View source for Montgomery normal form

[[Category:Normal forms]]

== The normal form ==

The '''Montgomery normal form'''<ref>Peter L. Montgomery. “Speeding the Pollard and elliptic curve methods of factorization.” ''Mathematics of Computation,'' vol. 48, no. 177, Jan 1987, pp. 243-264. https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866113-7/</ref> of an [[elliptic curve]] is a slightly different normalization from that of Weierstraß.

:<math>By^2=x^3+Ax^2+x</math>

It has in common with the [[Weierstraß normal form]] the same important property of being symmetric about the ''x''-axis, and hence it is just as useful and possibly more convenient or efficient in some cases for computing the [[point group operation]] by using many if not most of the same techniques.

== What is a Montgomery curve? ==

The concept of a ''Montgomery curve''<ref>Craig Costello and Benjamin Smith. “Montgomery curves and their arithmetic: The case of large characteristic fields.” ''Cryptology ePrint Archive,'' vol. 2017, no. 212. https://eprint.iacr.org/2017/212</ref><ref>Daniel J. Bernstein and Tanja Lange. “Montgomery curves and the Montgomery ladder.” ''Cryptology ePrint Archive,'' vol. 2017, no. 293. https://eprint.iacr.org/2017/293</ref> is very curious, if that is at all the intention to imply that some elliptic curves with rational coefficients are Montgomery and others are not. Because any elliptic curve described by a Weierstraß equation with rational coefficients ''a'' and ''b''

:<math>y^2=x^3 + ax + b</math>

may be transformed by simple rational operations into a so-called “Montgomery curve” with rational ''A'', ''B'', and ''C''.

:<math>By^2=(x+C)^3 + A (x+C)^2 + (x+C)</math>

This is a simple ''y''-axis scaling together with an ''x''-axis shift which preserves the exact same geometry and group structure among the rational points as the curve would have in [[Weierstraß normal form]], even in a finite field as the case may be, which is in all probability well understood, notwithstanding a certain presumptuousness or overexuberance in the nuts-and-bolts applications of so-called [[Montgomery arithmetic]] directly to modern-day [[strong cryptography]] together with a certain degree of ignorance of traditional or long-standing classical mathematical literature of the Victorian era.