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I know that the NSA has recommended cryptography algorithms, such as their A/B suite algorithms and their Commercial National Security Algorithms Suite. I also know that they change their recommendations for key size, an example being in 2019 when they changed the key size recommended for RSA and Diffie Hellman from 2048 to 3072.

Are there any examples of changes like this that have been directly prompted by the publication of a mathematical paper or the creation of an algorithm? When people ask me about the applications of prime numbers to cryptography I want to be able to cite a specific time where math has changed real word cryptography recommendations.

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There are many examples in cryptography. Much of progress in cryptography is driven by academic research.

The Advanced Encryption Standard (AES) was a result of a request for research from the National Institute of Standards and Technology. Two cryptographers proposed a new algorithm, which was eventually adopted as the standard. A bit over a year ago NIST released a report estimating that there was $250 billion in economic benefit from the AES standard. This can be directly traced back to academic research. This doesn't involve prime numbers or number theory, though.

Key sizes and algorithm design typically comes out of academia. The RSA algorithm was created by academics who published a research paper proposing the algorithm (which in turn built on prior research from others, such as Hellman and Diffie), and has been extremely widely used to protect digital infrastructure -- including secure browsing and other uses.

Research has played a major role in setting parameters and key sizes for cryptographic algorithm; see, e.g., https://en.wikipedia.org/wiki/Key_size for a number of examples.

Caveat: many of the researchers doing this work might self-identify as computer scientists more than mathematicians. But the lines between theoretical computer science and applied mathematics in this area are blurry.

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