Abstract
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.
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Notes
- 1.
Added in revision: We stress that, in contrast, our solution uses uni-directional communication. On the other hand, our solution holds only for a limited range of parameters; see discussion at the end of Sect. 3.
- 2.
Here we assume that 3t is a prime power. Actually, we use the first power of 2 that is greater than 3t. Clearly, this inaccuracy has a negligible effect on the construction.
- 3.
A Toeplitz matrix, \(T=(t_{i,j})\), satisfies \(t_{i,j}=t_{i+1,j+1}\), for every i, j.
- 4.
The proof uses the fact that any (non-zero) linear combination of rows (or columns) in a random Toeplitz matrix is uniformly distributed. The first condition is proved by observing that the probability that a non-zero combination of the rows of the \(2\ell \)-by-\(4\ell \) matrix has Hamming weight smaller than \(\ell '\) is upper-bounded by \((2^{2\ell }-1)\cdot \sum _{i=0}^{\ell '-1}{{4\ell }\atopwithdelims ()i}\cdot 2^{-4\ell }\), which is o(1) for some \(\ell '=\varOmega (\ell )\). The second condition is proved by observing that the probability that there exist \(\ell ''\) columns that yield a submatrix (of the last \(\ell \) rows) that is not full rank is upper-bounded by \({{4\ell }\atopwithdelims (){\ell ''}}\cdot (2^{\ell ''}-1)\cdot 2^{-\ell }\), which is o(1) for some \(\ell ''=\varOmega (\ell )\).
- 5.
Recall that the crossover probability is the probability that a bit is complemented in the transmission process.
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Decatur, S., Goldreich, O., Ron, D. (2020). A Probabilistic Error-Correcting Scheme that Provides Partial Secrecy. In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_1
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