arXiv subjects
Publications and source records attributed to Şükrü Yalçınkaya.
We briefly explain how to implement the morphisms in our paper ``Natural representations of black box groups encrypting $SL_2(\mathbb{F})$" and provide some examples.
This paper is a compressed summary of some principal definitions and concepts in the approach to the black box algebra being developed by the authors. We suggest that black box algebra could be useful in cryptanalysis of homomorphic encryption schemes, and that homomorphic encryption is an area of research where cryptography and black box algebra may benefit from exchange of ideas.
Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\rm SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the isomorphisms \[ \mathsf{Y} \longleftrightarrow {\rm SL}_2(\mathsf{K}), \] where $\mathsf{K}$ is a black box field encrypting $\mathbb{F}$. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.
Given a black box group $\mathsf{Y}$ encrypting $\rm{PSL}_2(\mathbb{F})$ over an unknown field $\mathbb{F}$ of unknown odd characteristic $p$ and a global exponent $E$ for $\mathsf{Y}$ (that is, an integer $E$ such that $\mathsf{y}^E=1$ for all $\mathsf{y} \in \mathsf{Y}$), we present a Las Vegas algorithm which constructs a unipotent element in $\mathsf{Y}$. The running time of our algorithm is polynomial in $\log E$. This answers the question posed by Babai and Beals in 1999. We also find the characteristic of the underlying field in time polynomial in $\log E$ and linear in $p$. Furthermore, we construct, in probabilistic time polynomial in $\log E$, 1. a black box group $\mathsf{X}$ encrypting $\rm{PGL}_2(\mathbb{F}) \cong\rm{SO}_3(\mathbb{F})$, its subgroup $\mathsf{Y}^\circ$ of index $2$ isomorphic to $\mathsf{Y}$ and a probabilistic polynomial in $\log E$ time isomorphism $\mathsf{Y}^\circ \longrightarrow \mathsf{Y}$; 2. a black box field $\mathsf{K}$, and 3. polynomial time, in $\log E$, isomorphisms \[ \rm{SO}_3(\mathsf{K}) \longrightarrow \mathsf{X} \longrightarrow \rm{SO}_3(\mathsf{K}). \] If, in addition, we know $p$ and the standard explicitly given finite field $\mathbb{F}$ isomorphic to $\mathbb{F}$ then we construct, in time polynomial in $\log E$, isomorphism \[ \rm{SO}_3(\mathbb{F})\longrightarrow \rm{SO}_3(\mathsf{K}). \] Unlike many papers on black box groups, our algorithms make no use of additional oracles other than the black box group operations. Moreover, our result acts as an $\rm{SL}_2$-oracle in the black box group theory. We implemented our algorithms in GAP and tested them for groups such as $\rm{PSL}_2(\mathbb{F})$ for $|\mathbb{F}|=115756986668303657898962467957$ (a prime number).
For the black box groups $X$ encrypting ${\rm{PGL}}_2(q)$, $q$ odd, we propose an algorithm constructing a subgroup encrypting ${\rm{Sym}}_4$ and subfield subgroups of $X$. We also present the analogous algorithms for black box groups encrypting ${\rm{(P)SL}}_2(q)$.
The paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. As the starting point of our programme, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic and then apply them to black box groups X encrypting groups (P)SL(2,q) in small odd characteristics. We propose an algorithm constructing a black box field K isomorphic to F_q, and an isomorphism from (P)SL(2,K) to X. The algorithm runs in time quadratic in the characteristic of the underlying field and polynomial in log q. Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory.