Mathematics
Small crowds hide big coincidences
In a room of 23 people, the odds of a shared birthday already top 50%, an exact 50.7% by the classic birthday-problem formula, because the odds climb far faster than the head count itself. Most people guess you'd need something like half of 365, about 183 people, before a match feels likely.
Sources & further reading
WikipediaBirthday problemAn encyclopedia entry laying out the classic probability puzzle with the exact numbers.WikipediaBirthday attackExplains how the same birthday-coincidence math becomes a real attack on digital security codes.National Institute of Standards and Technology (NIST)SP 800-107 Rev. 1, Recommendation for Applications Using Approved Hash AlgorithmsThe U.S. government's official cryptography guidance explaining how many bits of safety margin a security code needs.University of Alabama in Huntsville (Random Services), via LibreTexts12.6 The Birthday Problem, in Probability, Mathematical Statistics, and Stochastic ProcessesAn open-access probability textbook giving the formal proof behind the puzzle.Google Research and CWI AmsterdamSHAttered: the first collision for full SHA-1The real demonstration project that broke a widely used security fingerprint by finding two files with the same code.
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