Is the generator point in the curve in secp256k1?

Reda Bourial

12/15/23, 7:38 PM

Here is the fixed script

# Generator point coordinates
x=55066263022277343669578718895168534326250603453777594175500187360389116729240
y=32670510020758816978083085130507043184471273380659243275938904335757337482424
# Order
n=115792089237316195423570985008687907853269984665640564039457584007908834671663
# from the curve equation y^2=x^3+7
should_be_zero= (y**2)-(x**3+7)
print("by applying the equation, this should be equal to 0 :",should_be_zero)
# The remainder from the division by n should be 0
print("should_be_zero divided by n leaves:",should_be_zero%n)

the output

by applying the equation, this should be equal to 0 : -166977061698153803977729810299616665720111080589888563362701662779994291659332409807309461070447932090244771419528434792678509158779752908144538176572381887934774683088169260414743338484604182122883788458741320363571878334796108231
should_be_zero divided by n leaves: 0

133

1 + 5

elliptic-curves

modular-arithmetic

secp256k1

poncho

12/15/23, 7:51 PM

You do realize for any valid $x$ coordinate, there are two possible $y$ coordinates (and both $y$ coordinates correspond to valid points). Perhaps your 'curve' function returns the other one...

Reda Bourial

12/15/23, 9:17 PM

thank you for pointing that out but i've tried the other point with no success, i will try to rephrase my question to eliminate that posibility.

fgrieu

12/15/23, 9:34 PM

Hint: the curve's equation $y^2=x^3+7$ is in the [***BASE FIELD***](https://www.secg.org/sec1-v2.pdf#subsubsection.2.1.1). $n$ is the order of the [***ELLIPTIC CURVE GROUP***](https://www.secg.org/sec1-v2.pdf#subsubsection.2.2.1). These are like car and pizza. Parameters for [secp25k1](https://www.secg.org/sec2-v2.pdf#subsubsection.2.4.1).

Reda Bourial

12/15/23, 10:18 PM

could you be more explicit please ? \[moderator note: as stated [there](https://crypto.stackexchange.com/help/on-topic), this group's practice is to only give hint on homework questions; see e.g. this [meta](https://crypto.meta.stackexchange.com/q/1115/555)\].

fgrieu

12/16/23, 5:53 AM

Enough with analogies. The curve's equation $y^2=x^3+7$ is with $x$, $y$, the addition and multiplication operators for a [field](https://en.wikipedia.org/wiki/Field_(mathematics)) noted $\mathbb F_p$ in [this reference](https://www.secg.org/sec1-v2.pdf#subsubsection.2.1.1). Your code is trying to verify it in another field $\mathbb F_n$, where $n$ is the order (number of elements) of the [Elliptic Curve](https://www.secg.org/sec1-v2.pdf#subsubsection.2.2.1) group [secp256k1](https://www.secg.org/sec2-v2.pdf#subsubsection.2.4.1). That's reason enough for the verification to fail.

Score:2

Crypto

Reda Bourial

12/15/23, 11:45 PM

The generator point belongs to the curve when using the right parameters.

+ 1

DannyNiu

12/16/23, 10:43 AM

That is, the modulus for the coordinates is $p$, which is different from the order (number of elements) of the curve $n$. (This detail better be included in the answer.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Is the generator point in the curve in secp256k1?

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