Hamming Weight of Product of Random Sparse Polynomials

Consider two random binary vectors x, y ∈ {0, 1}
ⁿ
of which each coordinate is 1 with small probability ε independently from the others. It is easy to see that the Hamming weight of their sum x ⊕ y is strongly concentrated around the expectation by the standard probability analysis for independent random variables such as the Chernoff bounds, that is, wt(x ⊕ y) is in (1 ± 0.1)E[wt}}(x ⊕ s y)] with probability 1 - exp(-Ω(εn)), where E[w(x ⊕ )] = 2ε (1 - ε). In this paper, we focus on the Hamming weight of their “product” x · y defined as a product of polynomials over the ring F
₂
[t]/(t
ⁿ
-1}, which is utilized for construction of the post-quantum public-key encryption scheme HQC [3]. In the case of the product, the standard analysis does not work because of correlations among the coordinates of x · y. We prove weak concentration bounds for the Hamming weight of x · y from Chebyshev's inequality by analyzing the correlations. For example, we can show that x · y is in (1 ± 0.1)E[wt (x · y)] with probability 1 - O(ε
³
n) for ε = Ω (1/√n )$ from the bounds, where E[wt} (x · y)] = (1 - (1-2ε
²
)
ⁿ
)n/2.