How to Securely Compute with Noisy Leakage in Quasilinear Complexity

Since their introduction in the late 90’s, side-channel attacks have been considered as a major threat against cryptographic implementations. This threat has raised the need for formal leakage models in which the security of implementations can be proved. At Eurocrypt 2013, Prouff and Rivain introduced the noisy leakage model which has been argued to soundly capture the physical reality of power and electromagnetic leakages. In their work, they also provide the first formal security proof for a masking scheme in the noisy leakage model. However their work has two important limitations: (i) the security proof relies on the existence of a leak-free component, (ii) the tolerated amount of information in the leakage (aka leakage rate) is of O(1 / n) where n is the security parameter (i.e. the number of shares in the underlying masking scheme). The first limitation was nicely tackled by Duc, Dziembowski and Faust one year later (Eurocrypt 2014). Their main contribution was to show a security reduction from the noisy leakage model to the conceptually simpler random-probing model. They were then able to prove the security of the well-known Ishai-Sahai-Wagner scheme (Crypto 2003) in the noisy leakage model. The second limitation was addressed in a paper by Andrychowicz, Dziembowski and Faust (Eurocrypt 2016) which makes use of a construction due to Ajtai (STOC 2011) to achieve security in the strong adaptive probing model with a leakage rate of \(O(1/\log n)\). The authors argue that their result can be translated into the noisy leakage model with a leakage rate of O(1) by using secret sharing based on algebraic geometric codes. In terms of complexity, the protected program scales from |P| arithmetic instructions to \(\tilde{O}(|P| \, n^2)\). According to the authors, this \(\tilde{O}(n^2)\) blow-up could be reduced to \(\tilde{O}(n)\) using packed secret sharing but no details are provided. Moreover, such an improvement would only be possible for a program of width at least linear in n. The issue of designing an explicit scheme achieving \(\tilde{O}(n)\) complexity blow-up for any arithmetic program is hence left open.

In this paper, we tackle the above issue: we show how to securely compute in the presence of noisy leakage with a leakage rate \(\tilde{O}(1)\) and complexity blow-up \(\tilde{O}(n)\). Namely, we introduce a transform that turns any program P composed of arithmetic instructions on some filed \(\mathbb {F}\) into a (functionally equivalent) program \(\varPi \) composed of \(|\varPi | = O(|P| n \log n)\) arithmetic instructions which can tolerate some (quasi-constant) amount of noisy leakage on its internal variables (while revealing negligible information). We use a polynomial encoding allowing quasilinear multiplication based on the fast Number Theoretic Transform (NTT). We first show that our scheme is secure in the random-probing model with leakage rate \(O(1/\log n)\). Using the reduction by Duc et al. this result can be translated in the noisy leakage model with a \(O(1/|\mathbb {F}|^2 \log n)\) leakage rate. However, a straight application of this reduction is not satisfactory since our construction requires \(|\mathbb {F}| = O(n)\). In order to bypass this issue (which is shared with the construction of Andrychowicz et al.), we provide a generic security reduction from the noisy leakage model at the logical-instruction level to the random-probing model at the arithmetic level. This reduction allows us to prove the security of our construction in the noisy leakage model with leakage rate \(\tilde{O}(1)\).