Section: Overall Objectives

Design of Provably Secure Primitives and Protocols

Since the beginning of public-key cryptography, with the seminal Diffie-Hellman paper, many suitable algorithmic problems for cryptography have been proposed and many cryptographic schemes have been designed, together with more or less heuristic proofs of their security relative to the intractability of the underlying problems. However, many of those schemes have thereafter been broken. The simple fact that a cryptographic algorithm withstood cryptanalytic attacks for several years has often been considered as a kind of validation procedure, but schemes may take a long time before being broken. An example is the Chor-Rivest cryptosystem, based on the knapsack problem, which took more than 10 years to be totally broken by Serge Vaudenay, whereas before this attack it was believed to be strongly secure. As a consequence, the lack of attacks at some time should never be considered as a full security validation of the proposal.

A completely different paradigm is provided by the concept of “provable” security. A significant line of research has tried to provide proofs in the framework of computational complexity theory (a.k.a. “reductionist” security proofs): the proofs provide reductions from a well-studied problem (factoring, RSA or the discrete logarithm) to an attack against a cryptographic protocol.

At the beginning, researchers just tried to define the security notions required by actual cryptographic schemes, and then to design protocols which could achieve these notions. The techniques were directly derived from complexity theory, providing polynomial reductions. However, their aim was essentially theoretical. They were indeed trying to minimize the required assumptions on the primitives (one-way functions or permutations, possibly trapdoor, etc), without considering practicality. Therefore, they just needed to design a scheme with polynomial-time algorithms, and to exhibit polynomial reductions from the basic mathematical assumption on the hardness of the underlying problem into an attack of the security notion, in an asymptotic way. However, such a result has no practical impact on actual security. Indeed, even with a polynomial reduction, one may be able to break the cryptographic protocol within a few hours, whereas the reduction just leads to an algorithm against the underlying problem which requires many years. Therefore, those reductions only prove the security when very huge (and thus maybe unpractical) parameters are in use, under the assumption that no polynomial time algorithm exists to solve the underlying problem. For many years, more efficient reductions have been expected, under the denomination of either “exact security" or “concrete security”, which provide more practical security results, with concrete efficiency properties.

Unfortunately, in many cases, even just provable security is at the cost of an important loss in terms of efficiency for the cryptographic protocol. Thus, some models have been proposed, trying to deal with the security of efficient schemes: some concrete objects are identified with ideal (or black-box) ones. For example, it is by now usual to identify hash functions with ideal random functions, in the so-called “random-oracle model”. Similarly, block ciphers are identified with families of truly random permutations in the “ideal cipher model”. Another kind of idealization has also been introduced in cryptography, the black-box group, where the group operation, in any algebraic group, is defined by a black-box: a new element necessarily comes from the addition (or the subtraction) of two already known elements. It is by now called the “generic model”, extended to the bilinear and multi-linear setting. Some works even require several ideal models together to provide some new validations.

But still, such idealization cannot be instantiated in practice, and so one prefers to get provable security, without such ideal assumptions, under new and possibly stronger computational assumptions. As a consequence, a cryptographer has to deal with the four following important steps, which are all our main goals:

computational assumptions,

which are the foundations of the security. We thus need to have a strong evidence that the computational problems are reasonably hard to solve.

security model,

which makes precise the security notions one wants to achieve, as well as the means the adversary may be given. We contribute to this point, in several ways:

• by providing security models for many primitives and protocols;

• by enhancing some classical security models;

• by considering new means for the adversary, such as side-channel information.

design

of new schemes/protocols, or more efficient, with additional features, etc.

security proof,

which consists in exhibiting a reduction.