Algebraic Methods for Stronger Crypto

Projektinformationen

ALGSTRONGCRYPTO

ID Finanzhilfevereinbarung: 740972

DOI

10.3030/740972

Projekt abgeschlossen

EK-Unterschriftsdatum 23 Juni 2017

Startdatum 1 Oktober 2017

Enddatum 30 September 2022

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 2 447 439,00

EU-Beitrag

€ 2 447 439,00

Koordiniert durch

STICHTING NEDERLANDSE WETENSCHAPPELIJK ONDERZOEK INSTITUTEN
Netherlands

Periodic Reporting for period 4 - ALGSTRONGCRYPTO (Algebraic Methods for Stronger Crypto)

Berichtszeitraum: 2022-04-01 bis 2022-09-30

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

ALGSTRONGCRYPTO aims to advance frontiers in design and analysis of high-security cryptography.
We wish to enhance efficiency, functionality, and, last-but-not-least, fundamental understanding of cryptographic security against powerful adversaries with access to a possible future quantum computer.

We consider two areas of application.
First, public-key cryptography (PKC), particularly its security.
PKC, a cornerstone of internet security, provides message confidentiality without requiring communicating parties to agree on a secret key in advance; a receiver's public key suffices for encryption, whereas the receiver's matching secret key enables decryption.
The PKC concept also encompasses digital signatures.

Second, secure multiparty computation (MPC), which allows a network of parties who do not necessarily trust each other to jointly process mutually confidential input data, without having to exchange those data
and while ensuring both correctness of the processing and privacy of the individual inputs.
This underpins privacy-protecting decentralized data-processing, an area with significant societal relevance. Zero knowledge proofs (ZKP), a special case, force each party in a complex cryptographic system
to behave correctly, without harming individual security or privacy interests.
In the project we give special attention to security and communication-efficiency PKC, MPC and ZKP that withstand attacks aided by quantum computers.


Our approach here is to develop completely novel methods by deepening, strengthening and broadening the algebraic foundations of the field; a bottom-up approach instead of a top-down one.
Concretely, we are inspired by the arithmetic codex, an abstract, algebraic device we have previously developed and which has proven itself useful in a host of multi-party cryptography scenarios and which has a rich matematical theory.
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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

We are generally content with the project outcomes, both the planned research and the
unexpected, fruitful new research lines discovered along the way. Many articles were published,
predominantly in the top-tier conferences and journals.
The project attracted many talented young international researchers.
Below we give selected highlights.

SPD$$\mathbb {Z}_{2^k}$$: Efficient MPC mod $$2^k$$ for Dishonest Majority.
Ronald Cramer, Ivan Damgård, Daniel Escudero, Peter Scholl, Chaoping Xing.
CRYPTO 2018. We introduce novel techniques to achieve MPC modulo powers-of-p
instead of over a field, with noticeable efficiency benefits in important applications. In those cases, it can be a replacement for the popular SPDZ protocol.



Blackbox Secret Sharing Revisited: A Coding-Theoretic Approach with Application to Expansionless Near-Threshold Schemes. Ronald Cramer, Chaoping Xing. EUROCRYPT 2020. We give a new method for blackbox secret-sharing,
achieving an error-free and constant-rate rate solution.


Efficient Information-Theoretic Secure Multiparty Computation over $$\mathbb {Z}/p^k\mathbb {Z}$$ via Galois Rings. Mark Abspoel, Ronald Cramer, Ivan Damgård, Daniel Escudero, Chen Yuan. TCC 2019. A follow-up to CDESX19, but now in
the information-theoretic case (threshold, non-asymptotic).

Random Self-reducibility of Ideal-SVP via Arakelov Random Walks. Koen de Boer, Léo Ducas, Alice Pellet-Mary, Benjamin Wesolowski. CRYPTO 2020.
We show, for certain cyclotomic lattices, a worst-case to average-case reduction for ideal-SVP
(i.e. shortest-vector finding in such lattices) by using Arakelov theory.

Compressed $$\varSigma $$-Protocol Theory and Practical Application to Plug & Play Secure Algorithmics. Thomas Attema, Ronald Cramer. CRYPTO 2020.
We introduce an abstract paradigm for logarithmic-communication zero knowledge
based on arithmetic secret sharing and adaptation of powerful protocol
compression technique.



Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over ℤ/p^kℤ. Mark Abspoel, Ronald Cramer, Ivan Damgård, Daniel Escudero, Matthieu Rambaud, Chaoping Xing, Chen Yuan. ASIACRYPT 2020.
Another follow-up to CDESX19: the
the information-theoretic case (asymptotic, passive adversary), with a general treatment of lifting multiplicative secret sharing to work over rings.


Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time.
Ronald Cramer; Léo Ducas; Benjamin Wesolowski. JACM 2021.
Journal version of EUROCRYPT 2017 paper. We show that, under a suitable number-theoretic hypothesis, the shortest vector problem in cyclotomic lattices is considerably less hard than in generic lattices.

Compressed Σ-Protocols for Bilinear Group Arithmetic Circuits and Application to Logarithmic Transparent Threshold Signatures. Thomas Attema, Ronald Cramer, Matthieu Rambaud. ASIACRYPT 2021. We show a useful generalization of [AC20], with application to threshold signatures.

Compressing Proofs of k-Out-Of-n
Partial Knowledge. Thomas Attema, Ronald Cramer, and Serge Fehr. CRYPTO 2021.
We show how to compress a well-known ZKP for proving knowledge
of k out of n secrets.

Asymptotically-Good Arithmetic Secret Sharing over Z/(p^\ell Z) with Strong Multiplication and Its Applications to Efficient MPC. Ronald Cramer, Matthieu Rambaud, and Chaoping Xing. CRYPTO 2021.
Using advanced methods from algebraic geometry, we show to to lift
asymptotically-good arithmetic secret sharing over finite fields to work over Galois-rings.
The final follow-up to CDESX19 (information-theoretic, asymptotic, active adversary).

A Compressed Σ-Protocol Theory for Lattices. Thomas Attema, Ronald Cramer, Lisa Kohl. CRYPTO 2021.
We further enhance the framework from [AC20] and show to instantiate it from lattice-based cryptography.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

We give main examples of our novel methodologies.

Our line of work on MPC over rings (and arithmetic codices over rings) instead of finite fields is a new paradigm with important practical consequences; previously, only MPC operating directly over finite fields was typically considered. By (non-trivally) constructing MPC directly working modulo powers-of-two and by avoiding significant loss in efficiency, substantial gains in efficiency of MPC can be achieved in many applications involving natural integer operations.
Technically, it uses ideas from commutative algebra and algebraic geometry previously not exploited in the area. A particularly attractive result we achieved
in this line has been adopted in a very popular open source toolkit for MPC.

The unification, through the use of arithmetic secret sharing, of the well-established Sigma protocol theory with recent ideas on so-called Bullet Proofs provides an abstract, modular methodology for highly-efficient zero-knowledge protocol design with promising potential of practical application. In a series of works we cover the basic theory for our line of work, including extensions to make it suitable for a post-quantum cryptographic platform based on lattices.

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