Side-channel analysis made its way into Eurocrypt this year thanks to two talks, the first of which given by FranÃ§ois-Xavier Standaert on a new model to prove security of implementations in. When talking about provable security in the context of side-channel analysis, there is one prominent model that comes to mind: the

*d*-probing model, where the adversary is allowed to probe*d*internal variables (somewhat related to*d*wires inside an actual implementation) and learn them. Another very famous model, introduced ten years later, is the noisy leakage model in which the adversary is allowed probes on all intermediate variables (or wires) but the learnt values are affected by errors due to noise. To complete the picture, it was proved that security in the probing model implies security in the noisy leakage one.
The work of Barthe, Dupressoir, Faust, GrÃ©goire, Standaert and Strub is motivated precisely by the analysis of these two models in relation to how they specifically deal with parallel implementation of cryptosystems. On one hand, the probing model admits very simple and elegant description and proofs' techniques but it is inherently oriented towards serial implementations; on the other hand, the noisy leakage model naturally includes parallel implementations in its scope but, admitting the existence of noise in leakage functions, it lacks simplicity. The latter is particularly important when circuits are analysed with automated tools and formal methods, because these can rarely deal with errors.

The contribution of the paper can then be summarised in the definition of a new model trying to acquire the pros of both previous models: the Bounded Moment leakage Model (BMM). The authors show how it relates to the probing model and give constructions being secure in their model. In particular, they prove that BMM is strictly weaker than the probing model in that security in the latter implies security in the former but they give a counterexample that the opposite does not hold. The informal definition of the model given during the talk is the following:

An implementation is secure at order o in the BMM if all mixed statistical moments of order up to o of its leakage vectors are independent of any sensitive variable manipulated.

A parallel multiplication algorithm and a parallel refreshing algorithm are the examples brought to show practical cases where the reduction between models stated before holds, the statement of which is the following:

The falsity of the converse is shown in a slightly different setting, namely the one of continuous leakage: the adversary does not just learn values carried by some wires by probing them, but such an operation can be repeated as many times as desired and the probes can be moved adaptively. Clearly this is a much stronger adversary in that accumulation of knowledge over multiple probing sessions is possible, which is used as a counterexample to show that security in the continuous BMM does not imply security in the continuous probing model. The refreshing scheme mentioned above can easily be broken in the latter after a number of iterations linear in the number of shares, but not in the former as adapting the position of the probes does not help: an adversary in the BMM can already ask for leakage on a bounded function of all the shares.A parallel implementation is secure at order o in the BMM if its serialisation is secure at order o in the probing model.

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