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Fast Bounded-Independence Functions and Their Duals

Authors Martijn Brehm , Yuval Ishai , Nicolas Resch

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Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
  • Theory of computation → Error-correcting codes
  • Theory of computation → Cryptographic protocols
Keywords
  • Linear codes
  • hash function families
  • efficient encoding
  • secure computation

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITC.2026.8

Author Details

Martijn Brehm
  • Informatics Institute, University of Amsterdam, The Netherlands
Yuval Ishai
  • Technion, Haifa, Israel
  • AWS, New York, NY, USA
Nicolas Resch
  • Informatics Institute, University of Amsterdam, The Netherlands

Funding

  • Ishai, Yuval: Research supported by ISF grant 3527/24 and BSF grant 2022370. This work is not associated with Amazon.
  • Resch, Nicolas: Research supported by an NWO (Dutch Research Council) grant with number C.2324.0590.

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