This paper is concerned with two aspects of cryptography in which the autho
r has been working. One is the Data Encryption Standard (DES), developed at
IBM and now in wide use for commercial cryptographic applications. This is
a "private key" system; the communicants share a secret key, and the eaves
dropper will succeed if he can guess this key among its quadrillions of pos
sibilities. The other is the Diffie-Hellman key exchange protocol, a typica
l "public key" cryptographic system. Its security is based on the difficult
y of taking "discrete logarithms" (reversing the process of exponentiation
in a finite field). We describe the system and some analytic attacks agains
t it.