In a time when the security of personal data is paramount, few may think to scrutinize the intricacies of their tax returns. For many, these documents are straightforward, revealing little to be concerned about. However, recent revelations about vulnerabilities in encryption methods have raised alarms for those who rely on digital communication. An alarming discovery made by information technology security researcher Hanno Bck highlights how an attacker could have easily intercepted the encrypted communications between a laptop and printer during the printing process of these tax returns in recent years.

In early 2022, Bck made significant strides in understanding these vulnerabilities, culminating in a preprint paper published in 2023 in the International Association for Cryptologic Researchs Cryptology ePrint Archive. His work uncovered that some encryption methods could be compromised through a technique rooted in the groundbreaking work of the 17th-century French mathematician, Pierre de Fermat, who is perhaps best known for his enigmatic last theorem that stumped mathematicians for centuries.

Fermat, who laid the groundwork for various scientific domains, made notable contributions to number theory and probability. His explorations into prime numbersthose integers greater than one that are only divisible by one and themselveshave proven essential in contemporary cryptographic systems. In light of Bck's findings, it has become evident that Fermats methodologies, formulated centuries ago, can now serve as a tool to challenge the security of modern encryption.

Understanding the Mechanics of Modern Encryption

Modern encryption systems are predicated on complex mathematical problems, serving as a digital padlock where the problem itself (the lock) cannot be deciphered without access to specific information (the key). One of the most utilized procedures in this domain is RSA cryptography, which heavily relies on prime numbers. The difficulty of factoring large composite numbers into their prime components makes RSA a reliable method for securing data.

To illustrate how RSA cryptography functions, consider a simplified example in which an individual wishes to send the word SCIENCE encrypted as a seven-letter code. To achieve this, the sender employs a large seven-digit number, such as 6,743,214, to shift the letters of SCIENCE by the respective digits in the sequence. Thus, the letter S shifts six places to become Y, C becomes J after moving seven positions, and so forth. The resultant encrypted message would read as CJMHPDI. This transformation ensures that an unauthorized listener would face substantial difficulty in deciphering the message.

However, for the intended recipient to retrieve the original content, they must possess either the key itself (6,743,214) or a means to calculate it. This reliance on the key introduces vulnerabilities; should an attacker manage to eavesdrop during the exchange of keys, they could potentially compromise the data. The RSA system mitigates these risks by enabling the sender and receiver to collaboratively generate a key using publicly available information while secretly employing large prime numbers to ensure security. The eavesdropper, lacking access to the original prime factors, would thus be incapable of reconstructing the key from the intercepted products.

The Fermat Factorization Method

In an era long before encryption was a concern, Fermat was engaged in exploring how to factor numbers into their prime components, purely driven by mathematical curiosity. His work led to the creation of a method for factorizing numbers that are products of two prime numbers. Fermat famously illustrated this technique with the number 2,027,651,281.

The Fermat factorization method operates on a straightforward principle. Taking the square root of the number yields an odd result, which is then rounded up. This figure is subsequently squared, with the original number subtracted from the result. The process is repeated until a perfect square is found, allowing for the application of the third binomial formula to factor the original number into its prime elements.

For instance, through a series of calculations involving 45,041 and 1,020, Fermat eventually identified that the factors of 2,027,651,281 are 44,021 and 46,061, both of which are prime numbers. This methodology, albeit methodical and requiring patience, underscores the profound impact of Fermat's work on contemporary cryptographic practices.

Exposing Vulnerabilities in Modern Systems

Despite its historical significance, Bck's findings revealed troubling vulnerabilities in encryption systems currently in use. His research indicated that many printers employed encryption protocols that were not sufficiently robust, often selecting prime factors that were too closely spaced, thus making them susceptible to Fermat's factorization method. This opened the door for potential exploits, allowing unauthorized parties to circumvent the encryption securing sensitive documents transmitted via networks.

Following Bck's revelations in 2022, affected companies quickly issued alerts and patches to bolster their security measures. However, this situation serves as a stark reminder for organizations worldwide to reassess their encryption standards. Looking ahead, the rise of quantum computing poses an entirely new set of challenges for encryption, as these advanced systems could factor large numbers at speeds that outstrip traditional computing capabilities. Fermat could hardly have envisioned that more than 380 years after his contributions, his work would play a critical role in discussions surrounding the safety of digital communications.

This article originally appeared in Spektrum der Wissenschaft and is published with permission.