Gregory V. Bard

Nicolas T. Courtois, Gregory V. Bard, Daniel Hulme

One of the most famous conjectures in computer algebra is that matrix multiplication might be feasible in not much more than quadratic time. The best known exponent is 2.376, due to Coppersmith and Winograd. Many attempts to solve this problems in the literature work by solving, fixed-size problems and then apply the solution recursively. This leads to pure combinatorial optimisation problems with fixed size. These problems are unlikely to be solvable in polynomial time. In 1976 Laderman published a method to multiply two 3x3 matrices using only 23 multiplications. This result is non-commutative, and therefore can be applied recursively to smaller sub-matrices. In 35 years nobody was able to do better and it remains an open problem if this can be done with 22 multiplications. We proceed by solving the so called Brent equations [7]. We have implemented a method to converting this very hard problem to a SAT problem, and we have attempted to solve it, with our portfolio of some 500 SAT solvers. With this new method we were able to produce new solutions to the Laderman's problem. We present a new fully general non-commutative solution with 23 multiplications and show that this solution is new and is NOT an equivalent variant of the Laderman's original solution. This result demonstrates that the space of solutions to Laderman's problem is larger than expected, and therefore it becomes now more plausible that a solution with 22 multiplications exists. If it exists, we might be able to find it soon just by running our algorithms longer, or due to further improvements in the SAT solver algorithms.

Alan T. Sherman, Linda Oliva, Enis Golaszewski et al.

For two days in February 2018, 17 cybersecurity educators and professionals from government and industry met in a "hackathon" to refine existing draft multiple-choice test items, and to create new ones, for a Cybersecurity Concept Inventory (CCI) and Cybersecurity Curriculum Assessment (CCA) being developed as part of the Cybersecurity Assessment Tools (CATS) Project. We report on the results of the CATS Hackathon, discussing the methods we used to develop test items, highlighting the evolution of a sample test item through this process, and offering suggestions to others who may wish to organize similar hackathons. Each test item embodies a scenario, question stem, and five answer choices. During the Hackathon, participants organized into teams to (1) Generate new scenarios and question stems, (2) Extend CCI items into CCA items, and generate new answer choices for new scenarios and stems, and (3) Review and refine draft CCA test items. The CATS Project provides rigorous evidence-based instruments for assessing and evaluating educational practices; these instruments can help identify pedagogies and content that are effective in teaching cybersecurity. The CCI measures how well students understand basic concepts in cybersecurity---especially adversarial thinking---after a first course in the field. The CCA measures how well students understand core concepts after completing a full cybersecurity curriculum.