In Elliptic Curve Digital Signature Algorithm (ECDSA) I often see 2 different written equations of it:

  1. Elliptic curve point multiplication by a scalar, $Q_{A}=d_{A}\times G$, source
  2. Modular exponentiation, $y:=g^{x}\mod p$, source

Both methods work other than one is multiplication the other is exponent. Are there any advantage doing one or another?

  • $\begingroup$ Did you read the Wikipages properlty? $\endgroup$ – kelalaka Dec 18 '19 at 14:40

You are referring to two different protocols.

The second source is linked to the DSA (Digital Signature algorithm). This uses modular exponentiation in a group of prime order over the integers.

The first one is a version of the DSA over Elliptic curves, namely ECDSA (Elliptic Curve Digital Signature Algorithm).

They basically work the same. You have a group, with a defined operation, and you work on elements of this group with scalars. The difference between classical DSA and ECDSA is that the first one has multiplication as a group law (hence $g \times g \times \ldots \times g = g^k$ where k is the number of multiplications) whereas the second one has addition law (for a point $P$ of the curve, $P+P+\ldots+P = kP$)

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    $\begingroup$ Yes. Further, some authors note the group used in ECDSA multiplicatively, like $h=g^k$ (without modulo), or $Q=k⋅G$, or $Q=k\,G$, or $Q=kG$ (as this answer does in its last equation), or $Q=[k]G$, where the first source in the question uses $Q=k\times G$. In all cases, that denotes obtaining the result of (virtually) combining $k$ copies of $g$ or $G$ using $k-1$ group operations (sometime with the convention that $k=0$ yields the group's neutral as result, or/and a negative $k$ is first made positive by reducing $k$ modulo the order). $\endgroup$ – fgrieu Dec 18 '19 at 17:08

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