## Cryptology ePrint Archive: Report 2006/020

**Scrambling Adversarial Errors Using Few Random Bits, Optimal Information Reconciliation, and Better Private Codes**

*Adam Smith*

**Abstract: **When communicating over a noisy channel, it is typically much easier to deal with random, independent errors with a known distribution than with adversarial errors. This paper looks at how one can use schemes designed for random errors in an adversarial context, at the cost of relatively few additional random bits and without using unproven computational assumptions.

The basic approach is to permute the positions of a bit string using a permutation drawn from a $t$-wise independent family, where $t=o(n)$. This leads to two new results:

1. We construct *computationally efficient* information reconciliation protocols correcting $pn$ adversarial binary Hamming errors with optimal communication and entropy loss $n(h(p)+o(1))$ bits, where $n$ is the length of the strings and $h()$ is the binary entropy function. Information reconciliation protocols are important tools for dealing with noisy secrets in cryptography; they are also used to synchronize remote copies of large files.

2. We improve the randomness complexity (key length) of efficiently decodable capacity-approaching "private codes" from $\Theta(n\log n)$ to $n+o(n)$.

We also present a simplified proof of an existential result on private codes due to Langberg (FOCS '04).

**Category / Keywords: **cryptographic protocols / Information reconciliation, fuzzy cryptography, error-correcting codes, private codes, information theory, derandomization, combinatorial cryptography

**Date: **received 17 Jan 2006, last revised 23 Jan 2006

**Contact author: **adam smith at weizmann ac il

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20060123:135618 (All versions of this report)

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