Mage: Cracking Elliptic Curve Cryptography with Cross-Axis Transformers

MAGE: CRACKING ELLIPTIC CURVE CRYPTOGRAPHY WITH CROSS-AXIS TRANSFORMERS ∗ Lily Erickson EmerGen LLC Minneapolis, MN lilyerickson@emergenlabs.com ABSTRACT "Cryptography is the art of making discrete data appear statistically random." -Unattributed 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[1]. 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. Keywords AI · Cryptography · Security ∗Citation: Authors. Title. Pages.... DOI:000000/11111. arXiv:2512.12483v3 [cs.CR] 1 Jan 2026 EmerGen LLC 1 Introduction The security of modern ECC algorithms lies in the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). To keep things simple, the ’irreversibility’ of ECC algorithms lies largely in the modulo operand. Each byte cannot exceed a value of 255, and so the large products created by the curve multiplication must necessarily wrap back around to 0 once it exceeds the maximum value allowed by a standard byte. It is very easy to calculate this going forwards. r = a%n = a −n ∗ j a n k However it’s extremely difficult to reverse this process without knowing the precise quotient (i.e., the number of times the value wrapped around modulo N), and we are already multiplying these values by enormous numbers. The formula is not without fail though. In order to understand the potential of modern machine learning algorithms in this domain, it is first imperative to explore the two existing vulnerabilities in cryptographic algorithms that we intend to tackle before proceeding. A cryptographic Side Channel is a data leakage vector (eg: timing attack, power consumption during operations, etc) that exposes information about and compromises the integrity of a cryptographic computation. The Cryptographically Secure Pseudo-random Number Generator (CSPRNG) is considered one of the defacto methods of generating on-the-fly, cryptographically secure information. Normal random number generators are susceptible to timing attacks, however CSPRNG attempts to obfuscate the creation of the initial private key, and in doing so hopes to protect against side-channel and environment based attacks. Machine learning models however, are Universal Function Approximators - there is a very real chance that they are capable of modeling the underlying cipher itself, bypassing this safeguard. If there is a latent pattern within the ECDLP that a machine learning model could recognize, we should be able to detect it with enough data. A Rainbow Table is a partially or completely solved array or mapping of all known input (and/or output) permutations for a given equation. The National Institute of Standards and Technology (NIST) offers guidance that current 256-bit ECC keys have 128 bits of security, meaning it would take on the order of 2128 computational steps to break it. A 256-bit ECC key is a 32-character Uint8Array (each character is an integer between 0-255), meaning that a complete, unoptimized Rainbow Table for all known public -> private key mappings for a 256-bit curve would take up ∼7.41 ∗1057 ZB (zettabytes) of data. For comparison, it’s estimated that by the end of 2025, earth’s datasphere will only be ∼172 ZB[2]. However, for any problem in which a solved rainbow table with pre-computed mappings exists, the computation required to

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