r Variable Properties

In this file we introduce some generally helpful lemmas for r as introduced in Elligator.Elligator1.Variables.

References

See [bernstein2013a], Section 3.2, Theorem 1.

theorem Elligator.Elligator1.r_ne_zero {F : Type u_1} [Field F] [Fintype F] {s : F} {q : ℕ} (hs_ne_zero : s ≠ 0) (hq_card : Fintype.card F = q) (hq_mod : q % 4 = 3) : r s ≠ 0

theorem Elligator.Elligator1.four_add_r_ne_zero {F : Type u_1} [Field F] [Fintype F] {s : F} {q : ℕ} (hs_ne_zero : s ≠ 0) (sq_ne_pm_two : (s ^ 2 - 2) * (s ^ 2 + 2) ≠ 0) (hq_card : Fintype.card F = q) (hq_mod : q % 4 = 3) : 4 + r s ≠ 0

theorem Elligator.Elligator1.r_sq_sub_two_eq_c_sq_add_inv_c_sq {F : Type u_1} [Field F] [Fintype F] {s : F} {q : ℕ} (hs_ne_zero : s ≠ 0) (hq_card : Fintype.card F = q) (hq_mod : q % 4 = 3) : have r := r s; have c := c s; r ^ 2 - 2 = c ^ 2 + 1 / c ^ 2

theorem Elligator.Elligator1.r_sub_two_ne_zero {F : Type u_1} [Field F] [Fintype F] {s : F} {q : ℕ} (hs_ne_zero : s ≠ 0) (sq_ne_pm_two : (s ^ 2 - 2) * (s ^ 2 + 2) ≠ 0) (hq_card : Fintype.card F = q) (hq_mod : q % 4 = 3) : r s - 2 ≠ 0