Methods and devices for estimating secret values
A secret value estimation device is provided for determining an estimate of at least one secret value used by at least one cryptographic mechanism implemented in a cryptographic system from a statistical distribution of a set of multivariate leakage traces determined by a leakage traces statistical distribution unit. Each leakage trace being a vector comprises a plurality of random values, the number of said random values being an integer number superior or equal to 1, the statistical distribution being a function of parametric linear combinations of a set of leakage model basis vectors representing a multivariate leakage model, the number of basis vectors being an integer number superior or equal to 1, and the linear combinations being defined by a matrix of real values.
1. A device for determining an estimate of at least one secret value used by at least one cryptographic mechanism, from a statistical distribution of a set of multivariate leakage traces,
each leakage trace being a vector comprising a plurality of random values,
the number of said random values being an integer number greater than or equal to 1,
said statistical distribution being a function of parametric linear combinations of a set of leakage model basis vectors representing a multivariate leakage model,
the number of said basis vectors being an integer number greater than or equal to 1,
said parametric linear combinations being defined by a matrix of real values comprising at least one unknown value,
wherein said device is configured to simultaneously determine (i) said estimate of at least one secret value and (ii) an estimate of said at least one unknown value of said matrix of real values defining said parametric linear combinations, using a multivariate estimation algorithm that maximizes a probability of successful secret values recovery and that is an expectation maximization algorithm.
2. The device of claim 1 , wherein it comprises an analysis unit configured to determine said set of multivariate leakage traces from at least a leakage dimensionality and a set of acquired side-channel leakage information,
wherein said number of said random values comprised in a vector of each leakage trace is equal to said leakage dimensionality.
3. The device of claim 1 , wherein it comprises a processing unit configured to determine said statistical distribution of a set of multivariate leakage traces from at least a leakage dimensionality, a leakage model dimensionality, and a leakage model.
4. The device of claim 3 , wherein said processing unit is further configured to determine said statistical distribution of a set of multivariate leakage traces depending on a noise of a known covariance matrix.
5. The device of claim 4 , wherein said leakage dimensionality of said multivariate leakage traces depends on the noise probability distribution within each leakage trace, and the known covariance matrix is the covariance matrix of said noise distribution.
6. The device of claim 1 , wherein said expectation maximization algorithm iteratively alternates between an expectation step and a maximization step, until a convergence of estimated parameters to local maxima.
7. The device of claim 1 , wherein said device is further configured to deliver a security performance metric from said estimate of at least one secret value.
8. The device of claim 1 , wherein said device is configured to determine the number of said set of multivariate leakage traces depending on the signal to noise ratio (SNR) or/and on a target security performance metric.
9. The device of claim 1 , wherein said secret value is a cryptographic secret key used in said at least one cryptographic mechanism to encrypt and/or decrypt data and wherein said cryptographic secret key comprises a set of bits, each bit having a leakage property.
10. The device of claim 9 , wherein the leakage properties of the bits are similar.
11. A method of estimating at least one secret value used by at least one cryptographic mechanism from a statistical distribution of a set of multivariate leakage traces,
each leakage trace being a vector comprising a plurality of random values,
the number of said random values being an integer number greater than or equal to 1,
said statistical distribution being a function of parametric linear combinations of a set of leakage model basis vectors representing a multivariate leakage model,
the number of said basis vectors being an integer number greater than or equal to 1,
said parametric linear combinations being defined by a matrix of real values comprising at least one unknown value,
wherein the method comprises simultaneously determining (i) said estimate of at least one secret value and (ii) an estimate of said at least one unknown value of said matrix of real values defining said parametric linear combinations, using a multivariate estimation algorithm that maximizes a probability of successful secret values recovery and that is an expectation maximization algorithm.
12. The method of claim 11 , wherein said expectation maximization algorithm iteratively alternates between an expectation step and a maximization step, until a convergence of estimated parameters to local maxima.