Paper 2023/1247
Representations of Group Actions and their Applications in Cryptography
Abstract
Cryptographic group actions provide a flexible framework that allows the instantiation of several primitives, ranging from key exchange protocols to PRFs and digital signatures. The security of such constructions is based on the intractability of some computational problems. For example, given the group action $(G,X,\star)$, the weak unpredictability assumption (Alamati et al., Asiacrypt 2020) requires that, given random $x_i$'s in $X$, no probabilistic polynomial time algorithm can compute, on input $\{(x_i,g\star x_i)\}_{i=1,\dots,Q}$ and $y$, the set element $g\star y$. In this work, we study such assumptions, aided by the definition of group action representations and a new metric, the $q$linear dimension, that estimates the "linearity'' of a group action, or in other words, how much it is far from being linear. We show that under some hypotheses on the group action representation, and if the $q$linear dimension is polynomial in the security parameter, then the weak unpredictability and other related assumptions cannot hold. This technique is applied to some actions from cryptography, like the ones arising from the equivalence of linear codes, as a result, we obtain the impossibility of using such actions for the instantiation of certain primitives. As an additional result, some bounds on the $q$linear dimension are given for classical groups, such as $\mathcal{S}_n$, $\mathrm{GL}(\mathbb{F}^n)$ and the cyclic group $\mathbb{Z}_n$ acting on itself.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Published elsewhere. Minor revision. Finite Fields and Their Applications
 DOI
 10.1016/j.ffa.2024.102476
 Keywords
 Oneway group actionsweakly pseudorandomweakly unpredictablerepresentations
 Contact author(s)

giuseppe dalconzo @ polito it
antonio discala @ polito it  History
 20240802: last of 3 revisions
 20230817: received
 See all versions
 Short URL
 https://ia.cr/2023/1247
 License

CC BY
BibTeX
@misc{cryptoeprint:2023/1247, author = {Giuseppe D'Alconzo and Antonio J. Di Scala}, title = {Representations of Group Actions and their Applications in Cryptography}, howpublished = {Cryptology ePrint Archive, Paper 2023/1247}, year = {2023}, doi = {10.1016/j.ffa.2024.102476}, note = {\url{https://eprint.iacr.org/2023/1247}}, url = {https://eprint.iacr.org/2023/1247} }