Abstract Algebra Theory and Applications - Computer Science ...

104 CHAPTER 6 INTRODUCTION TO CRYPTOGRAPHYRSA cryptosystem (1978). It is not known how secure any of these systems are.The trapdoor knapsack cryptosystem, developed by Merkle and Hellman, has beenbroken. It is still an open question whether or not the RSA system can be broken.At the time of the writing of this book, the largest number factored is 135 digitslong, and at the present moment a code is considered secure if the key is about400 digits long and is the product of two 200-digit primes. There has been a greatdeal of controversy about research in cryptography in recent times: the NationalSecurity Agency would like to keep information about cryptography secret, whereasthe academic community has fought for the right to publish basic research.Modern cryptography has come a long way since 1929, when Henry Stimson,Secretary of State under Herbert Hoover, dismissed the Black Chamber (the StateDepartment’s cryptography division) in 1929 on the ethical grounds that “gentlemendo not read each other’s mail.”Exercises1. Encode IXLOVEXMATH using the cryptosystem in Example 1.2. Decode ZLOOA WKLVA EHARQ WKHA ILQDO, which was encoded usingthe cryptosystem in Example 1.3. Assuming that monoalphabetic code was used to encode the following secretmessage, what was the original message?NBQFRSMXZF YAWJUFHWFF ESKGQCFWDQ AFNBQFTILO FCWP4. What is the total number of possible monoalphabetic cryptosystems? Howsecure are such cryptosystems?5. Prove that a 2 × 2 matrix A with entries in Z 26 is invertible if and only ifgcd(det(A), 26) = 1.6. Given the matrixA =( 3 42 3use the encryption function f(p) = Ap + b to encode the message CRYP-TOLOGY, where b = (2, 5) t . What is the decoding function?7. Encrypt each of the following RSA messages x so that x is divided into blocksof integers of length 2; that is, if x = 142528, encode 14, 25, and 28 separately.(a) n = 3551, E = 629, x = 31(c) n = 120979, E = 13251,x = 142371),(b) n = 2257, E = 47, x = 23(d) n = 45629, E = 781,x = 2315618. Compute the decoding key D for each of the encoding keys in Exercise 7.