Brute-forcing a 256-bit Elliptic Curve Digital Signature Algorithm (ECDSA) with Grover's algorithm, instead of using Shor's algorithm, is a topic of interest in the field of quantum computing. However, it's important to note that Grover's algorithm is not specifically designed to break cryptographic schemes like ECDSA. It is a search algorithm that provides a quadratic speedup over classical algorithms for unstructured search problems.
Shor's algorithm, on the other hand, is a quantum algorithm specifically designed to factor large numbers efficiently, which can be applied to break the underlying mathematical problem that ECDSA relies on, namely integer factorization and the discrete logarithm problem. If Shor's algorithm were to be applied to ECDSA, it would render the cryptographic scheme insecure by efficiently finding the private key from the public key.
In contrast, Grover's algorithm can be used to speed up the search for a preimage or collision in a hash function, but it does not break the underlying cryptographic problem of ECDSA. ECDSA relies on the difficulty of the discrete logarithm problem, which is not directly addressed by Grover's algorithm.
When it comes to brute-forcing a 256-bit ECDSA key using Grover's algorithm, the complexity reduction is only a square root. Grover's algorithm provides a quadratic speedup, meaning it reduces the time complexity from 2^N to 2^(N/2), where N is the number of bits. In the case of a 256-bit key, the reduction would be from 2^256 to 2^128, which is still an extremely large number and considered computationally infeasible.
Therefore, using Grover's algorithm alone is not a practical solution for breaking a 256-bit ECDSA key. Shor's algorithm, which is specifically designed for factoring and the discrete logarithm problem, would be the more appropriate algorithm for attacking ECDSA with quantum computing.