There is no substitute for modular multiplication in the question's cryptosystems. Some languages like Python make that easy for educational purposes, only.
In RSA and DSA, and to a lesser degree ECC crypto on secp256k1 or secp256r1, one needs to compute $b^e\bmod m$ for large $e$. The fastest algorithms (e.g. sliding window exponentiation) perform about $\log_2 e$ modular squaring and like $\approx0.2\,\log_2 e$ modular multiplications. However there are other algorithms only marginally more costly (e.g. Montgomery's ladder) that may be better from the standpoint of security against side channels.
Each modular multiplication or squaring modulo $m$, for the above or (in ECC) point addition or multiplication by a scalar, has computational cost growing at most like $(\log m)^2$ when using the algorithms learned in primary school adapted to computer words instead of digits. That can be lowered to $(\log m)^{\approx1.6}$ with Karatsuba or $(\log m)^{\approx1.5}$ with Toom-3, but in ECC the modulus $m$ is not large enough that it pays much ($m$ is a 'only' some hundreds bit in ECC, rather than some thousands in RSA/DSA).
When developing signature or encryption using secp256k1 or secp256r1 from scratch for educational purposes, there typically are phases:
- Getting something that works, by building point addition and doubling in Cartesian coordinates on top of modular multiplication, then point multiplication, then signature or/and encryption.
- Getting it to work much faster by using a better representation of point, e.g. projective coordinates (which allows to deffer the costly modular inversion until the end of point multiplication)
- Getting it to work securely, which is very hard and typically requires rewriting a lot of things from the grounds up.
- Performance optimizations. These can be attempted at any stage, but ponder the saying "premature optimization is the root of all evil".
Some online references on modular multiplication and exponentiation (not ECC):
