7 research outputs found
Expansion in SL_d(Z/qZ), q arbitrary
Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it
generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G)
with respect to the generating set pi_q(S) form a family of expanders, where
pi_q is the projection map Z->Z/qZ
Growth in solvable subgroups of GL_r(Z/pZ)
Let and let be a subset of \GL_r(K) such that is
solvable. We reduce the study of the growth of $A$ under the group operation to
the nilpotent setting. Specifically we prove that either $A$ grows rapidly
(meaning $|A\cdot A\cdot A|\gg |A|^{1+\delta}$), or else there are groups $U_R$
and $S$, with $S/U_R$ nilpotent such that $A_k\cap S$ is large and
$U_R\subseteq A_k$, where $k$ is a bounded integer and $A_k = \{x_1 x_2...b x_k
: x_i \in A \cup A^{-1} \cup {1}}$. The implied constants depend only on the
rank $r$ of $\GL_r(K)$.
When combined with recent work by Pyber and Szab\'o, the main result of this
paper implies that it is possible to draw the same conclusions without
supposing that is solvable.Comment: 46 pages. This version includes revisions recommended by an anonymous
referee including, in particular, the statement of a new theorem, Theorem
Expansion in perfect groups
Let Ga be a subgroup of GL_d(Q) generated by a finite symmetric set S. For an
integer q, denote by Ga_q the subgroup of Ga consisting of the elements that
project to the unit element mod q. We prove that the Cayley graphs of Ga/Ga_q
with respect to the generating set S form a family of expanders when q ranges
over square-free integers with large prime divisors if and only if the
connected component of the Zariski-closure of Ga is perfect.Comment: 62 pages, no figures, revision based on referee's comments: new ideas
are explained in more details in the introduction, typos corrected, results
and proofs unchange
Cryptographic Hash Functions and Expander Graphs: The End of the Story?
Cayley hash functions are a family of cryptographic hash functions constructed from the Cayley graphs of non-Abelian finite groups. Their security relies on the hardness of mathematical problems related to long-standing conjectures in graph and group theory. We recall the Cayley hash design and known results on the underlying problems. We then describe related open problems, including the cryptanalysis of relevant parameters as well as new applications to cryptography and outside, assuming either that the problem is “hard” or easy.SCOPUS: cp.kInternational Conference on The New Codebreakers - Essays Dedicated to David Kahn on the Occasion of His 85th Birthday, 2010; LuxembourgISBN: 978-366249300-7Volume Editors: Quisquater J.-J.Ryan P.Y.A.Naccache D.Publisher: Springer Verlaginfo:eu-repo/semantics/publishe