Weierstrass form
E831077
Weierstrass form is a standardized algebraic representation of elliptic curves that simplifies their analysis and implementation in areas such as cryptography and number theory.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic curve representation
ⓘ
elliptic curve model ⓘ mathematical concept ⓘ |
| allows | definition of group law on elliptic curve ⓘ |
| belongsTo | classical analysis tradition ⓘ |
| contrastedWith |
Edwards form
NERFINISHED
ⓘ
Hessian form ⓘ Montgomery form NERFINISHED ⓘ |
| definedOver |
complex numbers
ⓘ
field ⓘ finite field ⓘ number field ⓘ |
| enables | efficient arithmetic formulas on elliptic curves ⓘ |
| ensures | curve is nonsingular ⓘ |
| equivalentUpToIsomorphismTo | any elliptic curve over a field of characteristic not 2 or 3 ⓘ |
| hasAffineChart | equation in variables x and y ⓘ |
| hasDomain | projective plane ⓘ |
| hasGeneralEquation | y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 ⓘ |
| hasParameter |
a_1
ⓘ
a_2 ⓘ a_3 ⓘ a_4 ⓘ a_6 ⓘ |
| hasProperty | birationally equivalent to other elliptic curve models ⓘ |
| hasShortEquation | y^2 = x^3 + ax + b ⓘ |
| hasVariant |
general Weierstrass form
NERFINISHED
ⓘ
long Weierstrass form ⓘ short Weierstrass form NERFINISHED ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| namedAfter | Karl Weierstrass NERFINISHED ⓘ |
| relatedTo |
elliptic curve discriminant
ⓘ
j-invariant ⓘ |
| requiresCondition | discriminant nonzero ⓘ |
| specialCaseOf | plane cubic curve ⓘ |
| usedFor |
ECDH
NERFINISHED
ⓘ
ECDSA NERFINISHED ⓘ classification of elliptic curves up to isomorphism ⓘ computing invariants of elliptic curves ⓘ elliptic curve cryptographic protocols ⓘ point addition on elliptic curves ⓘ scalar multiplication on elliptic curves ⓘ |
| usedIn |
algebraic geometry
ⓘ
computational number theory ⓘ elliptic curve cryptography NERFINISHED ⓘ elliptic curve theory ⓘ number theory ⓘ public-key cryptography ⓘ |
| usedInStandard |
ANSI X9.62 elliptic curve standards
NERFINISHED
ⓘ
FIPS 186 elliptic curve specifications NERFINISHED ⓘ SEC 2 recommended elliptic curves ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.