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Ghasemi-Kopparty Problem on Sparse $S$-Decoding Polynomials

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ghasemi-kopparty-sparse-decoding-polynomialsTheoretical computer scienceposed by Mahdi Ghasemi, Swastik Kopparty, 2026recorded: solved

1 attempt · no person has looked

Statement

Can SS-decoding polynomials modulo a product of kk primes be built with only k+1k+1 nonzero coefficients, the minimum their own lower bound allows? Yes, via a general framework for special prime products. The consequence is that for any constant ss there is an ss-server private information retrieval protocol with communication exp⁡(O((log⁡n)1/s(log⁡log⁡n)1−1/s))\exp(O((\log n)^{1/s}(\log\log n)^{1-1/s})) on an nn-bit database, where previous constructions at that communication needed 2O(s)2^{O(s)} servers.

Context

the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15

An open problem posed at ITCS 2026 with a matching lower bound already in hand, inside the matching-vector PIR line running from Efremenko to Ghasemi-Kopparty-Sudan.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.5 Pro with Aparna Gupte, Seyoon Ragavan ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.5 Pro
    people
    Aparna Gupte, Seyoon Ragavan

    Stated in the abstract itself rather than buried in an acknowledgement: the main result for constant ss and its proof were discovered in a GPT-5.5 Pro conversation prompted by the authors.

    the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15

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