How to use Double Compression point with scalar Double-and-Add in Elliptic curve

Cisco Saeed

2/9/24, 9:43 AM

refer to this paper Khabbazian Paper I am trying to use double point compression in scalar for example double-and-add with k=27 which need to reduce the multiplications process for example scalar double-and-add algo consumes 4 doubling and 3 addition for P1(3,10) based on this example example

I am thinking to take two points P1(3,10) P2(7,12) and compress to get P3(x1,x2,y1+y2) P3(3,7,22)which will be represent it in extra bit 0 because 22 is even number.

I don't know how this double compression point can reduce the number of multiplications process for K=27 or it's considered just as compress the point to save more space in memory?

0 + 2

elliptic-curves

Cisco Saeed

2/9/24, 1:26 PM

It based on this paper https://www.researchgate.net/publication/3049465_Double_Point_Compression_with_Applications_to_Speeding_Up_Random_Point_Multiplication

fgrieu

2/9/24, 5:00 PM

The question's P1 and P2 are for curve $y^2\equiv x^3+a\,x+b\pmod p$ with $p=23$, $a=1$, $b=1$ of [that course](https://www.site.uottawa.ca/%7Echouinar/Handout_CSI4138_ECC_2002.pdf). Independently: the question refers to [that paper](https://doi.org/10.1109/TC.2007.47)'s _"Double Point Compression"_ representing two points $(x_1,y_1)$ and $(x_2,y_2)$ as three field elements and a bit, usually $(x_1,x_2,y_1+y_2)$ (the bit distinguishes another exceptional encoding; the question gets that bit wrong). That compression is _not_ the [standard one](https://crypto.stackexchange.com/a/104140/555).

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