Section: New Results
Symmetric cryptosystems
Participants : Céline Blondeau, Christina Boura, Baudoin Collard, Anne Canteaut, Pascale Charpin, Stéphane Jacob, Gohar Kyureghyan.
From outside, it might appear that symmetric techniques become obsolete after the invention of publickey 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 highspeed or lowcost 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 lowcost 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 lowcost ciphers and of secure hash functions. Most teams in the research community are actually working on the design and on the analysis (cryptanalysis and optimization of the performance) of such primitives.
Hash functions.
Following the recent attacks against almost all existing hash functions (MD5, SHA0, SHA1...), we have initiated a research work in this area, especially within the Saphir2 ANR project and with several PhD theses. Our work on hash functions is twofold: we have designed two new hash functions, named FSB and Shabal, which have been submitted to the SHA3 competition, and we have investigated the security of several hash functions, including the previous standards (SHA0, SHA1...) and some other SHA3 candidates.
Recent results:

study of the algebraic properties of the recent hash function proposals, including the SHA3 candidates Keccak and Luffa. This work includes a theoretical study of the algebraic degree of iterated functions composed of parallel applications of a smaller function [24] .

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 [31] , [44] .
Stream ciphers.
Our research work on stream ciphers is a longterm work which has been developed within the 4year ANR RAPIDE project. It includes an important cryptanalytic effort on stream ciphers.
Recent results:

Evaluation of the bias of paritycheck relations in the context of cryptanalysis of combination generators with constituent devices which generate period sequences [13] .

Cryptanalysis of the recent stream cipher proposal Armadillo [21] .
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.
Recent results:

Differential cryptanalysis with multiple differentials, multiple differential cryptanalysis on the lightweight block cipher Present [23] .

Use of tools from error correcting theory in linear cryptanalysis [36] .

Determination of the data complexity (i.e., of the required number of plaintextsciphertexts) and of the success probability of all statistical attacks against block ciphers [12] .
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 projectteam, 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, characterizing 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 Sboxes which are wellsuited 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 Sboxes ensuring an (almost) optimal resistance to differential cryptanalysis.
Recent results:

Study of the properties of the family of power functions with exponents ${2}^{t}1$. This family notably includes the cube function ${x}^{3}$ and the inverse function over a finite field with characteristic 2. In this work, the whole Walsh spectrum of ${x}^{7}$ is determined [11] .

Construction and study of the properties of new families of permutation polynomials over the field with ${2}^{m}$ elements; study of permutations with a linear structure: [14] .

Study of the algebraic properties (e.g. the algebraic degree) of the inverses of APN power permutations [47] .