7 votes

Why is the bilinearity of an elliptic curve pairing shown as multiplicative rather than additive?

For any group, we can write the operation either additively or multiplicatively. If we decide to write it additively, we write the operation as $a + b = c$ If we decide to write it multiplicatively, ...
poncho's user avatar
  • 147k
5 votes

When incrementing a private key by 1, by how much is the public key Incremented?

If you have a secp256k1 keypair and you increment the private key by 1, what's a fast way to compute the new public key? Using notation close to sec1 and the parameters of secp256k1, if the private ...
fgrieu's user avatar
  • 141k
3 votes

Is it possible to generate an elliptic curve (with the hard discrete logarithm problem) by iterating only a finite field, but not its $j$-invariant?

Yes, this is straightforward enough. Write down you elliptic curve with coefficients in $\mathbb Q$. For any $p$ you can then use the same equation with the coefficients interpreted as elements of $\...
Daniel S's user avatar
  • 23.8k
3 votes
Accepted

Base point in Montgomery curve

Any member of a group is a generator of a subgroup, thus any point on a Montgomery curve is a "Generator Point (Base Point)" for some subgroup of the curve. If in a Montgomery curve group we ...
fgrieu's user avatar
  • 141k
3 votes
Accepted

Elliptic Curve Cryptography: Point Multiplication by 3 on secp256k1 Curve

A simple way to derive a point tripling method in Cartesian coordinates for secp256k1 is per $3P=(2P)+P$, and towards this computing $2P$ by point doubling in projective coordinates with Z=1 on input,...
fgrieu's user avatar
  • 141k
2 votes
Accepted

Montgomery Curve Point Multiplication in Projective Coordinates

On the Montgomery Curve $y^2\equiv x^3+10x^2+x\pmod{83}$ with $G$ having $(x,y)=(3,28)$, and using projective coordinates $(X,Y,Z)$ defined by $x\,Z=X$ and $y\,Z=Y$, the question is correctly stating ...
fgrieu's user avatar
  • 141k
2 votes

How to know if an ECC public key is y or -y

Straightening the question's vocabulary: on secp256k1 every point $Q$ has an opposite point (or additive inverse point) noted $-Q$. For the $n-1$ points $Q$, the opposite $Q'=-Q$ of $Q$ has the ...
fgrieu's user avatar
  • 141k
2 votes
Accepted

Is there any reference about the half-trace when m is even in F(2^m)

Good reference for this include section II.2.4 of Elliptic Curves in Cryptography (Blake, Seroussi, Smart) or section 11.2.6 of the Handbook of elliptic and hyperelliptic curve cryptography (Cohen and ...
Daniel S's user avatar
  • 23.8k
1 vote

Why is the bilinearity of an elliptic curve pairing shown as multiplicative rather than additive?

Poncho's answer gets to the point in that in this context (say Weil pairing on elliptic curves) the values of the pairing are in multiplicative group of a field. We could write the right hand side of ...
Jyrki Lahtonen's user avatar

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