If a key is 43 bits long, how much longer will it take to crack it by exhaustive search if extended to 50 bits?

• A. 7 times as long
• B. 14 times as long
• C. 128 times as long
• D. 256 times as long

Answer: (C)

Brute-forcing scales with the size of the key space, which for an n-bit key is 2^n. Moving from 43 bits to 50 bits expands the key space from 2^43 to 2^50. The time to exhaustively search is proportional to that space, so the factor increase is 2^50 / 2^43 = 2^(50−43) = 2^7 = 128. Therefore, it will take 128 times longer to crack a 50-bit key than a 43-bit key, assuming the same hardware and conditions. The exponential growth means even a small bit increase yields a large jump in effort. If you think in terms of average-case timing, it’s still a factor of 128.