6 research outputs found
Extremal behavior of reduced type of one dimensional rings
Let be a domain that is a complete local algebra in
dimension one. In an effort to address the Berger's conjecture, a crucial
invariant reduced type was introduced by Huneke et. al. In this article,
we study this invariant and its max/min values separately and relate it to the
valuation semigroup of . We justify the need to study in the context
of numerical semigroup rings and consequently investigate the occurrence of the
extreme values of for the Gorenstein, almost Gorenstein, and far-flung
Gorenstein complete numerical semigroup rings. Finally, we study the finiteness
of the category of maximal Cohen Macaulay modules and the
category of reflexive modules for rings which are of
maximal/minimal reduced type and provide many classifications.Comment: 23 pages. Comments are welcom
Traceable PRFs: Full Collusion Resistance and Active Security
The main goal of traceable cryptography is to protect against unauthorized redistribution of cryptographic functionalities. Such schemes provide a way to embed identities (i.e., a mark ) within cryptographic objects (e.g., decryption keys in an encryption scheme, signing keys in a signature scheme). In turn, the tracing guarantee ensures that any pirate device that successfully replicates the underlying functionality can be successfully traced to the set of identities used to build the device.
In this work, we study traceable pseudorandom functions (PRFs). As PRFs are the workhorses of symmetric cryptography, traceable PRFs are useful for augmenting symmetric cryptographic primitives with strong traceable security guarantees. However, existing constructions of traceable PRFs either rely on strong notions like indistinguishability obfuscation or satisfy weak security guarantees like single-key security (i.e., tracing only works against adversaries that possess a single marked key).
In this work, we show how to use fingerprinting codes to upgrade a single-key traceable PRF into a fully collusion resistant traceable PRF, where security holds regardless of how many keys the adversary possesses. We additionally introduce a stronger notion of security where tracing security holds even against active adversaries that have oracle access to the tracing algorithm. In conjunction with known constructions of single-key traceable PRFs, we obtain the first fully collusion resistant traceable PRF from standard lattice assumptions. Our traceable PRFs directly imply new lattice-based secret-key traitor tracing schemes that are CCA-secure and where tracing security holds against active adversaries that have access to the tracing oracle
RandomPoints package for Macaulay2
We present {\tt RandomPoints}, a package in \emph{Macaulay2} designed mainly
to identify rational and geometric points in a variety over a finite field. We
provide tools to estimate the dimension of a variety. We also present methods
to obtain non-vanishing minors of a given size in a given matrix, by evaluating
the matrix at a point.Comment: 10 pages, comments welcome. Package by Sankhaneel Bisui, Zhan Jiang,
Sarasij Maitra, Th\'ai Th\`anh Nguy\^en, Frank-Olaf Schreyer, Karl Schwede.
The current version can be found here
https://github.com/Macaulay2/Workshop-2020-Cleveland/blob/FastLinAlg/FastLinAlg/M2/RandomPoints.m
Partial Trace Ideals, Torsion and Canonical Module
For any finitely generated module with non-zero rank over a commutative
one dimensional Noetherian local domain, the numerical invariant was
introduced and studied in the author's previous work "Partial Trace Ideals and
Berger's Conjecture". We establish a bound on it which helps capture
information about the torsion submodule of when has rank one and it
also generalizes the discussion in the mentioned previous article. We further
study bounds and properties of in the case when is the canonical
module . This in turn helps in answering a question of S. Greco and
then provide some classifications. Most of the results in this article are
based on the results presented in the author's doctoral dissertation "Partial
Trace Ideals, The Conductor and Berger's Conjecture".Comment: 14 pages, Comments are welcom