# ECDSA multiplication vs exponentiation

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?

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

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, $$P+P+\ldots+P = kP$$)
• 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). – fgrieu Dec 18 '19 at 17:08