## Reducing Logarithms in Totally Non-Maximal Imaginary Quadratic Orders to Logarithms in Finite Fields (Extended Abstract) (1999)

Citations: | 8 - 5 self |

### BibTeX

@MISC{Hühnlein99reducinglogarithms,

author = {Detlef Hühnlein and Tsuyoshi Takagi and C To},

title = {Reducing Logarithms in Totally Non-Maximal Imaginary Quadratic Orders to Logarithms in Finite Fields (Extended Abstract)},

year = {1999}

}

### OpenURL

### Abstract

Since nobody can guarantee that the computation of discrete logarithms in elliptic curves or IF p remains intractible for the future it is important to study cryptosystems based on alternative groups. A promising candidate, which was proposed by Buchmann and Williams [8], is the class group Cl(\Delta) of an imaginary quadratic order O \Delta . This ring is isomorphic to the endomorphism ring of a non-supersingular elliptic curve over a finite field. While in the meantime there was found a subexponential algorithm for the computation of discrete logarithms in Cl(\Delta) [16], this algorithm only has running time L \Delta [ 1 2 ; c] and is far less efficient than the number field sieve with L p [ 1 3 ; c] to compute logarithms in IF p . Thus one may choose the parameters smaller to obtain the same level of security. It is an open question whether there is an L \Delta [ 1 3 ; c] algorithm to compute discrete logarithms in arbitrary Cl(\Delta). Recently there were proposed cry...