Paper 2015/513
Computing Individual Discrete Logarithms Faster in $GF(p^n)$
Aurore Guillevic
Abstract
The Number Field Sieve (NFS) algorithm is the best known method to compute discrete logarithms (DL) in finite fields $\mathbb{F}_{p^n}$, with $p$ medium to large and $n \geq 1$ small. This algorithm comprises four steps: polynomial selection, relation collection, linear algebra and finally, individual logarithm computation. The first step outputs two polynomials defining two number fields, and a map from the polynomial ring over the integers modulo each of these polynomials to $\mathbb{F}_{p^n}$. After the relation collection and linear algebra phases, the (virtual) logarithm of a subset of elements in each number field is known. Given the target element in $\mathbb{F}_{p^n}$, the fourth step computes a preimage in one number field. If one can write the target preimage as a product of elements of known (virtual) logarithm, then one can deduce the discrete logarithm of the target. As recently shown by the Logjam attack, this final step can be critical when it can be computed very quickly. But we realized that computing an individual DL is much slower in medium and largecharacteristic nonprime fields $\mathbb{F}_{p^n}$ with $n \geq 3$, compared to prime fields and quadratic fields $\mathbb{F}_{p^2}$. We optimize the first part of individual DL: the \emph{booting step}, by reducing dramatically the size of the preimage norm. Its smoothness probability is higher, hence the runningtime of the booting step is much improved. Our method is very efficient for small extension fields with $2 \leq n \leq 6$ and applies to any $n > 1$, in medium and large characteristic.
Note: Minor modification of ASIACRYPT 2015 version (typo in Table 3 corrected).
Metadata
 Available format(s)
 Publication info
 A minor revision of an IACR publication in ASIACRYPT 2015
 DOI
 10.1007/9783662487976_7
 Keywords
 Discrete logarithmfinite fieldnumber field sieveindividual logarithm
 Contact author(s)
 aurore guillevic @ ucalgary ca
 History
 20160526: last of 2 revisions
 20150529: received
 See all versions
 Short URL
 https://ia.cr/2015/513
 License

CC BY
BibTeX
@misc{cryptoeprint:2015/513, author = {Aurore Guillevic}, title = {Computing Individual Discrete Logarithms Faster in $GF(p^n)$}, howpublished = {Cryptology ePrint Archive, Paper 2015/513}, year = {2015}, doi = {10.1007/9783662487976_7}, note = {\url{https://eprint.iacr.org/2015/513}}, url = {https://eprint.iacr.org/2015/513} }