We describe a reduction from the problem of unordered search(with a unique solution) to the problem of inverting a permutation. Since there is a straightforward reduction in the reverse direction, the problems are essentially equivalent.

The reduction helps us bypass the Bennett-Bernstein-Brassard-Vazirani hybrid argument (1997} and the Ambainis quantum adversary ... more >>>

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube.

We prove lower bounds on designs in spaces with a large group ... more >>>

A distance estimator is a code together with a randomized algorithm. The algorithm approximates the distance of any word from the code by making a small number of queries to the word. One such example is the Reed-Muller code equipped with an appropriate algorithm. It has polynomial length, polylogarithmic alphabet ... more >>>