On Some Computational and Security Aspects of the Blom Scheme

Abstract

We study some computational and security aspects of the (k, m)-Blom scheme. This key pre-distribution scheme is defined by a symmetric polynomial (formula presented) of degree m in each variable over a field Fp and a set of user identifiers (formula presented). The pre-shared keys that are distributed to users are the coefficients of the polynomials (formula presented) with distinct users identifiers ri, (formula presented). Implementing Stinson’s proof of unconditional security of (2, m)-Blom scheme we get the proof of unconditional security of (k, m)-Blom scheme. We establish that for disclosing of the (k, m)-Blom scheme we do not need to use all coefficients of m + 1 compromised polynomials. We show that for disclosure, it is enough to use just (formula presented) certain coefficients - exactly as many coefficients as there are present in a non-redundant representation of the original polynomial. Additionally, we estimate the number of multiplications in the field Fp sufficient to compute a pre-shared key and the shared key (formula presented) for communication within a privileged group of k users and to find the initial polynomial as well.