A multivariate quadratic public-key cryptography (MQ-PKC) is one of the most promising alternatives for classical PKC after the eventual coming of a quantum computer. We propose a new MQ-signature scheme, ELSA, based on a hidden layer of quadratic equations which is an important role in dramatically reducing the secret key size and computational complexity in signing. We prove existential unforgeability of our scheme against an adaptive chosen-message attack under the hardness of the MQ-problem induced by a public key of ELSA with a specific parameter set in the random oracle model. We analyze the security of ELSA against known attacks and derive a concrete parameter based on the security analysis. Performance of ELSA on a recent Intel processor is the fastest among state-of-the-art signature schemes including classical ones and Post-Quantum ones. It takes 6.3   μ s and 13.39   μ s for signing and verification, respectively. Compared to Rainbow, the secret size of the new scheme has reduced by a factor of 88% maintaining the same public key size.

Identifying how spatially distributed information becomes integrated in the brain is essential to understanding higher cognitive functions. Previous computational and empirical studies suggest a significant influence of brain network structure on brain network function. However, there have been few analytical approaches to explain the role of network structure in shaping regional activities and directionality patterns. In this study, analytical methods are applied to a coupled oscillator model implemented in inhomogeneous networks. We first derive a mathematical principle that explains the emergence of directionality from the underlying brain network structure. We then apply the analytical methods to the anatomical brain networks of human, macaque, and mouse, successfully predicting simulation and empirical electroencephalographic data. The results demonstrate that the global directionality patterns in resting state brain networks can be predicted solely by their unique network structures. This study forms a foundation for a more comprehensive understanding of how neural information is directed and integrated in complex brain networks.