Advances in Information and Computer Security by Unknown

Author:Unknown
Language: eng
Format: epub, pdf
ISBN: 9783030582081
Publisher: Springer International Publishing

3.2 Ideal Representation Without the Collisions Stated in Section 3.1

For simplicity, we use the setting in CSIDH in this subsection; i.e., we assume .

Corollary 1 says that if one uses the secret exponents in the intervals with in CSIDH, then there are collisions in the ideal representation. For example, CSIDH-512, which is a parameter set of CSIDH with a prime p about 512 bits proposed by Castryck et al. [4], uses the intervals , so it contains collisions.

On the other hand, for CSIDH-512, Beullens, Kleinjung, and Vercauteren [1] proposed a method to choose ideal classes uniformly. However, their method relies on knowledge of the structure of the ideal class group; in particular, it needs a list of secret exponents which represent the identity element of the ideal class group. To obtain the structure of the ideal class group, they used the algorithm due to Hafner and McCurley [15]. Since that algorithm is subexponential time in the discriminant of the target number field, their method can not be applied to a CSIDH when a large base field is used. Therefore, the ideal representation proposed in Castryck et al. [4] is still important.

For the general case, one way to avoid the collisions stated in Sect. 3.1 is to use different intervals for each in which there is at least one interval of the form . De Feo, Kieffer, and Smith [11] and Meyer, Campos, and Reith [19] proposed using different intervals for each for speeding up the computation of the action of the ideal classes. One can expect that this representation is “almost” surjective and uniform, from a similar argument to the one in §7.1 in [4] (for the case of using different intervals, see §5.4 in [25]).

We propose another representation that is more efficiently computable than the method described in the above paragraph. Our representation uses instead of and is of the form

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.