Degradation and Amplification of Computational Hardness
What happens when you use a partially defective bit- commitment protocol to commit to the same bit many times? For example, suppose that the protocol allows the receiver to guess the committed bit with advantage Open image in new window , and that you used that protocol to commit to the same bit more than Open image in new window times. Or suppose that you encrypted some message many times (to many people), only to discover later that the encryption scheme that you were using is partially defective, and an eavesdropper has some noticeable advantage in guessing the encrypted message from the ciphertext. Can we at least show that even after many such encryptions, the eavesdropper could not have learned the message with certainty?
In this work we take another look at amplification and degradation of computational hardness. We describe a rather generic setting where one can argue about amplification or degradation of computational hardness via sequential repetition of interactive protocols, and prove that in all the cases that we consider, it behaves as one would expect from the corresponding information theoretic bounds. In particular, for the example above we can prove that after committing to the same bit for n times, the receiver’s advantage in guessing the encrypted bit is negligibly close to Open image in new window .
Our results for hardness amplification follow just by observing that some of the known proofs for Yao’s lemmas can be easily extended also to handle interactive protocols. On the other hand, the question of hardness degradation was never considered before as far as we know, and we prove these results from scratch.
KeywordsSecurity Parameter Commitment Scheme Oblivious Transfer Interactive Protocol Computational Setting
- 1.Canetti, R., Halevi, S., Steiner, M.: Hardness amplification of weakly verifiable puzzles. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 17–33. Springer, Heidelberg (2005)Google Scholar
- 4.Goldreich, O., Nisan, N., Wigderson, A.: On Yao’s xor-lemma. Electronic Colloquium on Computational Complexity (ECCC) 2(50) (1995)Google Scholar
- 6.Holenstein, T., Renner, R.: One-Way Secret-Key Agreement and Applications to Circuit Polarization and Immunization of Public-Key Encryption. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 478–493. Springer, Heidelberg (2005)Google Scholar
- 8.Kilian, J.: Founding Cryptography on Oblivious Transfer. In: STOC 1988, pp. 30–31. ACM Press, New York (1988)Google Scholar
- 10.Rabin, M.O.: How to exchange secrets by oblivious transfer. Technical Report TR-81, Harvard (1981)Google Scholar
- 12.Yao, A.C.: Theory and applications of trapdoor functions. In: 23rd Annual Symposium on Foundations of Computer Science, November 1982, pp. 80–91. IEEE Computer Society Press, Los Alamitos (1982)Google Scholar