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Some notes on the pseudorandomness of Legendre symbol and Liouville function

Johannes Grünberger, Arne Winterhof

Abstract

We improve bounds on the degree and sparsity of Boolean functions representing the Legendre symbol as well as on the $N$th linear complexity of the Legendre sequence. We also prove similar results for both the Liouville function for integers and its analog for polynomials over $\mathbb{F}_2$, or more general for any (binary) arithmetic function which satisfies $f(2n)=-f(n)$ for $n=1,2,\ldots$

Some notes on the pseudorandomness of Legendre symbol and Liouville function

Abstract

We improve bounds on the degree and sparsity of Boolean functions representing the Legendre symbol as well as on the th linear complexity of the Legendre sequence. We also prove similar results for both the Liouville function for integers and its analog for polynomials over , or more general for any (binary) arithmetic function which satisfies for

Paper Structure

This paper contains 10 sections, 8 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Some arithmetic functions and their pseudorandomness
  3. Legendre symbol
  4. Liouville function for integers
  5. Liouville function for polynomials
  6. Bounds on degree and sparsity
  7. Bounds on the $N$th linear complexity
  8. $N$th lattice level
  9. Linear complexity
  10. Conclusion

Key Result

Theorem 1

Let $f$ be any (binary) arithmetic function with and $B$ be the Boolean function defined by and $B(0,0,\ldots,0)=c$ with $c\in \mathbb{F}_2$. Then we have and

Figures (3)

  • Figure 1: The distance of ${\rm spr}(B)$ from $2^{r-1}$ for all primes $2 < p < 10000$, where $B$ is a Boolean function corresponding to the Legendre symbol with modulus $p$ and $r = \lfloor\log_2 p\rfloor$.
  • Figure 2: The maximum distance of $L(\mathcal{L}_p, N)$ from $N/2$, $N = 1, \ldots, p + 1$, for all primes $p < 10000$ with $p\equiv \pm 1\bmod 8$.
  • Figure 3: $L(\mathcal{L}_{100049}, N) - \frac{N}{2}$ for $N = 1, \ldots, 100050$.

Theorems & Definitions (8)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 2
  • Theorem 3