Abstract Algebra Theory and Applications - Computer Science ...

98 CHAPTER 6 INTRODUCTION TO CRYPTOGRAPHYSystems that use two separate keys, one for encoding and another fordecoding, are called public key cryptosystems. Since knowledge of theencoding key does not allow anyone to guess at the decoding key, the encodingkey can be made public. A public key cryptosystem allows A and Bto send messages to C using the same encoding key. Anyone is capable ofencoding a message to be sent to C, but only C knows how to decode sucha message.6.1 Private Key CryptographyIn single or private key cryptosystems the same key is used for bothencrypting and decrypting messages. To encrypt a plaintext message, weapply to the message some function which is kept secret, say f. This functionwill yield an encrypted message. Given the encrypted form of the message,we can recover the original message by applying the inverse transformationf −1 . The transformation f must be relatively easy to compute, as mustf −1 ; however, f must be extremely difficult to guess at if only examples ofcoded messages are available.Example 1. One of the first and most famous private key cryptosystemswas the shift code used by Julius Caesar. We first digitize the alphabet byletting A = 00, B = 01, . . . , Z = 25. The encoding function will bef(p) = p + 3 mod 26;that is, A ↦→ D, B ↦→ E, . . . , Z ↦→ C. The decoding function is thenf −1 (p) = p − 3 mod 26 = p + 23 mod 26.Suppose we receive the encoded message DOJHEUD. To decode this message,we first digitize it:3, 14, 9, 7, 4, 20, 3.Next we apply the inverse transformation to get0, 11, 6, 4, 1, 17, 0,or ALGEBRA. Notice here that there is nothing special about either of thenumbers 3 or 26. We could have used a larger alphabet or a different shift.Cryptanalysis is concerned with deciphering a received or interceptedmessage. Methods from probability and statistics are great aids in decipheringan intercepted message; for example, the frequency analysis of the