The need for security in electronic communications is crucial in todays world. The foundation for providing this security rests on mathematics. In particular, a certain kind of mathematical function called a "hash function" is central in the design of cryptographic systems that protect electronic communications. But recently the most secure hash function in use today was shown to be vulnerable to attack. An article in the upcoming issue of the Notices of the AMS describes these attacks and sends out a call for a new mathematical theory to undergird future communications security systems.
A hash function is an easy-to-compute compression function that takes as input any string of computer bits and distills that string down to a fixed-length output string: Whether the input is an 8-character password or a 100-page document, the hash function outputs a string of a fixed length. An important feature of hash functions is that they must be hard---or at least computationally too expensive---to invert. So, for example, it should be hard, given the hash of a password, to recover the password itself.
One researcher calls hash functions the "duct tape" of cryptography because they are used everywhere for many different purposes: to authenticate messages, to ascertain software integrity, to create one-time passwords, and to support Internet communication protocols. The very ubiquity of hash functions makes any vulnerability found in them a widespread concern.
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