The symmetric determinantal complexity $\sdc(f)$ of a polynomial $f$ is the
least $m$ such that $f=\Det(M)$ for an $m\times m$ symmetric matrix $M$ of
affine-linear forms. We prove, over $\CC$, that
\[
\sdc\!\left(\sum_{i=1}^n x_i^n\right)
\ge \left(\frac{1}{2e}-o(1)\right)n^2 .
\]
The result is a symmetric companion to the author's non-symmetric ... more >>>

I refute the 1995 dream XOR lemma conjecture by Goldreich, Nisan, and Wigderson. I also give a counterexample to the XOR lemma for low-degree polynomials.

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The hardness of the Learning Parity with Noise (LPN) problem is a foundational assumption in cryptography, forming the basis of constructions ranging from symmetric-key primitives to public-key encryption and beyond. A central open question is whether the average-case hardness of LPN can be based on worst-case complexity assumptions, as has ... more >>>