On the relation between Brauer groups and discrete logarithms

Rok, strany: 2006, 199 - 227

In this article we want to make evident that Brauer groups of local and global fields play an important role in public key cryptography. In the first section we show that all ideal class groups resp. divisor class groups attached to curves over finite fields $Bbb{F}_q$ taken into consideration for DL-systems can be mapped to such Brauer groups via the Tate-Lichtenbaum pairing. It is well known that this pairing applied to elements of order $n$ can be computed with complexity polynomial in $mid Bbb F_q(μ_n)mid$ where $μ_n$ is the group of roots of unity of order $n$. The image under this map leads to cyclic algebras over local fields with residue field $Bbb F_q(μ_n)$. In the next section we study such algebras and explain the connection with discrete logarithms in finite fields as well as with elements in the Brauer group over global fields. This leads to a possibility to apply index-calculus methods described in the third section. As a result we get subexponential algorithms for the computation of discrete logarithms in finite fields as well as for the computation of the Euler totient function. For the convenience of the reader we list the needed definitions and results about Galois cohomology groups in an appendix.

ISO 690:

Frey, G. 2006. On the relation between Brauer groups and discrete logarithms. In Tatra Mountains Mathematical Publications, vol. 33, no.1, pp. 199-227. 1210-3195.

APA:

Frey, G. (2006). On the relation between Brauer groups and discrete logarithms. Tatra Mountains Mathematical Publications, 33(1), 199-227. 1210-3195.