Mage: Cracking Elliptic Curve Cryptography with Cross-Axis Transformers

Abstract

With the advent of machine learning and quantum computing, the 21st century has gone from a place of relative algorithmic security, to one of speculative unease and possibly, cyber catastrophe. Modern algorithms like Elliptic Curve Cryptography (ECC) are the bastion of current cryptographic security protocols that form the backbone of consumer protection ranging from Hypertext Transfer Protocol Secure (HTTPS) in the modern internet browser, to cryptographic financial instruments like Bitcoin. And there's been very little work put into testing the strength of these ciphers. Practically the only study that I could find was on side-channel recognition, a joint paper from the University of Milan, Italy and King's College, London\cite{battistello2025ecc}. These algorithms are already considered bulletproof by many consumers, but exploits already exist for them, and with computing power and distributed, federated compute on the rise, it's only a matter of time before these current bastions fade away into obscurity, and it's on all of us to stand up when we notice something is amiss, lest we see such passages claim victims in that process. In this paper, we seek to explore the use of modern language model architecture in cracking the association between a known public key, and its associated private key, by intuitively learning to reverse engineer the public keypair generation process, effectively solving the curve. Additonally, we attempt to ascertain modern machine learning's ability to memorize public-private secp256r1 keypairs, and to then test their ability to reverse engineer the public keypair generation process. It is my belief that proof-for would be equally valuable as proof-against in either of these categories. Finally, we'll conclude with some number crunching on where we see this particular field heading in the future.