The security of elliptic curve cryptography is closely related to the computational complexity of the elliptic curve discrete logarithm problem (ECDLP). Today, the best practical attacks against ECDLP are exponential-time generic discrete logarithm algorithms such as Pollard's rho method. A recent line of inquiry in index calculus for ECDLP started by Semaev, Gaudry, and Diem has shown that, under certain heuristic assumptions, such algorithms could lead to subexponential attacks to ECDLP. In this study, we investigate the computational complexity of ECDLP for elliptic curves in various forms — including Hessian, Montgomery, (twisted) Edwards, and Weierstrass representations — using index calculus. Using index calculus, we aim to determine whether there is any significant difference in the computational complexity of ECDLP for elliptic curves in various forms. We provide empirical evidence and insight showing an affirmative answer in this paper.

We have proposed a new Bayesian network model (BNM) framework for single-trial-EEG-based Brain-Computer Interface (BCI). The BNM was constructed in the following. In order to discriminate between left and right hands to be imaged from single-trial EEGs measured during the movement imagery tasks, the BNM has the following three steps: (1) independent component analysis (ICA) for each of the single-trial EEGs; (2) equivalent current dipole source localization (ECDL) for projections of each IC on the scalp surface; (3) BNM construction using the ECDL results. The BNMs were composed of nodes and edges which correspond to the brain sites where ECDs are located, and their connections, respectively. The connections were quantified as node activities by conditional probabilities calculated by probabilistic inference in each trial. The BNM-based BCI is compared with the common spatial pattern (CSP) method. For ten healthy subjects, there was no significant difference between the two methods. Our BNM might reflect each subject's strategy for task execution.