(a) Groups wrote implementations of Paillier in C, C++, Java, and Python. The shortest implementation was in Python; the second shortest was in C++ using the NTL large number library. (b) Groups were awarded 5 points for a correct decryption and 5 points for a correct encryption. Many students asked how the number to be decrypted was chosen. The number was simply a randomly-generated 10 digit n...

1. Note in the 2010 version of Professor Anderson’s Lecture 4 Notes, the characterization of the firm in a Robinson Crusoe economy is that it maximizes profit over feasible sets. This is incorrect, and the appropriate correction has been made. The correct description should coincide with our understanding of firms and agents from previous exchange economies and more general Arrow-Debreu economi...

6.1 Harmonic Oscillator Reif §6.1: A simple harmonic one-dimensional oscillator has energy levels given by E n = (n + 1 2)ω, where ω is the characteristic (angular) frequency of the oscillator and where the quantum number n can assume the possible integral values n = 0, 1, 2,. . .. Suppose that such an oscillator is in thermal contact with a heat reservoir at temperature T low enough so that kT...