For cryptographers, a cryptographic "break" is anything faster than a brute-force attack – i.e., performing one trial decryption for each possible key. In many cases, even though they are the best known methods, they are still infeasible with current technology. One example is the best attack known against 128-bit [[Advanced Encryption Standard|AES]], which takes only <math>2^{126}</math> operations.<ref name=":0">{{cite book |author=Biaoshuai Tao |title = Information Security and Privacy|volume = 9144|pages = 39–56|author2=Hongjun Wu |name-list-style=amp |date=2015|doi=10.1007/978-3-319-19962-7_3 |series = Lecture Notes in Computer Science|isbn = 978-3-319-19961-0}}</ref> Despite being impractical, theoretical breaks can sometimes provide insight into vulnerability patterns.
=== Traveling salesman problem ===
For several decades, the best known approximation to the [[traveling salesman problem]] in a [[metric space]] was the very simple [[Christofides algorithm]] which produced a path at most 50% longer than the optimum. (Many other algorithms could ''usually'' do much better, but could not provably do so.) In 2020, a newer and much more complex algorithm was discovered that can beat this by <math>10^{-34}</math> percent.<ref>{{cite arXiv |eprint=2007.01409 |class=cs.DS |title=A (Slightly) Improved Approximation Algorithm for Metric TSP |author1=Anna R. Karlin |author2=Nathan Klein |author3=Shayan Oveis Gharan |date=September 1, 2020}}</ref> Although no one will ever switch to this algorithm for its very slight worst-case improvement, it is still considered important because "this minuscule improvement breaks through both a theoretical logjam and a psychological one".<ref>{{cite web |author=Klarreich |first=Erica |date=8 October 2020 |title=Computer Scientists Break Traveling Salesperson Record |url=https://www.quantamagazine.org/computer-scientists-break-traveling-salesperson-record-20201008/ |website=Quanta Magazine}}</ref>
