Package montgomery-thm: Properties of Montgomery multiplication

Information

Files

• Package tarball montgomery-thm-1.1.tgz
• Theory file montgomery-thm.thy (included in the package tarball)

Theorems

⊦ ∀n r k a.
    ¬(n = 0) ∧ ¬(r = 0) ∧ a ≤ r * r ⇒ Montgomery.reduce n r k a < r + n

⊦ ∀n r k a.
    ¬(n = 0) ∧ ¬(r = 0) ∧ a ≤ n * r ⇒ Montgomery.reduce n r k a < 2 * n

⊦ ∀n r s k a.
    ¬(n = 0) ∧ r * s = k * n + 1 ⇒
    Montgomery.reduce n r k a mod n = a * s mod n

Input Type Operators

• →
• bool
• Number
  • Natural
    • natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ¬
    • ⊥
    • ⊤
• Number
  • Natural
    • *
    • +
    • <
    • ≤
    • bit0
    • bit1
    • div
    • mod
    • suc
    • zero
    • Montgomery
      • Montgomery.reduce

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ⊥ ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 * n = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀m. m * 1 = m

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀m. suc m = m + 1

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀m n. m * n = n * m

⊦ ∀m n. m + n = n + m

⊦ ∀n. 2 * n = n + n

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀m n. ¬(n = 0) ⇒ m mod n < n

⊦ ∀m n. ¬(m = 0) ⇒ m * n mod m = 0

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀a b n. b < a * n ⇒ b div a < n

⊦ ∀m n p. m + n < m + p ⇔ n < p

⊦ ∀m n p. m + p ≤ n + p ⇔ m ≤ n

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀n m. ¬(n = 0) ⇒ m mod n mod n = m mod n

⊦ ∀m n p. m * (n + p) = m * n + m * p

⊦ ∀m n. ¬(n = 0) ⇒ (m div n) * n + m mod n = m

⊦ ∀m n p. m * p < n * p ⇔ m < n ∧ ¬(p = 0)

⊦ ∀n r k a. Montgomery.reduce n r k a = (a + (a * k mod r) * n) div r

⊦ ∀n m p. ¬(n = 0) ⇒ (m mod n) * (p mod n) mod n = m * p mod n

⊦ ∀n a b. ¬(n = 0) ⇒ (a mod n + b mod n) mod n = (a + b) mod n

OpenTheory package summary generated by opentheory 1.2 (release 20121108)