn mb gno yqpz gm coztgnoz mug pzgm gno cmzt jeqjn mb gno joq eqdoj. ri epbo eqj mz q dpjpg gm noy rmgnoyj, qzv bmy q boe vqij p eqj q veoccoy mzfo rmyo pz ri mcv kuqygoyj qg lqsoy jgyoog. 2.3 Message two: opchhcdjr! wks lcl cr. cjrejqczw wksp idrbeidrcydh qgchhq djl ue icabr evmeyr apedr rbcjaq zpki wks. 6

7 References [1] Simon Singh, The Code Book: The Secret History of Codes and Code-breaking Popular Science; Fourth Estate; New Ed edition (8 Jun 2000). [2] Robert Harris, Enigma, Novel, Arrow Books Ltd; New Ed edition (2 May 1996). 7