On Ajtai’s Lower Bound Technique for R-way Branching Programs and the Hamming Distance Problem

In this report we study the proof employed by Miklos Ajtai[Determinism versus Non-Determinism for Linear Time RAMswith Memory Restrictions, 31st Symposium on Theory of Computation (STOC), 1999] when proving a non-trivial lower boundin a general model of computation for the Hamming Distanceproblem: given n elements: decide whether any two of them have"small" Hamming distance. Specifically, Ajtai was able to showthat any R-way branching program deciding this problem usingtime O(n) must use space Omega(n lg n).We generalize Ajtai's original proof allowing us to prove atime-space trade-off for deciding the Hamming Distance problem in the R-way branching program model for time between nand alpha n lg n / lg lg n, for some suitable 0 < alpha < 1. In particular we provethat if space is O(n^(1−epsilon)), then time is Omega(n lg n / lg lg n)

Secure Computing, Economy, and Trust: A Generic Solution for Secure Auctions with Real-World Applications

In this paper we consider the problem of constructing secure auctions based on techniques from modern cryptography. We combine knowledge from economics, cryptography and security engineering and develop and implement secure auctions for practical real-world problems. In essence this paper is an overview of the research project SCET--Secure Computing, Economy, and Trust-- which attempts to build auctions for real applications using secure multiparty computation. The main contributions of this project are: A generic setup for secure evaluation of integer arithmetic including comparisons; general double auctions expressed by such operations; a real world double auction tailored to the complexity and performance of the basic primitives '+' and

Secure Key Management in the Cloud

We consider applications involving a number of servers in the cloud that go through a sequence of online periods where the servers communicate, separated by offline periods where the servers are idle. During the offline periods, we assume that the servers need to securely store sensitive information such as cryptographic keys. Applications like this include many cases where secure multiparty computation is outsourced to the cloud, and in particular a number of online auctions and benchmark computations with confidential inputs. We consider fully autonomous servers that switch between online and offline periods without communicating with anyone from outside the cloud, and semi-autonomous servers that need a limited kind of assistance from outside the cloud when doing the transition. We study the levels of security one can - and cannot - obtain in this model, propose light-weight protocols achieving maximal security, and report on their practical performance

Fast Threshold ECDSA with Honest Majority

ECDSA is a widely adopted digital signature standard. A number of threshold protocols for ECDSA have been developed that let a set of parties jointly generate the secret signing key and compute signatures, without ever revealing the signing key. Threshold protocols for ECDSA have seen recent interest, in particular due to the need for additional security in cryptocurrency wallets where leakage of the signing key is equivalent to an immediate loss of money. We propose a threshold ECDSA protocol secure against an active adversary in the honest majority model with abort. Our protocol is efficient in terms of both computation and bandwidth usage, and it allows the parties to pre-process parts of the signature, such that once the message to sign becomes known, they can compute a secret sharing of the signature very efficiently, using only local operations. We also show how to obtain fairness in the online phase at the cost of some additional work in the pre-processing, i.e., such that the protocol either aborts during the pre-processing phase, in which case nothing is revealed, or the signature is guaranteed to be delivered to all honest parties

Optimal Time-Space Trade-Offs for Sorting

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem. Beame ha

Optimal Time-Space Trade-Offs for Non-Comparison-Based Sorting

We study the fundamental problem of sorting n integers of w bits on a unit-cost RAM with word size w, and in particular consider the time-space trade-off (product of time and space in bits) for this problem. For comparison-based algorithms, the time-space complexity is known to be Theta(n^2). A result of Beame shows that the lower bound also holds for non-comparison-based algorithms, but no algorithm has met this for time below the comparison-based Omega(n lg n) lower bound. We show that if sorting within some time bound T~ is possible, then time T = O(T~ + n lg* n) can be achieved with high probability using space S = O(n^2/T + w), which is optimal. Given a deterministic priority queue using amortizedtime t(n) per operation and space n^O(1), we provide a deterministicalgorithm sorting in time T = O(n (t(n) + lg* n)) with S = O(n^2/T+w). Both results require that w = n(lg lg n)^2, and with high probability for T >= n lg lg n.Our results imply that recent lower bounds for deciding element distinctness in o(n lg n) time are nearly tight

Optimal Time-Space Trade-Offs for Sorting

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem. Beame has shown a lower bound of n 2 ) for this product leaving a gap of a logarithmic factor up to the previously best known upper bound of O(n 2 log n) due to Frederickson. Since then, no progress has been made towards tightening this gap. The main contribution of this paper is a comparison based sorting algorithm which closes the gap by meeting the lower bound of Beame. The time-space product O(n 2 ) upper bound holds for the full range of space bounds between log n and n= log n. Hence in this range our algorithm is optimal for comparison based models as well as for the very powerful general models considered by Beame. 1. Introduction 1.1. Motivation and results The complexity of sorting is a classical problem in computer science which has provided a wide scope of both algorithms and lower bound..

A Sense of Security in Pervasive Computing

is the light on when the refrigerator door is closed