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DOI: 10.4230/LIPIcs.ITC.2026.8

Fast Bounded-Independence Functions and Their Duals

Authors: Martijn Brehm, Yuval Ishai, and Nicolas Resch

Published in: LIPIcs, Volume 385, 7th Conference on Information-Theoretic Cryptography (ITC 2026)

Abstract

We continue the study of fast functions, computable by linear-size circuits, that share useful properties of random functions. Motivated by cryptographic applications, we generalize and improve on previous results in this area, obtaining the following results: - For any constant t, we construct a fast t-wise independent hash function with algebraic degree log₂ t (over F₂), simultaneously optimizing both asymptotic circuit size and degree. - We simplify and improve a recent construction (ITCS 2026) of a family of fast codes with fast duals, both meeting the Gilbert-Varshamov bound. Unlike the previous construction, our construction has negligible failure probability, can accommodate general fields and rates, supports a systematic encoding, and admits fast universal encoders. - We strengthen the above to support stronger random-like properties, such as optimal combinatorial list-decoding. This is achieved by constructing, for any constant t, a family of fast linear functions that map any t linearly independent inputs to uniform and statistically independent outputs. Prior to our work, this was only known for t = 1. We demonstrate the usefulness of the above results to cryptography. This includes the first nontrivial protocols for perfectly secure multiparty computation whose circuit complexity scales linearly with the number of parties, as well as protocols for computing encrypted matrix-vector products with optimal asymptotic circuit complexity.

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Martijn Brehm, Yuval Ishai, and Nicolas Resch. Fast Bounded-Independence Functions and Their Duals. In 7th Conference on Information-Theoretic Cryptography (ITC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 385, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brehm_et_al:LIPIcs.ITC.2026.8,
  author =	{Brehm, Martijn and Ishai, Yuval and Resch, Nicolas},
  title =	{{Fast Bounded-Independence Functions and Their Duals}},
  booktitle =	{7th Conference on Information-Theoretic Cryptography (ITC 2026)},
  pages =	{8:1--8:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-426-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{385},
  editor =	{Dodis, Yevgeniy},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2026.8},
  URN =		{urn:nbn:de:0030-drops-271018},
  doi =		{10.4230/LIPIcs.ITC.2026.8},
  annote =	{Keywords: Linear codes, hash function families, efficient encoding, secure computation}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.28

Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound

Authors: Martijn Brehm and Nicolas Resch

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We initiate the study of what we term "fast good codes" with "fast good duals." Specifically, we consider the task of constructing a binary linear code C ≤ 𝔽₂ⁿ such that both it and its dual C^⟂ : = {x ∈ 𝔽₂ⁿ:∀ c ∈ C, ⟨ x,c⟩ = 0} are asymptotically good (in fact, have rate-distance tradeoff approaching the GV bound), and are encodable in O(n) time. While we believe such codes should find applications more broadly, as motivation we describe how such codes can be used the secure computation task of encrypted matrix-vector product, as studied by Behhamouda et al (CCS 2025). Our main contribution is a construction of such a fast good code with fast good dual. Our construction is inspired by the repeat multiple accumulate (RMA) codes of Divsalar, Jin and McEliece (Allerton, 1998). To create the rate 1/2 code, after repeating each message coordinate, we perform accumulation steps - where first a uniform coordinate permutation is applied, and afterwards the prefix-sum modulo 2 is applied - which are alternated with discrete derivative steps - where again a uniform coordinate permutation is applied, and afterwards the previous two coordinates are summed modulo 2. Importantly, these two operations are inverse of each other. In particular, the dual of the code is very similar, with the accumulation and discrete derivative steps reversed. Our analysis is inspired by a prior analysis of RMA codes due to Ravazzi and Fagnani (IEEE Trans. Info. Theory, 2009). The main idea is to bound the input-output weight-enumerator function: the expected number of messages of a given weight that are encoded into a codeword of a given weight. We face new challenges in controlling the behaviour of the discrete derivative matrix (which can significantly drop the weight of a vector), which we overcome by careful case analysis.

Cite as

Martijn Brehm and Nicolas Resch. Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brehm_et_al:LIPIcs.ITCS.2026.28,
  author =	{Brehm, Martijn and Resch, Nicolas},
  title =	{{Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{28:1--28:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.28},
  URN =		{urn:nbn:de:0030-drops-253157},
  doi =		{10.4230/LIPIcs.ITCS.2026.28},
  annote =	{Keywords: Binary error-correcting codes, dual codes, fast encoding, repeat-multiple-accumulate codes}
}

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Documents authored by Brehm, Martijn

Fast Bounded-Independence Functions and Their Duals

Abstract

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Linear Time Encodable Binary Code Achieving GV Bound with Linear Time Encodable Dual Achieving GV Bound

Abstract

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