Sam Blake

Sam Blake

In this paper we describe a deep learning--based probabilistic algorithm for integer factorisation. We use Lawrence's extension of Fermat's factorisation algorithm to reduce the integer factorisation problem to a binary classification problem. To address the classification problem, based on the ease of generating large pseudo--random primes, a corpus of training data, as large as needed, is synthetically generated. We will introduce the algorithm, summarise some experiments, analyse where these experiments fall short, and finally put out a call to others to reproduce, verify and see if this approach can be improved to a point where it becomes a practical, scalable factorisation algorithm.

Sam Blake

For over three decades, the pursuit of perfect periodic autocorrelation sequences has been constrained by Mow's conjecture, which posits that no perfect sequence over an $n$-phase alphabet can exist with a length greater than $n^2$. While a proof across all conceivable sequence classes remains an open problem, this paper establishes bounds for a prominent class of constructions relying on the Array Orthogonality Property (AOP). We show that sequences generated by pure bivariate polynomial index functions cannot exceed the $n^2$ Frank-Heimiller bound due to algebraic periodicity. Furthermore, we extend this result to floored rational index functions, proving that attempts to geometrically expand the array dimensions inherently result in destructive fractional phase scattering. Neutralising this scattering strictly forces a collapse of the phase space, re-establishing the $n^2$ limit. Finally, we define the boundaries of these theorems, noting their fundamental reliance on commutative algebras, and contrast them with recent sequence constructions demonstrating the existence of unbounded perfect sequences over non-commutative unit quaternions.

Sam Blake

We give a self-contained, modern exposition of Édouard Goursat's 1887 theorem on pseudo-elliptic integrals -- those integrals of the form $\int F(t)\,\d t/\sqrt{R(t)}$ with $R$ a cubic or quartic polynomial that, despite living on a genus-$1$ algebraic curve, admit elementary antiderivatives. After reviewing integration in finite terms and Liouville's theorem, we present Goursat's two main theorems with proofs phrased in the language of Möbius automorphisms of the underlying hyperelliptic curve. We then develop a cube-root analog: for integrals of the form $\int F(t)\,\d t/\sqrt[3]{R(t)}$ with $R$ cubic, an order-$3$ Möbius substitution cyclically permuting the roots of $R$ induces an eigendecomposition into three pieces. Two of the three eigenpieces (eigenvalues $1$ and $ω^2$, where $ω= e^{2πi/3}$) descend through a chain of substitutions to genus-$0$ curves and yield elementary antiderivatives; the middle eigenpiece (eigenvalue $ω$) descends only to the genus-$1$ curve $y^3 = x(x-K)$ and is generically transcendental.

David Oranchak, Sam Blake, Jarl Van Eycke

The case of the Zodiac Killer is one of the most widely known unsolved serial killer cases in history. The unidentified killer murdered five known victims and terrorized the state of California. He also communicated extensively with the press and law enforcement. Besides his murders, Zodiac was known for his use of ciphers. The first Zodiac cipher was solved within a week of its publication, while the second cipher was solved by the authors after 51 years, when it was discovered to be a transposition and homophonic substitution cipher with unusual qualities. In this paper, we detail the historical significance of this cipher and the numerous efforts which culminated in its solution.

Sam Blake

In this paper we introduce a scheme for detecting manipulated audio and video. The scheme is a synthesis of blockchains, encrypted spread spectrum watermarks, perceptual hashing and digital signatures, which we call an Embedded Blockchain. Within this scheme, we use the blockchain for its data structure of a cryptographically linked list, cryptographic hashing for absolute comparisons, perceptual hashing for flexible comparisons, digital signatures for proof of ownership, and encrypted spread spectrum watermarking to embed the blockchain into the background noise of the media. So each media recording has its own unique blockchain, with each block holding information describing the media segment. The problem of verifying the integrity of the media is recast to traversing the blockchain, block-by-block, and segment-by-segment of the media. If any chain is broken, the difference in the computed and extracted perceptual hash is used to estimate the level of manipulation.