Partage de secret et théorie algorithmique de l’information

Secret sharing deals with splitting a secret data between several participants. The participants are organized within an access structure consisting of groups that may gain access to the secret. The goal is to provide each participant with a piece of information such that the following two requirements are met: 1. Recoverability: a qualified group can recover the secret, 2. Privacy: any other group cannot obtain any information on the secret. Whenever these two requirements are fulfilled, the scheme is said to be unconditionally secure – security is guaranteed even against adversaries with unbounded computational power. We use Information Theory to formalize this definition and its properties (recoverability, privacy). Our work focuses on monotone access structures, for which any group containing a qualified subgroup is also qualified. Since Benaloh et al and Ito et al, we know that every access structure admits a perfect secret-sharing scheme. However, a possible issue is that the size of information pieces assigned as shares to participants can grow exponentially with the number of participants. Defining the efficiency of a scheme as the ratio between the maximum size of a share and the size of the secret, the best known lower bound – from a result of Csirmaz in 1994 – is (only) almost linear. The usual method for obtaining these lower bounds consists in applying linear inequalities for Shannon entropy, also called information inequalities. The research axis of my thesis is based upon the link – which turns out to be quite powerful – between secret sharing and information inequalities. First, I define the notion of quasi-perfect secret sharing, where the perfect secret-sharing requirements (recoverability, privacy) are slightly relaxed. To this aim, we have introduced two quantities for leakage: the missing information ratio as the maximal proportion of information a qualified group needs to recover the secret completely, and the information leak ratio as the maximal proportion of information forbidden groups can learn about the secret. In our model, these leaks are vanishing as the size of the secret grows. For our notion of quasi-perfect secret sharing, it turns out that strict recoverability is never an issue (see Section 5.1.2), and that we can assume the secret uniformly distributed (see Section 5.1.3). If the size of the secret should remain small, we also worked on fixed finite size for the secret and showed that the