Suppose you have a n vector of ASCII encoded characters, arbitrarily choose 1 <= n <= 1000. Choose a modulus N such that 128 <= N <= 1000. Also choose a pseudo-random number seed 1 <= s <= 2147483647. Next find a random n x n matrix that is invertible by Gaussian elimination over the integer field consisting of N elements. Suppose this matrix is M and its tridiagonal form is M’. Now suppose the plaintext is the n vector P and the ciphertext is the n vector C then we have for encryption:
C = M’P
Further assume the inverse of M’ is N’. For decryption we use the equation:
P = N’C
Where
M’N’ = N’M’ = I
Such that I is the n x n identity matrix.
This cipher is related to the classic Hill Cipher. This cipher is polyalphabetic. We show the results of one encryption and decryption using 10 ASCII ‘A’ characters, N = 999, and s = 1. As you can see each occurrence of the letter ‘A’ which is encoded as the decimal number 65 leads to different integer in the range 0 to 998 which has a maximum of 10 bits. The key consists of the 100 integers in the original 10 x 10 matrix. The application was implemented in C# using a Gaussian elimination over a number field algorithm from Henri Cohen’s A Course in Computational Algebraic Number Theory.
Author: jamespatewilliamsjr
My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.
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Numerical Explorations 