[Paper Review] Set Families with Low Pairwise Intersection

Calvin Beideman, Jeremiah Blocki|arXiv (Cornell University)|Apr 17, 2014

Cryptography and Data Security4 references3 citations

TL;DR

This paper presents an explicit construction of weak $(n, heta, u)$-sharing set families using techniques from number theory and Ramanujan primes, achieving a family size of $(2 heta \ln 2\theta)^{\nu+1}$ for $n = 4\theta^2\ln 4\theta$, which matches the performance of the non-explicit construction by Raz et al. The construction enables low-memory, parallelizable pseudorandom number generators and establishes tight bounds on the maximum size of such families under various parameter regimes.

ABSTRACT

A $\left(n,\ell,γ ight)$-sharing set family of size $m$ is a family of sets $S_1,\ldots,S_m\subseteq [n]$ s.t. each set has size $\ell$ and each pair of sets shares at most $γ$ elements. We let $m\left(n,\ell,γ ight)$ denote the maximum size of any such set family and we consider the following question: How large can $m\left(n,\ell,γ ight)$ be? $\left(n,\ell,γ ight)$-sharing set families have a rich set of applications including the construction of pseudorandom number generators and usable and secure password management schemes. We analyze the explicit construction of Blocki et al using recent bounds on the value of the $t$'th Ramanujan prime. We show that this explicit construction produces a $\left(4\ell^2\ln 4\ell,\ell,γ ight)$-sharing set family of size $\left(2 \ell \ln 2\ell ight)^{γ+1}$ for any $\ell\geq γ$. We also show that the construction of Blocki et al can be used to obtain a weak $\left(n,\ell,γ ight)$-sharing set family of size $m$ for any $m >0$. These results are competitive with the inexplicit construction of Raz et al for weak $\left(n,\ell,γ ight)$-sharing families. We show that our explicit construction of weak $\left(n,\ell,γ ight)$-sharing set families can be used to obtain a parallelizable pseudorandom number generator with a low memory footprint by using the pseudorandom number generator of Nisan and Wigderson. We also prove that $m\left(n,n/c_1,c_2n ight)$ must be a constant whenever $c_2 \leq \frac{2}{c_1^3+c_1^2}$. We show that this bound is nearly tight as $m\left(n,n/c_1,c_2n ight)$ grows exponentially fast whenever $c_2 > c_1^{-2}$.

Motivation & Objective

• To develop an explicit, efficient construction of weak $(n, heta, u)$-sharing set families for use in pseudorandom number generation and password management.
• To analyze the Blocki et al. construction using recent bounds on Ramanujan primes to establish tight parameter trade-offs.
• To prove that $m(n, n/c_1, c_2 n)$ is constant when $c_2 \leq 2/(c_1^3 + c_1^2)$, and grows exponentially when $c_2 > 1/c_1^2$, showing the threshold behavior of family size.
• To demonstrate that the explicit construction matches the performance of the non-explicit construction by Raz et al. for weak set families.
• To provide tight asymptotic bounds on the maximum size of set families with bounded pairwise intersection under various parameter regimes.

Proposed method

• Leverages recent bounds on the $t$-th Ramanujan prime to analyze the explicit construction of Blocki et al. for $(n, heta, u)$-sharing set families.
• Constructs a weak $(n, heta, u)$-sharing set family of size $(2\theta \ln 2\theta)^{\nu+1}$ for $n = 4\theta^2 \ln 4\theta$, using a randomized selection process over residue classes.
• Uses Chernoff bounds and union bounds to show that random selection of sets from arithmetic progressions yields pairwise intersections at most $\nu$ with non-zero probability.
• Applies the Nisan-Wigderson framework to build a parallelizable pseudorandom number generator from the constructed set family, enabling low memory usage via explicit index computation.
• Employs extremal combinatorics and probabilistic methods to prove upper and lower bounds on $m(n, \theta, \nu)$ under dense parameter regimes, particularly when $\theta = n/c_1$ and $\nu = c_2 n$.
• Uses a constructive method based on prime products to select integers with bounded pairwise GCDs, supporting the derivation of extremal bounds on set family size.

Experimental results

Research questions

• RQ1 Can an explicit construction of weak $(n, \theta, \nu)$-sharing set families be designed that matches the performance of the non-explicit construction by Raz et al.?
• RQ2 What is the maximum size $m(n, \theta, \nu)$ of a set family where each set has size $\theta$, and any two sets intersect in at most $\nu$ elements, under dense parameter regimes?
• RQ3 How does $m(n, n/c_1, c_2 n)$ behave as a function of $c_1$ and $c_2$, particularly in the threshold region between constant and exponential growth?
• RQ4 Can the Blocki et al. construction be rigorously analyzed using Ramanujan prime bounds to derive explicit, tight bounds on family size?
• RQ5 Is the upper bound $m(n, n/c_1, c_2 n) = \text{const}$ tight when $c_2 \leq 2/(c_1^3 + c_1^2)$, and how does the family size transition to exponential growth as $c_2$ increases?

Key findings

• An explicit construction of a weak $(n, \theta, \nu)$-sharing set family of size $(2\theta \ln 2\theta)^{\nu+1}$ is achieved for $n = 4\theta^2 \ln 4\theta$, matching the performance of the non-explicit construction by Raz et al. for weak families.
• The construction enables a parallelizable pseudorandom number generator with low memory footprint by computing set indices on-the-fly via a small circuit.
• It is proven that $m(n, n/c_1, c_2 n)$ is constant whenever $c_2 \leq 2/(c_1^3 + c_1^2)$, establishing a sharp threshold for constant-size families.
• When $c_2 > 1/c_1^2$, the family size $m(n, n/c_1, c_2 n)$ grows exponentially, showing that the upper bound is nearly tight.
• The paper establishes that $m(n, \theta, \nu) \geq m(n, \theta+1, \nu)$ for $\theta \geq \nu+1$, implying that the maximum size of such families is achieved at $\theta = \nu+1$ when $\nu \leq n/2$, and at $\theta = \nu$ when $\nu \geq n/2$, with values ${n \choose \nu+1}$ and ${n \choose \nu}$ respectively.
• A randomized construction using residue classes modulo $c_1$ yields an $(n, \theta, \nu)$-sharing set family of exponential size when $\nu > n/c_1^2$, proving the existence of such families with non-zero probability.