Let $n$ be the number of points in the curve group and $p$ the size of the field. The quoted estimate for Pollard $\rho$ is $\sqrt {\pi n/2}$ which is the number of elliptic curve group operations required. To sign a ECDSA message (assuming that you already have your key pair) is an elliptic curve scalar point multiplication which takes (using windowed double-and-add) $(1+\epsilon)\log n$ elliptic curve group operations. To verify is 2 elliptic curve scalar multiplications which is $(2+\epsilon)\log n$ elliptic curve group operations. If you have to generate a key pair, that adds another elliptic curve scalar multiplication.

The elliptic curve group operation will get harder with $p$ as well and the exact complexity is subject to a range of variant algorithms. An estimate of $O(\log^3p)$ for each elliptic curve group would not be hugely inaccurate though.

Let $n$ be the number of points in the curve group and $p$ the size of the field. The quoted estimate for Pollard $\rho$ is $\sqrt {\pi n/2}$ which is the number of elliptic curve group operations required. To sign a ECDSA message (assuming that you already have your key pair) is an elliptic curve scalar point multiplication which takes (using windowed double-and-add) $(1+\epsilon)\log n$ elliptic curve group operations. To verify is 2 elliptic curve scalar multiplications which is $(2+\epsilon)\log n$ elliptic curve group operations. If you have to generate a key pair, that adds another elliptic curve scalar multiplication.

The elliptic curve group operation will get harder with $p$ as well and the exact complexity is subject to a range of variant algorithms. An estimate of $O(\log^2p)$ for each elliptic curve group would not be hugely inaccurate though.

The quoted estimate for Pollard $\rho$ is $\sqrt {\pi p/2}$ which is the number of elliptic curve group operations required. To sign a ECDSA message (assuming that you already have your key pair) is an elliptic curve scalar point multiplication which takes (using windowed double-and-add) $(1+\epsilon)\log p$ elliptic curve group operations. To verify is 2 elliptic curve scalar multiplications which is $(2+\epsilon)\log p$ elliptic curve group operations. If you have to generate a key pair, that adds another elliptic curve group operation.

The elliptic curve group operation will get harder with $p$ as well and the exact complexity is subject to a range of variant algorithms. An estimate of $O(\log^3p)$ for each elliptic curve group would not be hugely inaccurate though.

Let $n$ be the number of points in the curve group and $p$ the size of the field. The quoted estimate for Pollard $\rho$ is $\sqrt {\pi n/2}$ which is the number of elliptic curve group operations required. To sign a ECDSA message (assuming that you already have your key pair) is an elliptic curve scalar point multiplication which takes (using windowed double-and-add) $(1+\epsilon)\log n$ elliptic curve group operations. To verify is 2 elliptic curve scalar multiplications which is $(2+\epsilon)\log n$ elliptic curve group operations. If you have to generate a key pair, that adds another elliptic curve scalar multiplication.

The elliptic curve group operation will get harder with $p$ as well and the exact complexity is subject to a range of variant algorithms. An estimate of $O(\log^3p)$ for each elliptic curve group would not be hugely inaccurate though.