Inspired by quantum annealing, digital annealing computers specified for annealing computations have been realized on a large scale, such as the Digital Annealer (DA) developed by Fujitsu and the CMOS Annealing Machine developed by Hitachi. With the progress achieved using these computers, it has become necessary to estimate the computational hardness of cryptographic problems. This paper focuses on lattice problems, such as the closest vector problem (CVP) and shortest vector problem (SVP), which are a class of optimization problems. These problems form the basis of the security of lattice-based cryptography, which is a prime candidate for the NIST post-quantum cryptography standardization. For these lattice problems, we propose methods for generating an Ising model and solving the Ising model on annealing computers with a bit representation as the input, which represents encodings to map each integer variable in the SVP into binary variables. We propose two methods for SVPs, a basic method and a variant incorporating approximately the concept of the classical lattice enumeration. In our experimental results obtained using the second-generation DA, we succeeded in finding a shortest nonzero lattice vector in 40- and 45-dimensional lattices in the Darmstadt SVP Challenge. The basic method with a hybrid bit representation was the fastest among our methods with a bit representation, and the expected running time was estimated as 664 and 13,750 seconds for the 40- and 45-dimensional lattices, respectively. These results provide a benchmark for solving the SVP with annealing computers.

We consider an air conditioning system design that minimizes the costs of electricity and equipment, while satisfying individual usage requirements. The problem of finding an optimal configuration can be formulated as a combinatorial optimization problem. When the cost of electricity is evaluated using simulation software, computational time is excessive. We therefore propose an efficient iterated local search method that uses sparse estimation and extreme statistics to reduce the computational time for evaluating the cost of electricity, and which makes use of the information obtained during a local search. In addition, Thompson's update probability is used for effective neighborhood selection.