On Taking Square Roots without Quadratic Nonresidues over Finite Fields

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.Comment: 14 page

Nominal predication and focus anchoring

It will be shown that verbs can be missing in predicative sentences by using the data from Chinese. Copula-less sentences in Chinese are subject to 'Generalized Anchoring Principle' (GAP), which requires that every clause be anchored at the interface for LF convergence. To satisfy GAP, clauses may be either tensed or focused. It is shown that copula-less sentences in Chinese are subject to focus anchoring. It will be further argued that whether a verb is needed in predication depends on the syntax of predicate nominals