This work introduces two algebraic variants of the Massey-Omura cryptosystem based on newly defined generalized (k,t)-Jacobsthal p-numbers and their extensions to finite groups. We first generalize the classical Jacobsthal recurrence and establish structural properties including periodicity, invertibility conditions, and recurrence behavior modulo finite integers. These results are then extended to group-theoretic settings, where we construct the corresponding (k,t)-Jacobsthal sequences in specific finite groups and derive their sequence periods. Leveraging these algebraic foundations, we propose two Massey-Omura-type encryption schemes in which private exponents are selected from the generalized Jacobsthal sequences. We formally prove the correctness of both constructions and analyze the implications of periodicity on exponent invertibility and protocol feasibility. The proposed schemes do not introduce new hardness assumptions beyond those inherent in the underlying platform group. Instead, they provide a mathematically structured alternative to classical exponent selection in three-pass protocols. The results highlight a new connection between recurrence-defined sequences and multiplicative exponentiation in finite groups, offering an algebraically motivated direction for exploring generalized exponent families in symmetric and non-abelian cryptosystems.