Substitution Ciphers

Transcription

1 crypttext 2018/1/9 13:06 page 7 #11 Chapter 2 Substitution Ciphers One common and popular type of cipher for newspaper games and puzzle books is a substitution cipher. In simple substitution ciphers, users agree upon a rearrangement, or permutation, of the alphabet letters, yielding a collection of correspondences to be used for converting plaintext letters into ciphertext letters. This rearrangement of the alphabet letters is often called the cipher alphabet. To say that the cipher alphabet is a permutation means that each possible plaintext letter in the original alphabet is paired with one and only one possible ciphertext letter, and vice versa. With more sophisticated substitution ciphers, messages and cipher alphabets can include numbers, punctuation marks, or mixtures of multiple characters, and substitutions can be made for entire words or phrases. As we will see in this chapter, simple substitution ciphers are not very secure (meaning they are easy to break). Despite this, substitution ciphers have a rich history of being used. One of the earliest known ciphers, the Hebrew Atbash, was a substitution cipher, as was a cipher developedand used by Julius Caesar that we will consider in Chapter 6, and a cipher incorporated in Edgar Allan Poe s The Gold Bug. Not all substitution ciphers are easy to break, though. The cipher famously created during World War II by the Navajo code talkers, which we will consider in this chapter, and which was not known to have ever been broken during a period of use that extended through the Korean War and into the early stages of the Vietnam War, was essentially a substitution cipher. 2.1 Keyword Substitution Ciphers One way to form a substitution cipher is to just use a random cipher alphabet. 7

2 crypttext 2018/1/9 13:06 page 8 #12 8 CHAPTER 2. SUBSTITUTION CIPHERS Example 2.1 Consider a substitution cipher with the following cipher alphabet. TVXZUWYADGKNQBEHROSCFJMPIL Using this cipher alphabet, the plaintext YOUTH IS WASTED ON THE YOUNG 1 encrypts to the ciphertext IEFCA DS MTSCUZ EB CAU IEFBY. This ciphertext can be decrypted by using the same cipher alphabet but with the correspondences viewed in the reverse order. A problem with using a random cipher alphabet is that it may be inconvenient or cumbersome for users to keep a record of. For instance, users wishing to use a substitution cipher with the cipher alphabet in Example 2.1 would most likely have to keep a written record of this alphabet. One solution to this problem would be for users to use a keyword in forming the cipher alphabet. In this section, we will demonstrate two methods for doing this Simple Keyword Substitution Ciphers For simple keyword substitution ciphers, users agree upon one or more keywords for the cipher. Spaces and duplicate letters in the keyword(s) are removed, and the resulting letters are then listed in order as the ciphertext letters that correspond to the first plaintext letters in alphabetical order. The remaining alphabet letters not included in the keyword(s) are then listed in alphabetical order to correspond to the remaining plaintext letters in alphabetical order. Example 2.2 Consider a simple keyword substitution cipher with the keywords WILL ROGERS. Removing the space as well as the duplicate letters in these keywords gives WILROGES, and results in the following cipher alphabet. WILROGESABCDFHJKMNPQTUVXYZ Using this cipher alphabet, the plaintext EVERYTHING IS FUNNY AS LONG AS IT IS HAPPENING TO SOMEBODY ELSE 2 encrypts to OUONYQSAHE AP GTHHY WP DJHE WP AQ AP SWKKOHAHE QJ PJFOIJRY ODPO. To decrypt a ciphertext that was formed using this cipher alphabet, we can use the same cipher alphabet but with the characters considered in the reverse order. For example, the ciphertext QSO VADD NJEONP AHPQAQTQO OHEWEOP AH LWNRAJKTDFJHWNY 1 George Bernard Shaw ( ), quote. 2 Will Rogers ( ), quote.

3 crypttext 2018/1/9 13:06 page 9 # KEYWORD SUBSTITUTION CIPHERS 9 FORALWD NOPOWNLS decrypts to THE WILL ROGERS INSTITUTE ENGAGES IN CARDIOPULMONARY MEDICAL RESEARCH. Example 2.2 reveals a problem with simple keyword substitution ciphers. Often with this type of cipher, notably when the keyword doesn t contain any letters near the end of the alphabet, the last several correspondences in the cipher alphabet are letters corresponding to themselves. Such correspondences are called collisions, and can make a cipher more vulnerable to cryptanalysis. Keyword columnar substitution ciphers can help to alleviate this problem Keyword Columnar Substitution Ciphers For keyword columnar substitution ciphers, users again agree upon one or more keywords, and remove spaces and duplicate letters in the keyword(s). The resulting letters are then listed in order in a row, with the alphabet letters not included in the keyword(s) listed in order in successive rows of the same size beneath the keyword letters. The cipher alphabet is then obtained by taking the columns of the resulting array of letters in order starting from the left, and placing these columns as rows under the plaintext letters. Example 2.3 Consider a keyword columnar substitution cipher with keywords ABE LINCOLN. Removing the space and duplicate letters gives ABELINCO, and placing these letters in a row, with the remaining alphabet letters listed in order in successive rows, yields the following array. A B E L I N C O D F G H J K M P Q R S T U V W X Y Z Transcribing this array by columns starting from the left yields the following cipher alphabet. ADQYBFRZEGSLHTIJUNKVCMWOPX Using this cipher alphabet, the plaintext IF I WERE TWO-FACED WOULD I BE WEARING THIS ONE 3 encrypts to EF E WBNB VWI-FAQBY WICLY E DB WBANETR VZEK ITB, and the ciphertext FICN KQINB ATY KBMBT PBANK ARI decrypts to FOUR SCORE AND SEVEN YEARS AGO. 4 3 Abraham Lincoln ( ), quote. 4 Abraham Lincoln, from The Gettysburg Address.

4 crypttext 2018/1/9 13:06 page 10 #14 10 CHAPTER 2. SUBSTITUTION CIPHERS 2.1 Exercises 1. Consider a substitution cipher with the following cipher alphabet. XQKMDBPSETCLORUJVAFWZGHNIY (a) Use this cipher to encrypt QUOTH THE RAVEN NEVERMORE. 5 (b) Use this cipher to encrypt AFEWWORDSONSECRETWRITING. (c) Decrypt WSD BXLL UB WSD SUZFD UB ZFSDA, which was formed using this cipher. 2. Create a substitution cipher with a random cipher alphabet and use it to encrypt a plaintext of your choice with at least 20 letters. 3. Decrypt XPUUMOXREFSXAMWUBERMQZWBERMERPXPUUMMUPEF DXFI, which was formed using a substitution cipher with the following cipher alphabet. XQKMDBPSETCLORUJVAFWZGHNIY 4. Consider a simple keyword substitution cipher with the keyword GILLIGAN. (a) Use this cipher to encrypt ATHREEHOURTOUR. (b) Use this cipher to encrypt ATALEOFAFATEFULTRIP. (c) Decrypt RGQUNJ WNJLD GUAETEOMNA BOR KGRY GMM, which was formed using this cipher. 5. Create a simple keyword substitution cipher and use it to encrypt a plaintext of your choice with at least 20 letters. 6. Decrypt TGLDUCHQ HOTL LSUOLS AGC QRHKSHG E CUPGHO AQ ADUFGT, which was formed using a simple keyword substitution cipher with the keywords APPALACHIAN STATE. 7. Consider a keyword columnar substitution cipher with the keywords MARSHAL DILLON. (a) Use this cipher to encrypt SATURDAY NIGHT IN DODGE CITY. (b) Use this cipher to encrypt CAPTAIN KIRK WAS IN ONE EPISODE. (c) Decrypt MGA HWDKCZ DLYUF MLLCMZCA VYIZ KWHCD, which was formed using this cipher. 5 Edgar Allan Poe ( ), from The Raven.

5 crypttext 2018/1/9 13:06 page 11 # CRYPTANALYSIS OF SUBSTITUTION CIPHERS Create a keyword columnar substitution cipher and use it to encrypt a plaintext of your choice with at least 20 letters. 9. Decrypt NZVAHX XHQZVCD PNQF SNZXWXNHNGXI NHC IQELLY AQQKDDKY, which was formed using a keyword columnar substitution cipher with the keywords NC STATE UNIVERSITY. 10. Create and describe a method different from those illustrated in this section for using a keyword to form the cipher alphabet for a substitution cipher. Give at least one example of your method. 11. Find some information about the Hebrew Atbash cipher, including how it worked, and write a summary of your findings. 12. Find some information about Edgar Allan Poe s interest in cryptography, including the various cryptographic challenges he offered, and write a summary of your findings. 13. Find a copy of Edgar Allan Poe s short story The Gold Bug, and write a summary of how a substitution cipher is integrated into the story. 14. Find a copy of Sir Arthur Conan Doyle s Sherlock Holmes mystery The Adventure of Dancing Men, and write a summary of how a substitution cipher is integrated into the story and how the cipher worked. 2.2 Cryptanalysis of Substitution Ciphers Considering the number of possible cipher alphabets, substitution ciphers seem impossible to break. With 26 letters, there are more than possible cipher alphabets. To test them all would be infeasible. However, as it turns out, most simple substitution ciphers are fairly easy to break through the use of frequency analysis. In fact, inadequate security of substitution ciphers has even altered the course of history. For example, the breaking of a substitution cipher led to the execution of Mary, Queen of Scots in In languages like English, it is known that certain letters and combinations of letters occur more often than others. In ordinary English, the letters that naturally occur the most often are, in order, E, T, A, O, I, N, and S. The frequency with which each of the 26 letters in our alphabet occurs in ordinary English is shown in Table 2.1 on page 12. Common digraphs (letter pairs), trigraphs (letter triples), and repeated letters in ordinary English are also known. The most common digraphs are TH, ER, ON, AN, RE, HE, IN, ED, andnd. The most common trigraphs are THE, AND, THA, ENT, ION, TIO, FOR, NDE, HAS, andnce. The most common