TIME-MEMORY TRADEOFF IN EXPONENTIATING A FIXED ELEMENT OF GF(qⁿ) REQUIRING A SHORT REFERENCE TO THE MEMORY.

Raising alpha **x to the yth power over GF(q**n) can be performed by calculating alpha **y modulo the minimum polynomial of alpha **x and then multiplying the result by an n multiplied by n matrix over GF(q). The elements of the matrix are only a function of x and of the generating polynomial of the field. This principle offers a time-memory trade-off when exponentiating a fixed element of GF(q**n), where the multiplications (not the squarings) involved in the standard square-and-multiply process are traded for a reference to a stored n multiplied by n matrix. The operations which make use of the stored data consume time which is equivalent to a single multiplication operation over the field, and are performed continuously, where the time-consuming part of the exponentiation process is performed independently of the stored data. It is then shown how the presented principle enables an efficient implementation over GF(q**n) of some variations of Diffie-Hellman public-key distribution system.