The birthday paradox illustrates the tremendous difference between the effort required for a pre-image attack and a collision attack. If you are trying to find someone who has the same birthday as you, you'll have to ask 253 people to have a 50 per cent chance that at least one of them shares that birthday. But if you only want to have two people with the same birthday regardless of what day that is, you can make do with far fewer people. Within a group of only 23 people, there is a 50 percent chance that two of them will share a birthday.
If an attack is successful with far fewer attempts than in the brute force method, the procedure is considered cracked. According to Schneier, this is exactly what the Chinese research group has accomplished: they are said to have developed a method of finding a collision with 269 operations instead of 280. The number of operations now necessary would then be lower by a factor of 2,048 (211).