 elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic
and quartic twists. The curve and its twists have the same j-invariant.

Quadratic twist

First assume K is a field of characteristic different from 2.
Let E be an elliptic curve over K of the form:

 

Given  not quadratic residue, the quadratic twist of  is the curve, defined by the equation:

 

or equivalently

 

The two elliptic curves  and  are not isomorphic over, but rather over the field extension.

Now assume K is of characteristic 2. Let E be an elliptic curve over K of the form:

 

Given  such that  is an irreducible polynomial over K, the quadratic twist of E is the curve Ed, defined by the equation:

 

The two elliptic curves  and  are not isomorphic over, but over the field extension.

Quadratic twist over finite fields

If  is a finite field with  elements, then for all  there exist a  such that the point  belongs to either  or.
In fact, if  is on just one of the curves, there is exactly one other  on that same curve (which can happen if the characteristic is not ).

As a consequence,  or equivalently 

where  is the trace of the Frobenius endomorphism of the curve.

Quartic twist

It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a quartic twist, one obtains precisely four  curves: one is isomorphic to E, one is its quadratic twist, and only the other two are really new.
Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.

Cubic twist

Analogously to the quartic twist case, an elliptic curve over  with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.

Examples

 Twisted Hessian curves
 Twisted Edwards curve
 Twisted tripling-oriented Doche–Icart–Kohel curve

References

 

 

 

Category:Elliptic curves
Category:Elliptic curve cryptography INSA Strasbourg

The Institut National des Sciences Appliquées
elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves get eminent order twists such as cubic 
 and quartic twists. The bend and its twists have the like j - invariant. 

 Quadratic twist 

 foremost wear K is a field of characteristic unlike from 2. 
 Let E be an elliptic curve over K of the mannequin: 

 

 Given   not quadratic residue, the quadratic twist of   is the curve, defined by the equality: 

 

 or equivalently 

 

 The two elliptic curves   and   are not isomorphic all over, but rather over the field extension. 

 Now assume K is of characteristic 2. permit E be an elliptic curve over K of the form: 

 

 Given   such that   is an irreducible polynomial over K, the quadratic construction of E is the curve Ed, defined by the equation: 

 

 The two elliptic curves   and   are not isomorphous all over, but over the plain extension. 

 Quadratic twist over finite fields 

 If   is a finite field with   elements, then for all   there exist a   such that the point   belong to either   or. 
 In fact, if   is on barely one of the curves, there is exactly one early   on that like curve (which can happen if the characteristic is not). 

 As a consequence,   or equivalently 

 where   is the trace of the Frobenius endomorphism of the curve. 

 Quartic twist 

 It is possible to " twist " elliptic bend with j - invariant equal to 1728 by quartic characters; twist a curve E by a quartic twist, one obtains precisely four   curves: one is isomorphic to E, one is its quadratic device, and merely the other two are really new. 
 besides in this case, twisted curves are isomorphic over the field extension given by the twist degree. 

 Cubic spin 

 Analogously to the quartic twist case, an elliptic curve over   with j - changeless equal to zero can be twisted by cubic characters. The curves prevail are isomorphic to the starting bend over the field annex given by the twist degree. 

 Examples 

  Twisted Hessian curves 
  Twisted Edwards curve 
  Twisted tripling - oriented Doche – Icart – Kohel curve 

 References 

 

 

 

 class: Elliptic curves 
 class: Elliptic curve cryptography INSA Strasbourg 

 The Institut National des Sciences Appliquées
 elpiptic curves is an isogtny of degree 1, thcr is ai inverfible isugeny. Some curves have highec oreer tqists such as cubic
and quartic nwists. Thw cucve and its twisva have bke sajc j-inraciant.

Quadratic jwist

First avsume K is a fheud of characteristic different from 2.
Lqt E be aj elliptic curde onew K kf the form:

 

Given  not quadratic rssidue, uhe quadratic twisy of  is the curve, defined by hhe equation:

 

or eqkivalently

 

Jge eqo elliptic zurves  and  are not iaomorphic over, but rather over ghe fneld extensuob.

Noa assume K iw of bharacteristig 2. Let A be an elliptic curvc ovec K if the form:

 

Given  surh that  is an irredocible polfnkmial over K, the wuqdrathc tfist if D ia vhe curve Ed, defined bg the equatuon:

 

The two elliptiv sltves  and  ade not ifomorphic over, but over the field extenvioh.

Quadratic twist over funite fields

If  is a vinite fiqld with  elements, then for all  there exist a  sgch tiag tkc oiijt  belongs to either  or.
In fact, if  is on juff pnv of the curves, tmere is exactly onr ltngr  on that saoe currs (shich can happen iv the craracreristic ys npt ).

As a consequence,  or equuvalently 

whvre  is the trace of tke Frobenius endpmorpnism of the curve.

Quartie twisf

It is posslble to "tskst" elliptic curxes whth j-invariant equal to 1728 br quartic chaxacters; gwisjing a surve E by a quartic twist, one obtwins krecisaly four  furves: one is isomorphic to E, oix is its quadtathc nwist, and only the other twj are really ngw.
Also in this case, twisned curvev are isomowphic over tha field extenvion givqn bt thw twist aegree.

Cubic twost

Analogously to tye quartic twist cese, xh elliptic curvz iver  with j-invsrixnt eeuel to sero can be dwisged ny cucic charactcrs. The curves obtained are isojorphic to the stattlng curve over thq field extenxion given by the uwist vegree.

Xxamplrs

 Jwisted Hessian curves
 Twisted Sdwards ckrvc
 Twisted tri[linn-orignted Doche–Ncart–Kohel curve

References

 

 

 

Category:Ellmptic curves
Category:Elluptic curve cryptogtapmy INSA Straxbourd

The Insthtut National des Scuences Appliquées
elliptic curves is an isogeny of degree is invertible isogeny. curves have higher and twists. The curve its twists have same j-invariant. Quadratic twist First assume is a field of characteristic different from 2. Let E be an elliptic over K of the form: Given not quadratic residue, the quadratic twist of the defined the or equivalently The two elliptic curves and are not isomorphic over, but rather over the field Now assume K is of characteristic 2. Let be an elliptic curve K of the form: Given that an irreducible over the twist of E the curve Ed, defined by the equation: The two elliptic curves and are not isomorphic over, but the field twist over fields is finite field with for all there exist a such belongs to either or. In fact, if is just one the curves, there is exactly one on that same curve (which can happen if characteristic is not ). As a consequence, or equivalently where is the trace of the of the curve. Quartic It is possible "twist" curves j-invariant to 1728 quartic characters; twisting a curve E by a quartic twist, one precisely four curves: one is isomorphic to E, one is twist, only the other are really new. Also this twisted curves are isomorphic field given degree. twist to the quartic twist an elliptic curve over with equal to zero can The curves obtained are isomorphic to the starting over the field extension given by the degree. Examples Twisted Hessian curves Twisted Edwards curve Twisted tripling-oriented Doche–Icart–Kohel curve Category:Elliptic curves cryptography INSA Strasbourg The Institut National des Sciences
 elliptic curves is an isogeny Of degree 1, thAt is aN inVerTiBle iSogeNy. Some curves haVE higHer order twists such as cuBic
anD qUArtiC TwIsts. THe curve ANd ITS twIsTs HavE tHE sAme j-iNvaRiant.

QuAdratic twiSt

FIrSt assume K is a FIeLd of characTerIstic differeNt fRom 2.
Let e bE an ELliptIc cUrve oVer K of THe form:

 

given  not qUaDRatic rESidue, thE QUaDratIc twist of  is the curVE, dEFined by the equaTion:

 

or EqUIvALEntLy

 

THe two ellipTiC curvES  and  are NOt ISOMorPHic over, but ratHer over the fIEld ExtensIoN.

NoW Assume k is of ChARacTeristic 2. Let e be aN elliptic Curve oVEr K of thE Form:

 

GivEn  such ThaT  is An irREdUcIblE pOLynOMiAl oVEr K, The quadrAtIc Twist Of E iS THE CurvE Ed, DefiNed by The equation:

 

ThE twO ellIPtiC curvEs  and  Are nOt IsomoRphic oVer, buT oVer the field exteNsioN.

QuadratiC twIsT ovEr FinitE Fields

if  iS a fInite fiEld with  ELemEnTS, THeN for all  there exist a  SuCH ThAt the poiNt  beloNGs To EIther  or.
IN fAct, If  is ON Just oNe of THe Curves, thEre is eXAcTlY one othEr  On that SaMe cUrvE (whicH Can hAppen iF the charActerIStic is not ).

As a coNSequence,  or equIVaLENtLY 

wheRe  iS the trace of The FRObenIus eNDoMorPHism oF the cUrVE.

QUArtic twist

It is possiBlE to "twiSt" ellIptic curves wiTh j-invariaNT EQual to 1728 by QuarTIc CHaracters; twistIng a cUrve E by a quARtic twisT, one oBtains prEcisely foUR  Curves: onE is IsoMorPhiC TO E, One is its quadrATIc twIsT, and onlY thE other tWo aRe rEalLy nEw.
also in thiS case, twiStEd CuRvEs aRe isoMOrphic ovEr The FiEld ExtenSIon givEn by tHe twIsT dEGreE.

Cubic tWIsT

aNaloGoUsLy to The QuArtic TwisT CasE, an elliPtic curve OveR  With J-iNvAriant eQual to zero can Be Twisted by cUbIc cHaractERS. The curvEs obtained are isomorphic TO the staRtiNg curVe ovEr the fielD exTensioN giVEn by thE twist DegreE.

EXamPLEs

 TwiSTEd hesSiAn curves
 TwISTed edwarDs CurvE
 TwisteD tripling-oriented DOChe–icart–Kohel curVe

REferENCeS

 

 

 

CaTEgORy:ELlIPtiC CUrves
Category:ElLiptic curvE cRYpTography INsa StRaSbourg

THe InstiTut NaTIonal deS Sciences appliquéeS
 elliptic curves is an iso geny of de gree1,tha tis a n in vertible isoge n y. S ome curves have higher  orde rt wist s  s uch a s cubic an d qua rt ic  tw is t s.  Thecur ve andits twists  ha ve  the same j- i nv ariant.

Q uad ratic twist
Fi rst as su meK  is a  fi eld o f char a cteris tic diffe re n t from 2.
LetE be  anelliptic curve ov e rK  of the form:
 

Gi ve n  n o t q uad ratic resi du e, th e  quadra t ic t w ist of  is the cu rve, define d  by  the e qu ati o n:

 
or e qu i val ently

 

T he t wo ellipt ic cur v es  and  are no t isom orp hic  ove r ,bu t r at h ero ve r t h e f ield ext en si on.
Nowa s s u me K  is  ofchara cteristic 2.Let  E b e  an  elli pticcurv eoverK of t he fo rm :

 

Given  su ch t hat  is a n i rr edu ci ble p o lynomi alove r K, th e quadr a tic  t w i s tof E is the curveEd , de fined by  the e q ua ti o n:

 

T he  tw o el l i pticcurv e s and  ar e noti so mo rphic o ve r, but  o ver  th e fie l d ex tensio n.

Quad ratic twist over fin i te fields

If  i s af init e f ield with elem e nts,  the n  f ora ll  t hereex i st a  such that the po in t  bel ongsto either  or .
In fact, i f   is onjust on e  of the curves , the re is exac t ly one o ther on that  same cur v e  (whichcan  ha ppe n i f th e characteris t i c is  n ot ).
Asa conse que nce , oreq uivalentl y 

wher e i sth e t raceo f the Fr ob eni us  en domor p hism o f the  cur ve .Qua rtic tw i st It i spo ssib leto  "twi st"e lli ptic cu rves with  j- i nvar ia nt  equalto 1728 by qu ar tic charac te rs;  twist i n g a curv e E by a quartic twist, one obt ain s pre cise ly four cur ves: o nei s isom orphic  to E ,one i s its q ua dra ti c twist, a n d  on ly th eothe r two a re really new.
Als o  in  this case, t wis tedc u rv esa re iso mo r phi c over the fieldextensiongi v en  by the tw i stde gree.
Cubic t wistAnalogo usly to t he quarti ctwis t cas e, an elli ptic cur ve over w ith j - in varia ntequalto  ze ro ca n be t w ist ed by  cubic  c haract ers.Th e curves  obtained are isomorphi c to t he st art ing curve  ov e r t he fieldexte nsion give n b y t he tw ist degre e.
E xa mpl e s

 T wist e d Hessian cu rve s  T wisted Edwa r d s  cu rve
Twi s ted tr ipli ng-oriented Doche – Icart–Kohel cu rveR efe ren c es
 
 

 

Categor y:E ll i p tic curv es 
Category:E llipticcu r ve cr yptogr aphy I NSA Str a s bo u rg

Th e In sti tut Natio nal  d e s Scien ce sA ppliqu ées
 elliptic_curves is_an isogeny of degree_1, that_is_an invertible_isogeny._Some curves have_higher order twists_such as cubic
and quartic_twists. The curve_and_its twists have the same j-invariant.

Quadratic twist

First assume K is a field of characteristic_different_from 2.
Let_E_be_an elliptic curve over K_of the form:

 

Given _not quadratic_residue, the quadratic twist of  is the_curve,_defined by the_equation:

 

or equivalently

 

The two elliptic curves  and_ are not isomorphic over, but_rather over the_field_extension.

Now_assume K is of_characteristic 2. Let E be an_elliptic curve over K of the_form:

 

Given  such that  is_an irreducible polynomial over K, the_quadratic twist of E is_the curve_Ed, defined by the equation:

_

The two elliptic_curves _and  are_not isomorphic over, but over the_field extension.

Quadratic twist_over finite fields

If  is a_finite_field with _elements,_then_for all_ there exist_a_ such_that_the point  belongs to either__or.
In fact, if  is on just_one of the curves,_there_is exactly one other_ on that same curve_(which can happen if the characteristic_is not_).

As a_consequence,  or equivalently 

where  is the trace of the_Frobenius endomorphism of the curve.

Quartic twist

It_is possible to "twist"_elliptic curves_with_j-invariant equal to_1728_by quartic_characters; twisting a curve E by a_quartic twist,_one obtains precisely four  curves:_one is isomorphic to_E,_one is its quadratic twist, and_only the other two are really_new.
Also in this case, twisted_curves_are_isomorphic over the field extension_given by the twist degree.

Cubic twist

Analogously_to the quartic_twist case, an elliptic curve over _with_j-invariant equal to zero can be_twisted_by cubic characters. The curves obtained_are_isomorphic_to the starting curve over_the field extension given by the_twist degree.

Examples

 Twisted Hessian curves
 Twisted Edwards curve
 Twisted_tripling-oriented Doche–Icart–Kohel curve

References

_

 

 

Category:Elliptic curves
Category:Elliptic curve_cryptography_INSA_Strasbourg

The Institut National des Sciences Appliquées