On the Difficulty of Computing Logarithms Over GF (q^m)

In “New Directions in Cryptography”, Diffie and Hellman propose a public key distribution (PKD) system based on exponentiation in a discrete arithmetic system. The security of this technique is crucially dependent on the difficulty of computing discrete logarithms (the inverse of the discrete exponential function). Until recently, the best known method for computing discrete logs required running time which grew exponentially in the word size. However, Adleman has recently observed that certain fast algorithms for factoring integers are also applicable to computing discrete logs over GF(q), the Galois field with q elements (q denotes a prime number). He also noted that the running time for the modified algorithm should be of the same form as for factoring, namely