In terms of brute-forcing the private key, both a 12-word
seed phrase and a 24-word
seed phrase offer the same level of security if they are generated using a strong random number generator.
The reason for this is that the private key derived from the seed phrase is essentially a 256-bit
number in the case of the Bitcoin protocol (specifically, the secp256k1 elliptic curve
used in Bitcoin). The seed phrase is used to deterministically generate the private key, but the private key itself has a fixed length of 256 bits.
When using a 12-word seed phrase, the seed phrase is used to generate the private key through a series of cryptographic operations. Similarly, when using a 24-word seed phrase, the same process is followed to generate the private key.
The extra words in a 24-word seed phrase are used to provide additional entropy and increase the number of possible seed phrases. This helps in protecting against the possibility of collisions (two different seed phrases generating the same private key), but it does not directly impact the security of the private key itself.
In summary, the security of the private key is determined by the length and randomness of the underlying bits used to generate it, rather than the number of words in the seed phrase. Both a 12-word
seed phrase and a 24-word
seed phrase can provide strong security if generated properly.