A system is described for reducing leakage of meaningful information from cryptographic operations. The system uses a pairwise independent hash function to generate a modified secret key SK′ having individual components. The system forms a modified secret key collection that includes SK′ and its individual components. The system then uses the modified secret key collection to decrypt a message. The decryption involves providing multiple partial operation results in separate respective steps. Leakage of meaningful information is reduced due to difficulty in piecing together meaningful information from information leaked by the separate partial operations. In one example, the hash function has the form HK(r)=ar+b, where a, b, and r are selected values, such as random numbers. In another example, the hash function has the form HK(r)=Ar*B, where A, B, and r are selected values.