Hidden Shift Quantum Cryptanalysis and Implications

International audienceAt Eurocrypt 2017 a tweak to counter Simon's quantum attack was proposed: replace the common bitwise addition, with other operations, as a modular addition. The starting point of our paper is a follow up of these previous results: First, we have developed new algorithms that improve and generalize Kuperberg's algorithm for the hidden shift problem, which is the algorithm that applies instead of Simon when considering modular additions. Thanks to our improved algorithm, we have been able to build a quantum attack in the superposition model on Poly1305, proposed at FSE 2005, largely used and claimed to be quantumly secure. We also answer an open problem by analyzing the effect of the tweak to the FX construction. We have also generalized the algorithm. We propose for the first time a quantum algorithm for solving the problem with parallel modular additions , with a complexity that matches both Simon and Kuperberg in its extremes. We also propose a generic algorithm to solve the hidden shift problem in non-abelian groups. In order to verify the theoretical analysis we performed, and to get concrete estimates of the cost of the algorithms, we have simulated them, and were able to validate our estimated complexities. Finally, we analyze the security of some classical symmetric constructions with concrete parameters, to evaluate the impact and practicality of the proposed tweak, concluding that it does not seem to be efficient

On Abelian and Homomorphic Secret Sharing Schemes

Abelian secret sharing schemes (SSS) are generalization of multi-linear SSS and similar to them, abelian schemes are homomorphic. There are numerous results on linear and multi-linear SSSs in the literature and a few ones on homomorphic SSSs too. Nevertheless, the abelian schemes have not taken that much attention. We present three main results on abelian and homomorphic SSSs in this paper: (1) abelian schemes are more powerful than multi-linear schemes (we achieve a constant factor improvement), (2) the information ratio of dual access structures are the same for the class of abelian schemes, and (3) every ideal homomorphic scheme can be transformed into an ideal multi-linear scheme with the same access structure. Our results on abelian and homomorphic SSSs have been motivated by the following concerns and questions. All known linear rank inequities have been derived using the so-called common information property of random variables [Dougherty, Freiling and Zeger, 2009], and it is an open problem that if common information is complete for deriving all such inequalities (Q1). The common information property has also been used in linear programming to find lower bounds for the information ratio of access structures [Farràs, Kaced, Molleví and Padró, 2018] and it is an open problem that if the method is complete for finding the optimal information ratio for the class of multi-linear schemes (Q2). Also, it was realized by the latter authors that the obtained lower bound does not have a good behavior with respect to duality and it is an open problem that if this behavior is inherent to their method (Q3). Our first result provides a negative answer to Q2. Even though, we are not able to completely answer Q1 and Q3, we have some observations about them

Private Permutations in Card-based Cryptography

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