Elliptic curves over prime fields in Montgomery form.

Montgomery curves are introduced in (Section 10.3.1 of) Montgomery's ``Speeding the Pollard and Elliptic Curve Methods of Factorization'' and described in perhaps more detail in Bernstein and Lange's ``Montgomery curves and the Montgomery ladder'' and Bernstein, Birkner, Joye, Lange, and Peters's ``Twisted Edwards Curves''.

A Montgomery curve over a prime field of size

where

- Montgomery-mul
- Scalar multiplication in the Montgomery group.
- Montgomery-add
- Group addition on a Montgomery curve.
- Montgomery-point-orderp
- Check if a point on a Montgomery curve has a certain order.
- Montgomery-add-commutativity
- Commutativity of Montgomery addition.
- Montgomery-mul-distributivity-over-scalar-addition
- Distributivity of scalar multiplication over scalar addition.
- Montgomery-add-associativity
- Assumption of associativity of Montgomery addition.
- Birational-montgomery-twisted-edwards
- Birational equivalence between Montgomery curves and twisted Edwards curves.
- Montgomery-curve
- Fixtype of elliptic curve over prime fields in Montgomery form.
- Montgomery-mul-nonneg
- Montgomery-not-point-with-x-minus1-when-a-minus-2-over-b-not-square
- Theorem about no points with abscissa -1 on a Montgomery curve.
- Point-on-montgomery-p
- Check if a point is on a Montgomery curve.
- Montgomery-neg
- Negation of a point of the Montgomery curve group.
- Montgomery-sub
- Subtraction of two points of the Montgomery group.
- Montgomery-add-closure
- Assumption of closure of Montgomery addition.
- Montgomery-only-point-with-y-0-when-aa-minus-4-non-square
- Theorem about the only point with zero ordinate for certain Montgomery curves.
- Montgomery-point-order-leastp
- Montgomery-distinct-x-when-nonzero-mul-in-order-range
- Multiplying a Montgomery point by two distinct and not opposite non-zero scalars in the range of the point order yields points with distinct abscissae.
- Montgomery-add-inverse-uniqueness
- Uniqueness of inverse (i.e. negation).
- Montgomery-distributivity-of-neg-over-add
- Distributivity of negation over addition.
- Montgomery-mul-associativity
- Associativity of scalar multiplication.
- Montgomery-points-with-same-x-have-same-or-neg-y
- Theorem about points on the curve with the same abscissa having the same or opposite ordinates.
- Montgomery-zero
- Neutral point of the Montgomery curve group.
- Montgomery-points-with-same-x-are-same-or-neg-point
- Theorem about points on the curve with the same abscissa being the same or opposite points.
- Montgomery-mul-of-mod-order
- Scalar multiplication modulo order.
- Montgomery-neg-inverse
- Property that negation is left and right inverse for addition.
- Montgomery-zero-identity
- Left and right identity properties of the neutral point.
- Point-on-montgomery-finite-when-not-zero
- A non-zero point on a Montgomery curve is finite.