By hard, we mean that no method much faster than trying elements of A using brute force is known to invert F effectively. This is collision-resistance. If there are 23 or more people in a room, the odds are better than .5 that two of them will have the same birthday. Assume that the 365 days of the year are all equally likely as birthdays, and that birthdays are random among people. So, actually, with 20 people in a room it is around an even bet that two will have the same birthday.
Hash values are small and fast to compute.
They also export a little better than pure cryptosystems do, yet can be used for encryption. Authentication Rather than using a challenge-response authentication method based on encrypting the challenge nonces, hashing may be used instead. Bob can authenticate Alice similarly. Unfortunately, several popular MD algorithms use as an intermediate value when computing the hash the same quantity that is used as the hash output at the end. This is not a problem with MD2, and is easily solved in any of several ways.