The covert set-cover problem with application to Network Discovery

We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size $OPT$ within $O(\log N)$ factor with high probability using $O(OPT \cdot \log^2 N)$ queries where $N$ is the input size. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown $n$-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an $O(\log^2 n)$-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of $\Omega (\sqrt{n\log n})$ and therefore our result achieves an exponential improvement

A result on the distribution of quadratic residues with applications to elliptic curve cryptography

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x + 1), f(x + 2), ..., f(x +κ)) are quadratic non-residue is ≈ 1/2κ. In particular for f(x) = x3 + ax + b corresponding to the elliptic curve y2 = x3 + ax + b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly distributed. Using this result we describe an efficient implementation of El-Gamal Cryptosystem. that requires efficient computation of a mapping between plain-texts and the points on the elliptic curve

Efficient Format Preserving Encrypted Databases

We propose storage efficient SQL-aware encrypted databases that preserve the format of the fields. We give experimental results of storage improvements in CryptDB using FNR encryption scheme

Improvements on the Johnson bound for Reed-Solomon Codes

For Reed-Solomon Codes with block length n and dimension k, the Johnson theorem states that for a Hamming ball of radius smaller than n − √ nk, there can be at most O(n2) codewords. It was not known whether for larger radius, the number of code words is polynomial. The best known list decoding algorithm for Reed-Solomon Codes due to Guruswami and Sudan [13] is also known to work in polynomial time only within this radius. In this paper we prove that when k < αn for any constant 0 < α < 1, we can overcome the barrier of the Johnson bound for list-decoding of Reed-Solomon Codes (even if the field size is exponential). More specifically in such a case, we prove that for Hamming ball of radius n − √ nk + c, (for any c> 0) there can be at most O(n c (1 − √ α) 2 +c+2) number of codewords. For any constant c, we describe a polynomial time algorithm to enumerate all of them, thereby also improving on the Guruswami-Sudan’s algorithm. Although the improvement is modest this provides evidence for the first time that the n − √ nk bound is not sacrosanct for such a high rate. We apply our method to obtain sharper bounds on a list recovery problem introduced by Guruswami and Rudra [11] where they establish super polynomial lower bounds on the output size when the list size exceeds ⌈ n k ⌉. We show that even for larger list sizes the problem can be solved in polynomial time for certain values of k. 2 √ α