Structure-Preserving Error-Correcting Codes for Polynomial Frames

TL;DR

This work tackles silent data corruption in polynomial-frame computations by introducing a structure-preserving reliability layer that operates directly in the encoded ring $R=\mathbb{Z}_{2^k}[X]/(X^N+1)$. It provides two concrete encoders—one for odd $N$ via a Hensel-lifted BCH with an idempotent encoder and one for $N=2^m$ via a repeated-root negacyclic code—along with an automorphism-based interleaver to disperse bursts, and a bounded-distance decoder that handles symbol errors and flagged erasures without re-encoding or data-format changes. The approach achieves per-frame failure targets around $10^{-9}$ at symbol-error rates $p\in[10^{-6},10^{-5}]$, with overheads of $0.20\%$–$1.56\%$ across $N\in\{1024,2048,4096,8192\}$ and tolerates bursts of roughly $32$–$72$ bytes after interleaving, making it practical for polynomial-frame pipelines in HE/PPML contexts. By aligning protection with ring semantics, the method preserves linear and, in the odd-length encoder case, multiplicative operations, enabling deployment without costly data reformatting and providing a robust, non-interactive alternative to traditional transport/retry or byte-oriented ECC schemes.

Abstract

Modern FFT/NTT analytics, coded computation, and privacy-preserving ML interface routinely move polynomial frames across NICs, storage, and accelerators. However, even rare silent data corruption (SDC) can flip a few ring coefficients and cascade through downstream arithmetic. Conventional defenses are ill-matched to current low-latency pipelines: detect-and-retransmit adds RTTs, while byte-stream ECC ignores the algebraic structure and forces format conversions. To that end, we propose a structure-preserving reliability layer that operates in the encoded data's original polynomial ring, adds a small amount of systematic redundancy, and corrects symbol errors/flagged erasures without round-trip or format changes. We construct two complementary schemes: one for odd length $N_{odd}$ via a Hensel-lifted BCH ideal with an idempotent encoder, and one for power-of-two length $N_{2^m}$ via a repeated-root negacyclic code with derivative-style decoding. In particular, to stay robust against clustered errors, a ring automorphism provides in-place interleaving to disperse bursts. Implementation wise, on four frame sizes $N\!=\!1024, 2048, 4096, 8192$, we meet a per-frame failure target of $10^{-9}$ at symbol error rates $10^{-6}\text{--}10^{-5}$ with $t\!=\!8\text{--}9$, incurring only $0.20\%\text{--}1.56\%$ overhead and tolerating $\sim\!32\text{--}72$\,B unknown-error bursts (roughly doubled when flagged as erasures) after interleaving. By aligning error correction with ring semantics, we take a practical step toward deployable robustness for polynomial-frame computations from an algebraic coding perspective.

Figures (7)

• Figure 1: Hash-and-check workflow: interactive detection only, RTT on the critical path.
• Figure 2: Plain BCH on bitstreams: good for transport, but not ring-compatible with HE.
• Figure 3: Repeated-root: general input domain, linear HE preserved.
• Figure 4: Idempotent encoder: shrinks plaintext domain to $Re$ but preserves add and mult.
• Figure 5: Block failure probability under i.i.d. symbol flips, shown for $N \in \{1024, 2048, 4096, 8192\}$ with $t = N/32$. Each curve captures the sharp transition as $p$ increases. The $N{=}1024$ case shows a discrete jump due to binomial granularity at small $t$.

Theorems & Definitions (25)

• definition 1: Primitive idempotents and ideal codes
• lemma 1: Idempotent encoder
• proof
• theorem 1: Preservation and decoding via the binary residue
• proof
• proposition 1: Injective domains
• proof
• proposition 2: No nontrivial idempotents when $N=2^m$
• proof
• definition 2: Repeated-root negacyclic ideal codes