Technical Summary for Mathematicians
This is an introduction to finite fields, rings, and groups, with applications to modern cryptography.
The examples used here are all ultimately derived from Z/nZ, by means of constructions like products, field extensions, and groups of invertibles.
The RSA cryptosystem, probably the most popular, is the discrete logarithm problem in the group (Z/pqZ)×.
It is conjecturally as difficult to solve this problem as it is to factor pq (as evidence, note that the order of any element divides lcm(p-1, q-1)), which is, in turn, conjecturally computationally difficult.
Of course, there are special values (of pq and the log base) which are low-order and so must be avoided.
Other well-investigated cryptosystems are based on the discrete log problem in GF(pd)×. Again, it is open whether such discrete log problems really provide "trapdoor" functions.
Further technical information and references can be found at the RSA FAQ.
The Woodrow Wilson Leadership Program in Mathematics
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