Computational Complexity and Property Testing: On the Interplay Between Randomness and Computation

In the course of research in computational learning theory, we found ourselves in need of an error-correcting encoding scheme for which relatively few bits in the codeword yield no information about the plain message. Being unaware of a previous solution, we came-up with the scheme presented here. Clearly, a scheme as postulated above cannot be deterministic. Thus, we introduce a probabilistic coding scheme that, in addition to the standard coding theoretic requirements, has the feature that any constant fraction of the bits in the (randomized) codeword yields no information about the message being encoded. This coding scheme is also used to obtain efficient constructions for the Wire-Tap Channel Problem. Appeared (under the title “A Probabilistic Error-Correcting Scheme”) as record 1997/005 of the IACR Cryptology ePrint Archive, 1997. In the current revision, the introduction was intentionally left almost intact, but the exposition of the main result (esp., its proof) was elaborated and made more reader-friendly. 1 Original Introduction (Dated April 1997) We believe that the following problem may be relevant to research in cryptography: Provide an error-correcting encoding scheme for which relatively few bits in the codeword yield no information about the plain message. Certainly, no deterministic encoding may satisfy this requirement, and so we are bound to seek probabilistic error-correcting encoding schemes. Specifically, in addition to the standard coding theoretic requirements (i.e., of correcting upto a certain threshold number of errors), we require that obtaining less than a threshold number of bits in the (randomized) codeword yield no information about the message being encoded. Below we present such a probabilistic encoding scheme. In particular, the scheme can (always) correct a certain constant fraction of errors, and has the property that fewer than a certain constant fraction of the bits (in the codeword) yield no information about the encoded message. Thus, using this encoding scheme over an insecure channel tampered by an adversary who can read and modify (only) a constant fraction of the transmitted bits, we establish correct and private communication between the legitimate end-points. c © Springer Nature Switzerland AG 2020 O. Goldreich (Ed.): Computational Complexity and Property Testing, LNCS 12050, pp. 1–8, 2020. https://doi.org/10.1007/978-3-030-43662-9_1

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