Montgomery curve
Introduction In mathematics, the '''Montgomery curve''' is a form of elliptic curve introduced by Peter L. Montgomery in 1987, different from the usual Weierstrass form. It is used for certain computations, and in particular in different cryptography applications.
Definition A Montgomery curve of equation A Montgomery curve over a field  is defined by the equation:for certain  and with .Generally this curve is considered over a finite field ''K'' (for example, over a finite field of  elements, ) with characteristic different from 2 and with  and , but they are also considered over the rationals with the same restrictions for  and .
Montgomery arithmetic It is possible to do some "operations" between the points of an elliptic curve:  "adding" two points  consists of finding a third one  such that ; "doubling" a point consists of computing  (For more information about operations see The group law) and below.A point  on the elliptic curve in the Montgomery form  can be represented in Montgomery coordinates , where  are projective coordinates and  for .Notice that this kind of representation for a point loses information: indeed, in this case, there is no distinction between the affine points  and  because they are both given by the point . However, with this representation it is possible to obtain multiples of points, that is, given , to compute .Now, considering the two points  and : their sum is given by the point  whose coordinates are:: : If , then the operation becomes a "doubling"; the coordinates of  are given by the following equations:: : : The first operation considered above (addition) has a time-cost of 3'''M'''+2'''S''', where '''M''' denotes the multiplication between two general elements of the field on which the elliptic curve is defined, while '''S''' denotes squaring of a general element of the field.The second operation (doubling) has a time-cost of 2'''M''' + 2'''S''' + 1'''D''', where '''D''' denotes the multiplication of a general element by a constant; notice that the constant is , so  can be chosen in order to have a small '''D'''.
Algorithm and example The following algorithm represents a doubling of a point  on an elliptic curve in the Montgomery form.It is assumed that . The cost of this implementation is 1M + 2S + 1*A + 3add + 1*4.Here M denotes the multiplications required, S indicates the squarings, and a refers to the multiplication by A.: : : ===Example===Let  be a point on the curve .In coordinates , with , .Then:: : : The result is the point  such that .
Addition Given two points ,  on the Montgomery curve  in affine coordinates, the point  represents, geometrically the third point of intersection between  and the line passing through  and . It is possible to find the coordinates  of , in the following way:1) consider a generic line  in the affine plane and let it pass through  and  (impose the condition), in this way, one obtains  and ;2) intersect the line with the curve , substituting the  variable in the curve equation with ; the following  equation of third degree is obtained:: As it has been observed before, this equation has three solutions that correspond to the  coordinates of ,  and . In particular this equation can be re-written as:: 3) Comparing the coefficients of the two identical equations given above, in particular the coefficients of the terms of second degree, one gets:: .So,  can be written in terms of , , , , as:: 4) To find the  coordinate of the point  it is sufficient to substitute the value  in the line . Notice that this will not give the point  directly. Indeed, with this method one find the coordinates of the point  such that , but if one needs the resulting point of the sum between  and , then it is necessary to observe that:  if and only if . So, given the point , it is necessary to find , but this can be done easily by changing the sign to the  coordinate of . In other words, it will be necessary to change the sign of the  coordinate obtained by substituting the value  in the equation of the line.Resuming, the coordinates of the point ,  are:: :
Doubling Given a point  on the Montgomery curve , the point  represents geometrically the third point of intersection between the curve and the line tangent to ; so, to find the coordinates of the point  it is sufficient to follow the same method given in the addition formula; however, in this case, the line ''y'' = ''lx'' + ''m'' has to be tangent to the curve at , so, if  with: then the value of ''l'', which represents the slope of the line, is given by:: by the implicit function theorem.So  and the coordinates of the point ,  are::
Equivalence with twisted Edwards curves Let  be a field with characteristic different from 2.Let  be an elliptic curve in the Montgomery form:: with , and let  be an elliptic curve in the twisted Edwards form:: with The following theorem shows the birational equivalence between Montgomery curves and twisted Edwards curve:'''Theorem''' (i) Every twisted Edwards curve is birationally equivalent to a Montgomery curve over .In particular, the twisted Edwards curve  is birationally equivalent to the Montgomery curve  where , and .The map:: : is a birational equivalence from  to , with inverse:: : : Notice that this equivalence between the two curves is not valid everywhere: indeed the map  is not defined at the points  or  of the .
Equivalence with Weierstrass curves Any elliptic curve can be written in Weierstrass form. In particular, the elliptic curve in the Montgomery form: :  can be transformed in the following way:divide each term of the equation for  by , and substitute the variables ''x'' and ''y'', with  and  respectively, to get the equation: To obtain a short Weierstrass form from here, it is sufficient to replace ''u'' with the variable :: finally, this gives the equation:: Hence the mapping is given as: : : In contrast, an elliptic curve over base field  in Weierstrass form: : can be converted to Montgomery form if and only if  has order divisible by four and satisfies the following conditions:#  has at least one root ; and#  is a quadratic residue in .When these conditions are satisfied, then for  we have the mapping: : : .
See also * Curve25519* Table of costs of operations in elliptic curves – information about the running-time required in a specific case
Notes
References * * *
External links * Genus-1 curves over large-characteristic fields