Error Correction Capabilities of Non-Linear Cryptographic Hash Functions

TL;DR

The paper investigates whether non-linear cryptographic hash functions can serve as forward error-correcting codes over noisy channels. It develops a joint error-correction and hash-check framework built from Systematic Random Non-Linear Codes (S-RNLC) and a GRAND-based decoder, proving achievability of channel capacity for NL-CHF at rate $k/n < C$, where $C$ is the AWGN capacity. Empirical results compare SHA-1/SHA-256 against S-RLC and S-RNLC, showing comparable BLER performance under GRAND decoding across tested rates and lengths. This work suggests a single-stage approach where NL-CHF replaces separate coding and authentication steps, broadening practical options for secure communications.

Abstract

Linear hashes are known to possess error-correcting capabilities. However, in most applications, non-linear hashes with pseudorandom outputs are utilized instead. It has also been established that classical non-systematic random codes, both linear and non-linear, are capacity achieving in the asymptotic regime. Thus, it is reasonable to expect that non-linear hashes might also exhibit good error-correcting capabilities. In this paper, we show this to be the case. Our proof is based on techniques from multiple access channels. As a consequence, we show that Systematic Random Non-Linear Codes (S-RNLC) are capacity achieving in the asymptotic regime. We validate our results by comparing the performance of the Secure Hash Algorithm (SHA) with that of Systematic Random Linear Codes (SRLC) and S-RNLC, demonstrating that SHA performs equally.

Figures (4)

• Figure 1: Reliable digital signature verification over a noisy channel, one source, Alice, one legitimate destination, Bob.
• Figure 2: Proposed non-linear cryptographic hash functions with error correction capabilities, e.g., SHA (upper figure) and a Systematic RLC decoded with GRAND (bottom figure).
• Figure 3: BLER vs. Eb/N0 for codes of $k=128$ and $n=288$ encoded with SHA1 as error correcting code and with Systematic RLC and Systematic RNLC. Here, both $M\in \mathbb{F}_{2}^{k}$ and $D\in \mathbb{F}_{2}^{n-k}$ are transmitted over the noisy channel. The joint error correction and hash check is performed with GRAND.
• Figure 4: BLER vs. Eb/N0 for codes of $k=200$ and $n=350$ encoded with SHA1 as error correcting code and with Systematic RLC and Systematic RNLC. Here, both $M\in \mathbb{F}_{2}^{k}$ and $D\in \mathbb{F}_{2}^{n-k}$ are transmitted over the noisy channel. The joint error correction and hash check is performed with GRAND.

Theorems & Definitions (9)

• Definition 1: Random oracle
• Definition 2: Cryptographic hash function
• Definition 3: Systematic Random Function
• Definition 4: Random Code
• Proposition 1: Random Codes are Good
• proof
• Theorem 1
• proof
• Remark 1