Final answer:
It is true that finding a collision for a hash function, on average, takes less time than finding a pre-image due to the birthday paradox. A collision occurs when two different inputs produce the same output, which statistically requires less computational effort.
Step-by-step explanation:
The statement that it requires half of the time on average for an attacker to find a collision for a hash function as compared to finding a pre-image is true. This concept is based on cryptographic hash functions, which are designed to take inputs of variable length and produce a fixed-length hash value. A pre-image attack involves trying to find any input that hashes to a specific output, which is a computationally difficult task due to the properties of hash functions.
A collision occurs when two different inputs produce the same hash output. Due to the birthday paradox, finding a collision can be significantly more efficient than finding a pre-image. The birthday paradox suggests that in a group of approximately the square root of the total number of possible hash values, there is a 50% chance that at least two individuals will share a birthday. Similarly, after hashing the square root of the number of possible outputs, there is a good chance of finding a collision, which on average requires less computational effort than finding a pre-image.