Data security: A problem in search of a mathematical theory

The need for security in electronic communications is crucial in today's 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.

SHA-1 is a secure hash algorithm (that is, a computer algorithm based on a hash function) that is the government standard and is very widely used; it was developed by the National Security Agency. Other older and even less secure hash algorithms are still in use in many applications. Cryptographers were already concerned about vulnerabilities that had been exposed in those older algorithms. But they were astonished when, at a cryptography meeting in 2005, researchers announced that they had found a way to attack SHA-1 in far fewer steps than was previously known.

For now, SHA-1 is still safe; the method announced by the researchers would take a huge amount of computational time and resources, and it is not clear how this would be carried out. But such attacks always grow more sophisticated, so cryptographers would like to replace SHA-1 as soon as possible. The NSA has a series of hash algorithms, beginning with SHA-256, that are secure; the issue is how to deploy them throughout the infrastructure.

In her Notices article "Find Me a Hash", Susan Landau describes these developments and the nature of the new vulnerability of SHA-1. Her central point is that the mathematical theory of hash functions needs much more development before researchers can come up with more secure hash algorithms for tomorrow's applications.


An advance copy of Landau's article may be found through the non-public link The article will appear in the March 2006 issue of the Notices of the AMS (

Founded in 1888 to further mathematical research and scholarship, the more than 30,000-member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

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Last reviewed: By John M. Grohol, Psy.D. on 21 Feb 2009
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