Anindya De

Trevisan's extractor in the presence of quantum side information

Randomness extraction involves the processing of purely classical information and is therefore usually studied with in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well-known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g., extract all the conditional (smooth) min-entropy of the source using a seed of length polylogarithmic in the input, or only require the seed to be weakly random.

Fast integer multiplication using modular arithmetic

We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an O(N • log N • 2O(log*N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürers algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürers algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.

Near-optimal extractors against quantum storage

We show that Trevisans extractor and its variants [22,19] are secure against bounded quantum storage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K-b), where K is the sources entropy and b the adversarys storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Θ((K-b)/Kγ) for any γ>0. In contrast, the previous best construction [21] could only extract (K/b)1/15 bits. Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications. Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in [21], which was the source of the multiplicative overhead.

Fast Integer Multiplication Using Modular Arithmetic

We give an

A size-free CLT for poisson multinomials and its applications

N\cdot \log N\cdot 2^{O(\log^*N)}

Computation of Optimal Break Point Set of Relays—An Integer Linear Programming Approach

time algorithm to multiply two

Beyond the Central Limit theorem: Asymptotic Expansions and Pseudorandomness for Combinatorial Sums

N

Efficient deterministic approximate counting for low-degree polynomial threshold functions

-bit integers that uses modular arithmetic for intermediate computations instead of arithmetic over complex numbers as in Furers algorithm, which also has the same and so far the best known complexity. The previous best algorithm using modular arithmetic (by Schonhage and Strassen) has complexity

A Robust Khintchine Inequality, and Algorithms for Computing Optimal Constants in Fourier Analysis and High-Dimensional Geometry

O(N \cdot \log N \cdot \log\log N)

A polynomial-time approximation scheme for fault-tolerant distributed storage

. The advantage of using modular arithmetic as opposed to complex number arithmetic is that we can completely evade the task of bounding the truncation error due to finite approximations of complex numbers, which makes the analysis relatively simple. Our algorithm is based upon Furers algorithm, but uses fast Fourier transform over multivariate polynomials along with an estimate of the least prime in an arithmetic progression to achieve this improvement in the modular setting. It can also be viewed as a