HEXAGONbeta

The Welch function is not third-order sum-free

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorycs.IT — Information Theory

Contributed by Deep Bhattacharjee

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Let n=2m+1n=2m+1 be odd and let $\F_{2^n}$ be the field with 2n2^n elements. A function on $\F_{2^n}$ is third-order sum-free when its values over any eight elements forming a three-dimensional affine subspace never add up to zero. The Welch function is the power map x↦x2m+3x\mapsto x^{2^m+3}; it is one of the classical almost perfect nonlinear functions, and Hou and Zhao conjectured that for n≥7n\ge7 it is not third-order sum-free. We prove this. The question comes down to a linearised polynomial g(x,y)g(x,y) in yy whose roots form a vector space Z(x)Z(x) over $\F_2$ containing 11 and xx, and the Welch function fails to be third-order sum-free exactly when Z(x)Z(x) has dimension three for some xx outside $\F_2$. We show that the dimension is always 22 or 33, and that it is 33 if and only if two explicit trace conditions hold: with u=x2mu=x^{2^m}, both $\Tr(1/(x+u))$ and $\Tr(xu/(x+u))$ equal 11. The proof applies the field automorphism z↦z2m+1z\mapsto z^{2^{m+1}}, whose square is the Frobenius map, to turn gg into an additive polynomial of degree eight whose roots always lie in $\F_{2^n}$. Counting the elements that satisfy both trace conditions gives exactly (2n+1−3Kn)/4(2^n+1-3K_n)/4 of them, where KnK_n is the binary Kloosterman sum of $\F_{2^n}$, and Weil's bound ∣Kn∣≤2n/2+1|K_n|\le 2^{n/2+1} makes this count positive for every n≥7n\ge7. The Welch function sums to zero over exactly (2n+1−3Kn)(2n−1)/168(2^n+1-3K_n)(2^n-1)/168 three-dimensional linear subspaces, and the count 2+(2n+1−3Kn)/42+(2^n+1-3K_n)/4 agrees with the numbers that Hou and Zhao computed for n≤15n\le15. Every statement is proved by hand.

Provenance statement

Deep Bhattacharjee chose the problem, Conjecture 4.10 of Hou and Zhao (arXiv:2609.31489), and directed the work. Claude (Anthropic), through Claude Code, searched the literature, found and checked the proofs, wrote the verification programs and drafted the paper. The proof is complete by hand in eight pages: the kernel Z(x) of the Hou–Zhao linearized polynomial has dimension 3 exactly when two traces equal 1, the number of such x is (2^n+1−3K_n)/4 with K_n the binary Kloosterman sum, and Weil's bound makes it positive for every n ≥ 7. C and Python programs check the criterion and every lemma at every element of F_(2^n) (n ≤ 21 and n ≤ 13), C++ counts the zero-sum subspaces from the definition for n ≤ 11, a Lean kernel certificate settles n = 5 and n = 7, and Lean 4 with Mathlib proves the algebraic identities of the proof. Code and proofs: release v1.1.0, https://github.com/creelie/eConjectureWelchSumFree/releases/tag/v1.1.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Fable 5.1 Medium

References

  1. A. Canteaut, P. Charpin and H. Dobbertin, Binary mm-sequences with three-valued crosscorrelation: a proof of Welch's conjecture , IEEE Trans. Inform. Theory 46 (2000), 4--8. 10.1109/18.817504DOI
  2. C. Carlet, On the vector subspaces of $\F_{2^n}$ over which the multiplicative inverse function sums to zero , Des. Codes Cryptogr. 93 (2025), 1237--1254. 10.1007/s10623-024-01531-6DOI
  3. C. Carlet, Two generalizations of almost perfect nonlinearity , J. Cryptology 38 (2025), Paper No. 20. 10.1007/s00145-025-09538-5DOI
  4. L. Carlitz, Kloosterman sums and finite field extensions , Acta Arith. 16 (1969), 179--194. 10.4064/aa-16-2-179-194DOI
  5. H. Dobbertin, Almost perfect nonlinear power functions on GF(2n)GF(2^n): the Welch case , IEEE Trans. Inform. Theory 45 (1999), 1271--1275. 10.1109/18.761283DOI
  6. X. Hou and S. Zhao, Further results on sum-freedom of binary and qq-ary functions , preprint, 2026. arXiv:2609.31489 (https://arxiv.org/abs/2609.31489)arXiv
  7. R. Lidl and H. Niederreiter, Finite Fields , 2nd ed., Encyclopedia Math. Appl. 20 , Cambridge University Press, Cambridge, 1997. 10.1017/CBO9780511525926DOI
  8. A. Weil, On some exponential sums , Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204--207. 10.1073/pnas.34.5.204DOI

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 22, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 26, 2026