On the communication complexity of XOR functions

An XOR function is a function of the form g(x,y) = f(x + y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g's quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g's quantum complexity is Theta(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness.Comment: 18 pages; v2: minor correction

On the parity complexity measures of Boolean functions

The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to the non-deterministic complexity of the function or its complement. We also show that they are polynomially related to an analogue of the block sensitivity. We further study parity decision trees in their relations with an intermediate variant of the decision trees, as well as with communication complexity.Comment: submitted to TCS on 16-MAR-200

Spectral Norm of Symmetric Functions

The spectral norm of a Boolean function $f:\{0,1\}^n \to \{-1,1\}$ is the sum of the absolute values of its Fourier coefficients. This quantity provides useful upper and lower bounds on the complexity of a function in areas such as learning theory, circuit complexity, and communication complexity. In this paper, we give a combinatorial characterization for the spectral norm of symmetric functions. We show that the logarithm of the spectral norm is of the same order of magnitude as $r(f)\log(n/r(f))$ where $r(f) = \max\{r_0,r_1\}$, and $r_0$ and $r_1$ are the smallest integers less than $n/2$ such that $f(x)$ or $f(x) \cdot parity(x)$ is constant for all $x$ with $\sum x_i \in [r_0, n-r_1]$. We mention some applications to the decision tree and communication complexity of symmetric functions

Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

Privacy-Aware Processing of Biometric Templates by Means of Secure Two-Party Computation

The use of biometric data for person identification and access control is gaining more and more popularity. Handling biometric data, however, requires particular care, since biometric data is indissolubly tied to the identity of the owner hence raising important security and privacy issues. This chapter focuses on the latter, presenting an innovative approach that, by relying on tools borrowed from Secure Two Party Computation (STPC) theory, permits to process the biometric data in encrypted form, thus eliminating any risk that private biometric information is leaked during an identification process. The basic concepts behind STPC are reviewed together with the basic cryptographic primitives needed to achieve privacy-aware processing of biometric data in a STPC context. The two main approaches proposed so far, namely homomorphic encryption and garbled circuits, are discussed and the way such techniques can be used to develop a full biometric matching protocol described. Some general guidelines to be used in the design of a privacy-aware biometric system are given, so as to allow the reader to choose the most appropriate tools depending on the application at hand