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We’re awaiting the release of OpenAI’s o3 model later this month. Its performance is impressive on very hard benchmarks like SWE-bench Verified, Frontier Math and the ARC AGI benchmark (discussed previously in this blog). And yet at the same time some behaviors of the frontier AI models are very concerning. Their performance on assorted math […]...

The security of RSA encryption depends on the difficulty of factoring the product of two large primes. If you can factor large numbers efficiently, you can break RSA. But if can break RSA, can you factor large numbers? Sorta. It’s conceivable that there is a way to break RSA encryption without having to recover the […] The post Converse of RSA...

Stegonography attempts to prevent messages from being read by unintended recipients by hiding the messages rather than (or in addition to) encrypting them. Stegonography is used when you not only want to keep your communication private, you want to hide the fact that you’ve communicated at all. Fun fact: The words stegonography and stegosaurus...

A blog carnival is a way to discover new blogs. Writers on a given topic, such as math, take turns hosting the carnival, featuring recent posts from various writers. Blog carnivals were once much more common, but most have faded away. The Carnival of Mathematics, however, is one of the oldest carnivals and still active, […] The post Carnival of...

The previous post showed that there are 10 Abelian groups that have 2025 elements. Implicitly that means there are 10 Abelian groups up to isomorphism, i.e. groups that are not in some sense “the same” even if they look different. Sometimes it is clear what we mean by “the same” and there’s no need to […] The post Up to isomorphism first appeared...