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AlmaCrypt Report Summary

Project ID: 669891
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - AlmaCrypt (Algorithmic and Mathematical Cryptology)

Reporting period: 2016-01-01 to 2017-06-30

Summary of the context and overall objectives of the project

Cryptology is a foundation of information security in the digital world. Today's internet is protected by a form of cryptography based on complexity theoretic hardness assumptions. Ideally, they should be strong to ensure security and versatile to offer a wide range of functionalities and allow efficient implementations. However, these assumptions are largely untested and internet security could be built on sand.

The main ambition of Almacrypt is to remedy this issue by challenging the assumptions through an advanced algorithmic analysis.

In particular, this proposal questions the two pillars of public-key encryption: factoring and discrete logarithms. Recent progress showed that in some cases, the discrete logarithm problem is considerably weaker than previously assumed. A main objective is to ponder the security of other cases of the discrete logarithm problem, including elliptic curves, and of factoring. We will study the generalization of the recent techniques and search for new algorithmic options with comparable or better efficiency.

We will also study hardness assumptions based on codes and subset-sum, two candidates for post-quantum cryptography. We will consider the applicability of recent algorithmic and mathematical techniques to the resolution of the corresponding putative hard problems, refine the analysis of the algorithms and design new algorithm tools.

Cryptology is not limited to the above assumptions: other hard problems have been proposed to aim at post-quantum security and/orto offer extra functionalities. Should the security of these other assumptions become critical, they would be added to Almacrypt's scope. They could also serve to demonstrate other applications of our algorithmic progress.

In addition to its scientific goal, Almacrypt also aims at seeding a strengthened research community dedicated to algorithmic and mathematical cryptology.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Main results of the time period:

1) A new method for solving polynomial systems of Boolean equations. The method was used to perform record computations in the Fukuoka challenge. The pre-existing record was at 66 variables using some dedicated hardware (FPGAs), with the method it was possible to find solutions to all the type I challenges from 67 up to 74 variables on a general purpose cluster. Note that the 74 variables challenge was the largest type I instance published by the Fukuoka team.

2) An algorithm for speeding up linear algebra on nearly sparse matrices using a new variation of block Wiedemann algorithm.

3) A new candidate cryptosystem based on the used of Mersenne primes.


In addition to the main results, some collaborative work has been started concerning discrete logarithms computations and preparatory steps have been taken toward the integration of lattice problems in Almacrypt after the end of 2017, as suggested by the project's reviewers.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The ability to efficiently solve Boolean systems of polynomial equations farther than what was previously known will have impact on the security and parameter choices of cryptosystems which are based on the hardness of this problem. This mainly includes multivariate cryptosystems which are some of the considered candidates for post-quantum cryptography. The exact impact will depend on the scheme of this type that will be submitted to the undergoing NIST call for post-quantum proposal.

The Mersenne system is also promising and we are currently considering the possibility of submitting it to the aforementioned NIST call.
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