Abstract Algebra and Number Theory—Sharing Theories And Ideas for Cryptography
In abstract algebra, we were introduced to the groups and rings Zn, representing the
integers modulo n, where n is an integer. Number theory is the study of the properties of whole
numbers, and some of it (our current topic) deals with the groups Zn.
One of the most basic equations in number theory is 1 = as + bt where a, b, s, t are all
integers and gcd(s,t) = 1. (Solving this equation involves the use of the Extended Euclidean
Algorithm, based on the Chinese Remainder Theorem.) This equation is sometimes used in
introductory cryptology to find the inverses of numbers modulo a prime number, for the
purposes of decryption.
Most cryptography in general today is based on PGP, which is done using Zp, where p is
a very large prime number. (For a general overview of how public cryptography works, see
Gallian’s “Contemporary Abstract Algebra”, pages 158-160.) Without the group properties
displayed by Zp we would not be able to harness the power of inverses and the cyclic nature of
doing mathematics modulo p for these uses.
[Suggested reading: “The Queen of Mathematics: A Historically Motivated Guide to Number
Theory” by Jay R. Goldman, ISBN 1-56881-006-7; “Invitation to Cryptology” by Thomas H.
Barr, ISBN 0-13088-976-8.]