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• Montgomery

Montgomery-mul-associativity

Associativity of scalar multiplication.

This involves heterogeneous entities, namely two scalars and a point. Multiplying the point by one scalar and the the other is equivalent to multiplying the scalars first and then the point.

Definitions and Theorems

Theorem: montgomery-mul-of-mul

(defthm montgomery-mul-of-mul
  (implies
       (and (montgomery-add-closure)
            (montgomery-add-associativity)
            (point-on-montgomery-p point curve)
            (integerp scalar)
            (integerp scalar1))
       (equal (montgomery-mul scalar
                              (montgomery-mul scalar1 point curve)
                              curve)
              (montgomery-mul (* scalar scalar1)
                              point curve))))

Theorem: montgomery-mul-of-mul-converse

(defthm montgomery-mul-of-mul-converse
  (implies
       (and (montgomery-add-closure)
            (montgomery-add-associativity)
            (point-on-montgomery-p point curve)
            (integerp scalar)
            (integerp scalar1))
       (equal (montgomery-mul (* scalar scalar1)
                              point curve)
              (montgomery-mul scalar
                              (montgomery-mul scalar1 point curve)
                              curve))))