Algebraic Solutions of Caesar and Multiplicative Ciphers

Transcription

1 Spring 2015 Chris Christensen MAT/CSC 483 Algebraic Solutions of Caesar and Multiplicative Ciphers The Caesar cipher and the multiplicative cipher each involve one algebraic operation addition modulo 26 and multiplication modulo 26, respectively. The enciphering operations are (where CT is ciphertext and pt is plaintext): Caesar cipher enciphering CT = pt + key mod 26 Multiplicative cipher enciphering CT = pt key mod 26 To decipher, the operation is repeated using the key inverse in place of the key. Caesar cipher deciphering pt = CT + key inverse mod 26 Multiplicative cipher deciphering pt = CT key inverse mod 26 For Caesar ciphers there are 26 possible keys. The keys and their inverses are: Key Additive Inverse modulo Key Additive Inverse modulo

2 For multiplicative ciphers there are only 12 possible keys. The keys and their inverses are: Key Multiplicative Inverse modulo Algebraic Solution of Caesar Ciphers CT = pt + key mod 26 Caesar cipher enciphering is a linear relationship among cipher text (CT), plaintext (pt), and key. If CT and pt are known, the modular equation above can be solved for key. Consider the ciphertext: VXMNA WLAHY CXPAJ YQHRB JORNU MCQJC MAJFB QNJER UHXWV JCQNV JCRLB XVYDC NABLR NWLNJ WMLUN ENAWN BB

3 Frequency analysis of the ciphertext shows: Because e is the most frequent plaintext letter in English, we might believe that ciphertext N corresponds to plaintext e. If so, e(5) N(14) e + key = N mod26 5+key = 14 mod key = mod 26 key = 9 mod 26 key inverse = 17mod 26 This suggests that the key is 9 and the key inverse is 17. Try them.

4 Algebraic Solution of Multiplicative Ciphers CT = pt key mod 26 Multiplicative cipher enciphering is also a linear relationship among cipher text (CT), plaintext (pt), and key. If CT and pt are known, the modular equation above can be solved for key. Consider the ciphertext: XTSVI XCEVI DOSAG FCXQI KSVAQ COXTS DIFKS OXIVJ NEXSV XCIDD QMEOX CVXFG OCPSI MSFCA IVCVX SDDCK SVASI KSVAQ Frequency analysis of the ciphertext shows:

5 Because e is the most frequent plaintext letter in English, we might believe that ciphertext S corresponds to plaintext e. If so, e(5) S(19) e + key = N mod26 5 key = 19 mod key = mod 26 key = 9 mod 26 key inverse = 3mod 26 This suggests that the key is 9 and the key inverse is 3. Here is the deciphering of the first 5-letter block. X T S V I mod t h e n a

6 Simple Substitution Ciphers Because there is an algebraic relationship between plaintext and ciphertext letters for both Caesar ciphers and multiplicative ciphers, there is a pattern in the key that can be solved for algebraically. What about simple substitution ciphers for which the key is randomly generated? Like Plaintext letters: abcdefghijklmnopqrstuvwxyz Ciphertext letters: EKMFLGDQVZNTOWYHXUSPAIBRCJ It is possible that by using frequency analysis we might be able to spot that plaintext e was enciphered by ciphertext L, but because there is no algebraic relationship between plaintext and ciphertext knowing that e became L would not provide any information about the correspondences between the remaining plaintext-ciphertext pairs. There is no pattern in the key; the key has been randomly generated.

7 Exercises 1. A message is known to be encrypted with a Caesar cipher. We suspect that plaintext e corresponds to ciphertext R. What is the additive key? Repeat the exercise with plaintext e corresponding to ciphertext B. 2. A message is known to be encrypted with a Caesar cipher. If plaintext e corresponds to ciphertext T, what is the additive key? To what ciphertext letter would plaintext o correspond? 3. A message is known to be encrypted with a Caesar cipher. If plaintext e corresponds to ciphertext C, what is the additive key? To what ciphertext letter would plaintext m correspond? 4. A message is known to be encrypted with a multiplicative cipher. We suspect that plaintext e corresponds to ciphertext G. What is the multiplicative key? Repeat the exercise with plaintext e corresponding to ciphertext A. 5. A message is known to be encrypted with a multiplicative cipher. If plaintext e corresponds to ciphertext O, what is the multiplicative key? To what ciphertext letter would plaintext d correspond? 6. A message is known to be encrypted with a multiplicative cipher. If plaintext e corresponds to ciphertext G, what is the multiplicative key? To what ciphertext letter would plaintext l correspond? 7. A message is known to be encrypted with a multiplicative cipher. We believe that plaintext e corresponds to ciphertext N, what can we conclude?