How to transform a point (x,y) from y^2=x^3+7 to y^2=x^3+2 using sextic operations

Josh666

7/13/24, 8:26 AM

Point on curve y^2=x^3+2 (to be found/result): Q2 = E2(68702062392910446859944685018576437177285905222869560568664822150761686878291, 78930926874118321017229422673239275133078679240453338682049329315217408793256)

Known parameters are:

p = 115792089237316195423570985008687907853269984665640564039457584007908834671663

E2 = EllipticCurve(GF(p), [0,2])

Point on curve y^2=x^3+7 (to be transformed): (53861016066807093961356075254651239904322369578872440483060364758223655134379, 66206189303543936050020216518583700194174653320251368756673339628741140159784)

Base Point of E2: P2 = E2(34450129095809207277443089178970023159365999968937291419691966854030888759742, 103113457269188258644933175729489183329932073011449500633910298163941611786454)

Order: ord2 = 3319

Just need a sage code that can transform a point (x,y) from y^2=x^3+7 to y^2=x^3+2 using sextic operations. Moreover, its known that this process can be done between isomorphic curves only and in this case isomorphism is over the extension field. Hope someone's gonna help with that.

0 + 2

elliptic-curves

finite-field

diffie-hellman

pairings

honzaik

7/14/24, 9:33 AM

I do not understand your question completely. You have two points on $E2$, the points are related: first point is $[229]P2$. The one point on $E1$ maps to a point on $E2$ which has coordinates in GF($p^2$) and not in GF($p$). The explicit isomorphism (defined over GF($p^2$)) can be computed using https://doc.sagemath.org/html/en/reference/arithmetic_curves/sage/schemes/elliptic_curves/ell_generic.html#sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic.isomorphisms The curves are not isomorphic over the base field.

Josh666

7/14/24, 11:35 AM

The curves E7 and E2 are isomorphic over extension field Fp6. The base point P2 is the base/generator point of order 3319 which is subgroup of curve E7 prime field Fp. Just need to know how to map the point on curve E7(mentioned above) to E2 to get Q2 as result. As far as I know φ:E7(x,y)⟶E2(μ^2x,μ^3y) towering is used.

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EN: How to transform a point (x,y) from y^2=x^3+7 to y^2=x^3+2 using sextic operations

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