Differential addition chains (also known as strong addition chains, Lucas chains, and Chebyshev chains) are addition chains in which every sum is already accompanied by a difference. Low-cost differential addition chains are used to efficiently exponentiate in groups where the operation a, b, a/b 7→ ab is fast: in particular, to perform x-coordinate scalar multiplication P 7→ mP on an elliptic ...

The quantum integer [n]q is the polynomial 1+q+q+ · · ·+q. Two sequences of polynomials U = {un(q)}∞n=1 and V = {vn(q)} ∞ n=1 define a linear addition rule ⊕ on a sequence F = {fn(q)}∞n=1 by fm(q) ⊕ fn(q) = un(q)fm(q)+vm(q)fn(q). This is called a quantum addition rule if [m]q⊕[n]q = [m + n]q for all positive integers m and n. In this paper all linear quantum addition rules are determined, and a...

We introduce Probabilistic Matrix Addition (PMA) for modeling real-valued data matrices by simultaneously capturing covariance structure among rows and among columns. PMA additively combines two latent matrices drawn from two Gaussian Processes (GPs) respectively over rows and columns. The resulting joint distribution over the observed matrix does not factorize over entries, rows, or columns, a...

There is growing interest in the use of mechanical energy to alter the molecular and supramolecular structure of polymers to create stress-responsive materials.1a-l Chemical reactions that are accelerated by force remain poorly understood, and there is a need for the rapid discovery of new mechanophores (i.e., stress-sensitive units). Screening putative mechanophores, however, is a slow process...

We introduce Probabilistic Matrix Addition (PMA) for modeling real-valued data matrices by simultaneously capturing covariance structure among rows and among columns. PMA additively combines two latent matrices drawn from two Gaussian Processes respectively over rows and columns. The resulting joint distribution over the observed matrix does not factorize over entries, rows, or columns, and can...

We introduce an efficient method for performing computation on encrypted data, allowing addition of ElGamal encrypted plaintexts. We demonstrate a solution that is robust and leaks no information to a minority of colluding cheaters. Our focus is on a three-player solution, but we also consider generalization to a larger number of players. The amount of work is exponential in the number of playe...