I am a bit confuse about the term, variable-base point/scalar multiplication, in Elliptic Curve Cryptography. What I have understood so far. It means that the base or point on EC is variable/unknown.

Am I right?


2 Answers 2


Yes, you are correct. There are various methods for scalar multiplication on elliptic curves.

Some of them are optimised for fixed base-point scalar multiplication, i.e., where you a-priori know that you will mostly/exclusively perform scalar multiplications with respect to a fixed base point on the curve. Thus, one can make (extensive) pre-computations (e.g., as done with the the fixed-base Comb multiplier). Think for instance of a scenario where you extensively issue ECDSA signatures, then you always do your scalar multiplication using to the same fixed base point.

Other methods are not optimised to some a-priori fixed point and thus are useful for variable base point scalar multiplication, which often arises in other cryptographic protocols where you do not know the point you will be using a-priori.

Clearly, there are many criteria to decide which multiplier you will use such as speed, side-channel resistance or memory footprint.


If $G$ is the distinguished point on your curve, you can perform pre-computations to speed up multiplication of $G$ by a scalar. For instance, if you need a transient key pair $\{K_{PRI},K_{PUB}\}$, you can use these pre-computed points to compute $K_{PUB} = [K_{PRI}]G$ quickly.

But if you need to compute a shared secret using another public key $K'_{PUB}$, you must compute $S = [K_{PRI}]K'_{PUB}$, and this will be much slower -- typically 10-20 times slower in a limited environment like a smartcard.


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