Sur les courbes algébriques et les variétés qui s’en déduisent (Introduction)

Binary cyclotomic generators

“…Behind this uniformity of all cyclotomic numbers is the Riemann Hypothesis for Curves over Finite Fields, which can be described as follows. This theorem, proven by Weil [20], not only indicates the uniformity of the form of cyclotomic formulae, but also can be used to set up bounds for pattern distributions in the output sequences of the cyclotomic generators. With the Weft Theorem one can see some cryptographic meanings of the genus of curves.…”

“…At one side of the second bridge are the autocorrelation property, the twodigit pattern distribution property of the output sequences, and the difference property and nonlinearity of the cryptographic functions of the NSG; while at the other side are cyclotomy-related problems and the Riemann Hypothesis for Curves over Finite Fields, which was proven to be true by Weft in 1948 [20].…”

Explicit Constructions

“…Suppose f (x, y) is absolutely irreducible, i.e., irreducible over every algebraic extension of GF (q). The famous theorem of Weil (1948), known as the Riemann hypothesis for curves over finite fields, states that…”

The Diophantine equation 𝑥⁴+2𝑦⁴=𝑧⁴+4𝑤⁴

“…By the work of André Weil [We,Corollaire 3 du Théorème 13], the numbers of F p -rational points on their projectivizations are given by…”