# Algebraic Methods for Stronger Crypto

## Periodic Reporting for period 2 - ALGSTRONGCRYPTO (Algebraic Methods for Stronger Crypto)

Reporting period: 2019-04-01 to 2020-09-30

Our field is cryptology. Our overarching objective is to advance significantly the frontiers in
design and analysis of high-security cryptography for the future generation.
Particularly, we wish to enhance the efficiency, functionality, and, last-but-not-least, fundamental understanding of
cryptographic security against very powerful adversaries.
Our approach here is to develop completely novel methods by
deepening, strengthening and broadening the
algebraic foundations of the field.
Concretely, our lens builds on
the arithmetic codex. This is a general, abstract cryptographic primitive whose basic theory we recently developed and
whose asymptotic part, which relies on algebraic geometry, enjoys crucial applications in surprising foundational results on
constant communication-rate two-party cryptography. A codex is a linear (error correcting) code that, when endowing its
ambient vector space just with coordinate-wise multiplication, can be viewed as simulating, up to some degree, richer
arithmetical structures such as finite fields (or products thereof), or generally, finite-dimensional algebras over finite fields.
Besides this degree, coordinate-localities for which simulation holds and for which it does not at all are also captured.
Our method is based on novel perspectives on codices which significantly
widen their scope and strengthen their utility. Particularly, we bring
symmetries, computational- and complexity theoretic aspects, and connections with algebraic number theory, -geometry,
and -combinatorics into play in novel ways. Our applications range from public-key cryptography to secure multi-party
computation.
Our proposal is subdivided into 3 interconnected modules:
(1) Algebraic- and Number Theoretical Cryptanalysis
(2) Construction of Algebraic Crypto Primitives
(3) Advanced Theory of Arithmetic Codices
A detailed overview has been given in Project Achievements.
For progress, see Project Achievements.
We expect that the new leads and novel methods found in the first half of the project (see Project Achievements) will continue to yield during the second.
We also expect to discover further novel leadsand methods in the second half of the project; however, this being a theory project, it is hard to predict their exact nature.