On the Leakage of Massey Secret Sharing Schemes under Linear Computations

Leakage attacks on secret sharing schemes exploit partial information about individual shares to recover the underlying secret. In coding theory, linear exact repair schemes (LERSs) enable the recovery of one codeword symbol from a small amount of information obtained from the remaining symbols, provided that the code has sufficiently low rate. This can be interpreted as recovering the secret from partial information, namely subfield symbols, of the shares. Recently, a randomized construction based on subfield subcodes was proposed for constructing LERS-derived leakage attacks against Massey secret sharing schemes based on general linear codes. We extend this framework to multiple shared secrets whose corresponding shares are related through linear computations, with leakage also allowed on the computation outcomes. More precisely, we consider N secrets, of which K $\le$ N are linearly independent input values and the remaining N -K secrets are determined by linear computations on these inputs. We analyse the existence of LERS-derived leakage that exploits this structure. We first study the case of addition and then generalize our construction to arbitrary linear computations. Our analysis applies to general linear codes of length n+1 and dimension k over F\_{q^m} with k $\le$ N n/(Km), and supports arbitrary linear computations, whereas the previous subfield subcode construction only applies to k $\le$ n/m -1. Consequently, exploiting the linear relations enables LERS based leakage which extend the range of code parameters vulnerable to such attacks. Finally, identical leakage functions can arise for certain linear relations, making this a more realistic yet still potentially powerful attack model. Finally, simulations indicate that identical leakage functions can be used for certain linear relations, yielding a more realistic attack model.