Some Remarks on the Herlestam-Johannesson Algorithm for Computing Logarithms over GF(2^p)

Abstract

At the 1981 IEEE Symposium on Information Theory, T. Herlestam and R. Johannesson presented a heurestic method for computing logarithms over GF(2^p). They reported computing logarithms over GF(2^{3 !}) with surprisingly few iterations and claimed that the running time of their algorithm was polynomial in p. If this were true, the algorithm could be used to cryptanalyze the Pohlig-Hellman cryptosystem, currently in use by Mitre Corporation for key distribution. The Mitre system operates in GF(2¹²⁷). However, the algorithm was not implemented for GF(2^p) for p > 31 because it would require multiple precision arithmetic. Consequently attempts to evaluate the possible threat to the Pohlig-Hellman cryptosystem have centered on modeling the algorithm so that some predictions could be made analytically about the number of iterations required to find logarithms over GF(2P) for p > 31.

This work performed at Sandia National Laboratories supported by the U. S. Department of Energy under contract number DE-AC04-76DP00789.