Abstract: Elements of the finite field GF(2.sup.m) are represented by a vector of m binary digits in such a way that multiplication can be performed by using the same logic function to compute each binary component of the product of two elements, squaring can be performed by logic circuitry that rotates the vector representing the element to be squared, and addition can be performed by logic circuitry that forms the modulo-two sum of the corresponding components of the two vectors representing the elements to be summed.

Abstract: A private message of m bits is conveyed from its sender to its receiver by transmission of a public message of m bits, transmission of a public reply of m bits, and another transmission of a public message of m bits such that only the intended receiver can easily recover the private message from the three public messages. This private message now available to both the sender and intended receiver also allows all subsequent private messages between these two points to require only one public message for each private message while maintaining the property that only the intended receiver of each message can easily recover the private message for each public message.