This paper is concerned with two aspects of cryptography in which the author 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 eavesdropper will succeed if he can guess this key among its quadrillions of possibilities. The other is the Diffie-Hellman key exchange protocol, a typical "public key" cryptographic system. Its security is based on the difficulty of taking "discrete logarithms" (reversing the process of exponentiation in a finite field). We describe the system and some analytic attacks against it.