[Paper Review] Power moments of Kloosterman sums

Ke Гонг, Willem Veys+1 more|arXiv (Cornell University)|Jun 18, 2013

Coding theory and cryptography5 references3 citations

TL;DR

This paper derives explicit formulas for power moments of Kloosterman sums over ℤ/qℤ when q is a powerful prime power, using Igusa zeta functions and p-adic analysis. It shows that for sufficiently large exponents r in q = p^r, the n-th moment S_n(p^r) vanishes when n is odd, and admits a closed-form expression when n is even, with optimal bounds on r depending on n and p.

ABSTRACT

In this paper we give an essential treatment for estimating arbitrary integral power moments of Kloosterman sums over the residue class ring. For prime moduli we derive explicit estimates, and for prime-power moduli we prove concrete formulas using computations with Igusa zeta functions.

Motivation & Objective

• To derive explicit formulas for the n-th power moments of Kloosterman sums over ℤ/qℤ when q is a prime power.
• To understand the distribution of Kloosterman sums K(1,v;q) as v varies, particularly for large q.
• To establish uniform, clean formulas for S_n(q) when q is sufficiently powerful relative to n.
• To resolve the complexity of higher moments by identifying conditions under which they simplify.
• To provide a foundation for applications in cryptography and coding theory requiring sharp bounds on the maximum absolute value of Kloosterman sums.

Proposed method

• Reduction of the general moment S_n(q) to the prime power case using multiplicative properties of Kloosterman sums.
• Application of Igusa zeta functions to compute power moments for p-adic local factors.
• Use of p-adic stationary phase methods and Taylor expansions to analyze solutions modulo p^r.
• Employment of Hensel's lemma and its refined variants to count solutions lifting from lower to higher p-adic levels.
• Analysis of singular and non-singular solutions in p-adic spaces, particularly for p=2.
• Computation of the order of partial derivatives to control lifting behavior and ensure vanishing contributions when conditions are not met.

Experimental results

Research questions

• RQ1 Under what conditions on q and n do the power moments S_n(q) of Kloosterman sums admit explicit closed-form expressions?
• RQ2 How does the size of the p-adic valuation v_p(q) affect the vanishing or non-vanishing of odd-power moments?
• RQ3 What is the precise threshold for r = v_p(q) such that the moment formula for even n becomes valid?
• RQ4 Why do the formulas break down when r is below the derived bounds, and how can this be detected via p-adic analysis?
• RQ5 Can the behavior of higher moments be uniformly described when q is a powerful prime power, especially for p=2?

Key findings

• For odd n, if r > log_p(n) + 1, then S_n(p^r) = 0, and this bound is optimal.
• For even n, if r > log_p(n/2) + 1 and p is odd, then S_n(p^r) = binom(n-1, n/2 - 1) * (p-1)/p * p^{(n/2 + 1)r}.
• For p = 2 and even n, if r > log_2(n) + 2, then S_n(2^r) = binom(n-1, n/2 - 1) * 2^{n/2 - 2} * 2^{(n/2 + 1)r}.
• The bounds on r in the theorems are optimal, as confirmed by computer experiments and failure of formulas when r is too small.
• The moments for p=2 require a more refined analysis due to singularities modulo 2, leading to a higher threshold in the exponent.
• When r is small, correction terms appear in the moment formulas that depend on n modulo p, indicating increased complexity.