Coins from 1912

New coins from old, smoothly

Given a (known) function f : [0, 1] → (0, 1), we consider the problem of simulating a coin with probability of heads f(p) by tossing a coin with unknown heads probability p, as well as a fair coin, N times each, where N may be random. The work of Keane and O’Brien (1994) implies that such a simulation scheme with IPp(N < ∞) = 1 exists iff f is continuous. Nacu and Peres (2005) proved that f is ...

Fast Simulation Of New Coins From Old

Let S ⊂ (0, 1). Given a known function f : S → (0, 1), we consider the problem of using independent tosses of a coin with probability of heads p (where p ∈ S is unknown) to simulate a coin with probability of heads f(p). We prove that if S is a closed interval and f is real analytic on S, then f has a fast simulation on S (the number of p-coin tosses needed has exponential tails). Conversely, i...

Moving coins

We consider combinatorial and computational issues that are related to the problem of moving coins from one configuration to another. Coins are defined as non-overlapping discs, and moves are defined as collision free translations, all in the Euclidean plane. We obtain combinatorial bounds on the number of moves that are necessary and/or sufficient to move coins from one configuration to anothe...

Detachable Electronic Coins

We propose a non-divisible electronic cash system that combines the restrictive blind signature technique and a batch signature scheme based upon the binary tree structure. Contemporary electronic cash solutions rely on the divisibility paradigm to achieve significant computational savings, and such techniques are considered to be the most efficient. We demonstrate an alternative approach that ...

Michael Abercrombie (1912-1979).