Section: New Results

Symmetric cryptosystems

Participants : Christina Boura, Baudoin Collard, Anne Canteaut, Pascale Charpin, Gohar Kyureghyan, María Naya-Plasencia, Joëlle Roué, Valentin Suder.

From outside, it might appear that symmetric techniques become obsolete after the invention of public-key cryptography in the mid 1970's. However, they are still widely used because they are the only ones that can achieve some major features as high-speed or low-cost encryption, fast authentication, and efficient hashing. Today, we find symmetric algorithms in GSM mobile phones, in credit cards, in WLAN connections. Symmetric cryptology is a very active research area which is stimulated by a pressing industrial demand for low-cost implementations (in terms of power consumption, gate complexity...). These extremely restricting implementation requirements are crucial when designing secure symmetric primitives and they might be at the origin of some weaknesses. Actually, these constraints seem quite incompatible with the rather complex mathematical tools needed for constructing a provably secure system.

The specificity of our research work is that it considers all aspects of the field, from the practical ones (new attacks, concrete specifications of new systems) to the most theoretical ones (study of the algebraic structure of underlying mathematical objects, definition of optimal objects). But, our purpose is to study these aspects not separately but as several sides of the same domain. Our approach mainly relies on the idea that, in order to guarantee a provable resistance to the known attacks and to achieve extremely good performance, a symmetric cipher must use very particular building blocks, whose algebraic structures may introduce unintended weaknesses. Our research work captures this conflict for all families of symmetric ciphers. It includes new attacks and the search for new building blocks which ensure both a high resistance to the known attacks and a low implementation cost. This work, which combines cryptanalysis and the theoretical study of discrete mathematical objects, is essential to progress in the formal analysis of the security of symmetric systems.

In this context, the very important challenges are the designs of low-cost ciphers and of secure hash functions. Most teams in the research community are actually working on the design and on the analysis (cryptanalysis and optimisation of the performance) of such primitives.

Hash functions.

Following the recent attacks against almost all existing hash functions (MD5, SHA-0, SHA-1...), we have initiated a research work in this area, especially within the Saphir-2 ANR project and with several PhD theses. Our work on hash functions is two-fold: we have designed two new hash functions, named FSB and Shabal, which have been submitted to the SHA-3 competition, and we have investigated the security of several hash functions, including the previous standards (SHA-0, SHA-1...) and some other SHA-3 candidates.

Recent results:

• Upper bounds on the degree of an iterated permutation from the degree of the inverse of the inner transformation; this result has been applied both to hash functions and to block ciphers. Most notably, this work leads to the best (theoretical) analysis of the hash function Keccak, which has been selected for the new SHA-3 standard [12] , [22] , [9] .

• Side-channel attacks on two SHA-3 candidates, Skein and Grøstl, when they are used with HMAC, and counter-measures [23] , [50] .

• Indifferentiability results for a broadened mode of operation including the modes based on block ciphers, and modes based on un-keyed functions [51] .

Block ciphers.

Even if the security of the current block cipher standard, AES, is not threaten when it is used in a classical context, there is still a need for the design of improved attacks, and for the determination of design criteria which guarantee that the existing attacks do not apply. This notably requires a deep understanding of all previously proposed attacks. Moreover, there is a high demand from the industry of lightweight block ciphers for some constrained environments. Several such algorithms have been proposed in the last few years and their security should be carefully analysed. Most of our work in this area is related to an ANR Project named BLOC.

Recent results:

• Algebraic analysis of some recent lightweight block ciphers, including LED and Piccolo [24] .

• Analysis of the security of the lightweight block cipher mCRYPTON [56] .

• Design of a new block cipher, named PRINCE, with a very low-latency, leading to instantaneous encryption (i.e., within one clock cycle) with a very competitive chip area [21] , [49] .

• Analysis of the differential properties of the AES Superbox [58] .

• Study of the significance of the related-key and known-key models for block ciphers [48] .

Stream ciphers.

The project-team has been involved in the international project eSTREAM, which aimed at recommending some secure stream ciphers.

Recent results:

• Generalisation of several improvements of the so-called correlation attacks against stream ciphers and study of their complexities [13] .

• Study of the bias of parity-check relations for combination generators used in stream ciphers [14] .

Cryptographic properties and construction of appropriate building blocks.

The construction of building blocks which guarantee a high resistance to the known attacks is a major topic within our project-team, for stream ciphers, block ciphers and hash functions. The use of such optimal objects actually leads to some mathematical structures which may be the origin of new attacks. This work involves fundamental aspects related to discrete mathematics, cryptanalysis and implementation aspects. Actually, characterising the structures of the building blocks which are optimal regarding to some attacks is very important for finding appropriate constructions and also for determining whether the underlying structure induces some weaknesses or not.

For these reasons, we have investigated several families of filtering functions and of S-boxes which are well-suited for their cryptographic properties or for their implementation characteristics. For instance, bent functions, which are the Boolean functions which achieve the highest possible nonlinearity, have been extensively studied in order to provide some elements for a classification, or to adapt these functions to practical cryptographic constructions. We have also been interested in functions with a low differential uniformity (e.g., APN functions), which are the S-boxes ensuring an (almost) optimal resistance to differential cryptanalysis.

Recent results:

• Study of the algebraic properties (e.g. the algebraic degree) of the inverses of APN power permutations [26] .

• Study of the planarity of some mappings, including products of linearized polynomials [25] , [16] .

• Definition of a new criterion for Sboxes and link with some recent algebraic attacks on the hash functuion Hamsi [29] , [9] .

• Survey of PN and APN mappings [42] .