# Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures

Fontein, F (2008). Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures. Advances in Mathematics of Communications, 2(3):293-307.

## Abstract

In discrete logarithm based cryptography, a method by Pohlig and Hellman allows solving the discrete logarithm problem efficiently if the group order is known and has no large prime factors. The consequence is that such groups are avoided. In the past, there have been proposals for cryptography based on cyclic infrastructures. We will show that the Pohlig-Hellman method can be adapted to certain cyclic infrastructures, which similarly implies that certain infrastructures should not be used for cryptography. This generalizes a result by Müller, Vanstone and Zuccherato for infrastructures obtained from hyperelliptic function fields.

We recall the Pohlig-Hellman method, define the concept of a cyclic infrastructure and briefly describe how to obtain such infrastructures from certain function fields of unit rank~one. Then, we describe how to obtain cyclic groups from discrete cyclic infrastructures and how to apply the Pohlig-Hellman method to compute absolute distances, which is in general a computationally hard problem for cyclic infrastructures. Moreover, we give an algorithm which allows to test whether an infrastructure satisfies certain requirements needed for applying the Pohlig-Hellman method, and discuss whether the Pohlig-Hellman method is applicable in infrastructures obtained from number fields. Finally, we discuss how this influences cryptography based on cyclic infrastructures.

In discrete logarithm based cryptography, a method by Pohlig and Hellman allows solving the discrete logarithm problem efficiently if the group order is known and has no large prime factors. The consequence is that such groups are avoided. In the past, there have been proposals for cryptography based on cyclic infrastructures. We will show that the Pohlig-Hellman method can be adapted to certain cyclic infrastructures, which similarly implies that certain infrastructures should not be used for cryptography. This generalizes a result by Müller, Vanstone and Zuccherato for infrastructures obtained from hyperelliptic function fields.

We recall the Pohlig-Hellman method, define the concept of a cyclic infrastructure and briefly describe how to obtain such infrastructures from certain function fields of unit rank~one. Then, we describe how to obtain cyclic groups from discrete cyclic infrastructures and how to apply the Pohlig-Hellman method to compute absolute distances, which is in general a computationally hard problem for cyclic infrastructures. Moreover, we give an algorithm which allows to test whether an infrastructure satisfies certain requirements needed for applying the Pohlig-Hellman method, and discuss whether the Pohlig-Hellman method is applicable in infrastructures obtained from number fields. Finally, we discuss how this influences cryptography based on cyclic infrastructures.

## Citations

4 citations in Web of Science®
3 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English August 2008 23 Oct 2008 15:54 05 Apr 2016 12:28 American Institute of Mathematical Sciences 1930-5338 Swiss National Science Foundation, grant no. 107887 First published in Advances in Mathematics of Communication in Volume 2, No. 3, 2008, 293–307, published by the American Institute of Mathematical Sciences and Shandong 10.3934/amc.2008.2.293 http://aimsciences.org/journals/displayPapers1.jsp?pubID=255 http://arxiv.org/abs/0803.2123
Permanent URL: http://doi.org/10.5167/uzh-3683

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