Section: New Results
Discrete Logarithm computations in finite fields with the NFS algorithm
The best discrete logarithm record computations in prime fields and large characteristic finite fields are obtained with Number Field Sieve algorithm (NFS) at the moment. This algorithm is made of four steps:
The two more time consuming steps are the relation collection step and the linear algebra step. The polynomial selection is quite fast but is very important since it determines the complexity of the algorithm. Selecting better polynomials is a key to improve the overall running-time of the NFS algorithm. The final step: individual discrete logarithm, was though to be quite fast but F. Morain and A. Guillevic showed that it has an increasing complexity with respect to the extension degree of the finite field. A. Guillevic proposed a new method to reduce considerably the complexity, with at most a factor two speed-up in the exponent [22] .
In 2015, F. Morain and A. Guillevic released with P. Gaudry and
R. Barbulescu a major discrete logarithm record in a quadratic finite
field GF
DL Record computation in a quadratic finite field GF
In order to compare the practical running time of discrete logarithm computation in prime fields and quadratic finite fields, F. Morain and A. Guillevic with P. Gaudry and R. Barbulescu launched a DL record in a 180dd finite field. The last DL record in a prime field was held by the CARAMEL team of Nancy, in 2014, in a 180 dd prime field. The parameters chosen for the quadratic finite field are the following.
The discrete logarithm computation was made modulo
The two polynomials used in the NFS algorithm were chosen to be the following:
We indeed designed a new polynomial selection method, that we called
the Conjugation method. It is very well suited for quadratic and cubic
finite fields GF
We finally computed the discrete logarithm in basis
The running time was very surprising: our record was much faster than the concurrent DL computation in a prime field of the same global size of 180dd, and even faster than the RSA modulus factorization of the same size.
Algorithm | relation collection | linear algebra | total |
NFS-IF | 5 years | 5.5 months | 5.5 years |
NFS-DL |
50 years | 80 years | 130 years |
NFS-DL |
157 days | 18 days (GPU) | 0.5 years |
Individual discrete logarithm computation
A big difference between prime fields and finite fields of small
extension such as GF