Editing Elliptic-curve cryptography (section)

=== Projective coordinates ===
A close examination of the addition rules shows that in order to add two points, one needs not only several additions and multiplications in <math>\mathbb{F}_q</math> but also an [[Modular multiplicative inverse|inversion]] operation. The [[Modular multiplicative inverse|inversion]] (for given <math>x \in \mathbb{F}_q</math> find <math>y \in \mathbb{F}_q</math> such that <math>x y = 1</math>) is one to two orders of magnitude slower<ref>{{cite journal|first1=Y. |last1=Hitchcock |first2=E. |last2=Dawson |first3=A. |last3=Clark |first4=P. |last4=Montague |url=http://anziamj.austms.org.au/V44/CTAC2001/Hitc/Hitc.pdf |title=Implementing an efficient elliptic curve cryptosystem over GF(p) on a smart card |year=2002 |journal=ANZIAM Journal |volume=44 |url-status=dead |archive-url=https://web.archive.org/web/20060327202009/http://anziamj.austms.org.au/V44/CTAC2001/Hitc/Hitc.pdf |archive-date=2006-03-27 }}</ref> than multiplication. However, points on a curve can be represented in different coordinate systems which do not require an [[Modular multiplicative inverse|inversion]] operation to add two points. Several such systems were proposed: in the ''projective'' system each point is represented by three coordinates <math>(X,Y,Z)</math> using the following relation: <math>x = \frac{X}{Z}</math>, <math>y = \frac{Y}{Z}</math>; in the ''Jacobian system'' a point is also represented with three coordinates <math>(X,Y,Z)</math>, but a different relation is used: <math>x = \frac{X}{Z^2}</math>, <math>y = \frac{Y}{Z^3}</math>; in the ''López–Dahab system'' the relation is <math>x = \frac{X}{Z}</math>, <math>y = \frac{Y}{Z^2}</math>; in the ''modified Jacobian'' system the same relations are used but four coordinates are stored and used for calculations <math>(X,Y,Z,aZ^4)</math>; and in the ''Chudnovsky Jacobian'' system five coordinates are used <math>(X,Y,Z,Z^2,Z^3)</math>. Note that there may be different naming conventions, for example, [[IEEE P1363]]-2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates.<!--TBD: insert formulas--> An additional speed-up is possible if mixed coordinates are used.<ref>{{Cite book |first1=H. |last1=Cohen |author1-link=Henri Cohen (number theorist)|first2=A. |last2=Miyaji |author2-link=Atsuko Miyaji|first3=T. |last3=Ono |title=Advances in Cryptology — ASIACRYPT'98 |chapter=Efficient Elliptic Curve Exponentiation Using Mixed Coordinates |year=1998 |series=Lecture Notes in Computer Science |volume=1514 |pages=51–65 |doi=10.1007/3-540-49649-1_6 |isbn=978-3-540-65109-3 }}</ref>