Abstract
The three Beale ciphers (published by J. B. Ward in 1885) remain a touchstone problem in historical cryptanalysis: cipher B2 decrypts against the United States Declaration of Independence to a coherent description of a buried treasure, while B1 and B3 have resisted 140 years of attack. Building on Gillogly’s observation that B1 decoded against the Declaration produces an anomalous alphabetic run, we report a fully reproducible, open-source re-examination that (i) re-establishes the B2 positive control under a single explicit scoring framework, (ii) systematically forecloses the principal composite-cipher hypotheses for B1 and B3—Caesar, short- and long-keyword Vigenère, autokey, Beaufort, columnar transposition over all periods to 25, and the complete enumeration of all 157,248 invertible Hill 2 × 2 matrices—none of which yields English plaintext, and (iii) quantifies the Gillogly anomaly with a generative “alphabet-walking” construction model that yields a likelihood ratio of at least 100:1 in favour of deliberate construction over encryption. We additionally document two methodological pitfalls that recur in book-cipher analysis: the index-of-coincidence baseline inflation produced by a skewed key’s first-letter distribution, and the dependence of significance on null-distribution construction. Our conclusions agree with the prevailing hoax verdict for B1 and B3; the contribution is the rigour, completeness, and reproducibility of the negative result, together with a calibrated statement of residual uncertainty. All code, data, and a continuous-integration test suite are publicly archived.
Acknowledgements
The author thanks the maintainers of Project Gutenberg, Wikisource, and Documenting the American South for the public-domain key texts, and the cryptologic-history community—in particular the published work of J. Gillogly, L. Kruh, C. Hammer, L. Campanelli, R. Wassmer, and the analyses of N. Pelling—without which this re-examination would have no foundation.
Disclosure of AI assistance
The analysis pipeline, statistical computations, figure generation, and a draft of this manuscript were produced with the assistance of an AI coding agent operating under the direction of the human author, using MATLAB and Python tooling. All methods are fully specified and reproducible from the archived code; no result in this paper depends on an unverifiable AI judgement. The AI system is a tool and is not an author.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Data availability statement
All code, cipher data, the key-text corpus, figure-generating scripts, phase-by-phase analysis reports, and a continuous-integration test suite are archived in the project repository at https://github.com/myhsieh1002/beale-cipher-analysis A permanent archival snapshot of the version corresponding to this publication has been deposited on Zenodo: https://doi.org/10.5281/zenodo.21193924
Additional information
Funding
Notes on contributors
Ming-Yu Hsieh
Ming-Yu Hsieh is a physician and clinical researcher who directs the Center for Evidence-Based Medicine at Chung Shan Medical University Hospital and serves as Vice Director of the School of Medicine at Chung Shan Medical University, Taichung, Taiwan. This study is a cross-disciplinary application of clinical-research methodology—pre-specified analysis, reproducibility, and adversarial review—to a classic problem in cryptologic history.
Facts Only
* Ming-Yu Hsieh is a physician and clinical researcher.
* Ming-Yu Hsieh is the Director of the Center for Evidence-Based Medicine at Chung Shan Medical University Hospital and Vice Director of the School of Medicine at Chung Shan Medical University in Taichung, Taiwan.
* The subject of the study is the three Beale ciphers published by J. B. Ward in 1885.
* Cipher B2 decrypts using the United States Declaration of Independence.
* Ciphers B1 and B3 have not been decrypted.
* The study tested B1 and B3 against Caesar, Vigenère (short- and long-keyword), autokey, Beaufort, and columnar transposition (periods up to 25).
* The study enumerated 157,248 invertible Hill 2 × 2 matrices.
* The research utilized MATLAB and Python tooling with AI coding assistance.
* Data and code are archived on GitHub and Zenodo.
* The study cites previous work by J. Gillogly, L. Kruh, C. Hammer, L. Campanelli, R. Wassmer, and N. Pelling.
Executive Summary
The Beale ciphers present a long-standing challenge in historical cryptanalysis. While the second cipher (B2) was successfully decrypted using the U.S. Declaration of Independence to describe buried treasure, the first and third ciphers (B1 and B3) have remained unsolved for 140 years. Recent re-examination employs a rigorous, open-source framework to test these remaining ciphers against a wide array of composite-cipher hypotheses, including various Vigenère variants and a comprehensive set of Hill matrices.
The results indicate that none of the tested encryption methods yield coherent English plaintext for B1 or B3. Furthermore, statistical modeling of an alphabetic anomaly in B1 suggests a 100:1 likelihood that the sequence was deliberately constructed to mimic a cipher rather than being a genuine encrypted message. These findings align with the prevailing conclusion that B1 and B3 are hoaxes. The analysis emphasizes methodological rigor and reproducibility, providing a calibrated statement of residual uncertainty and public access to all computational pipelines.
Full Take
This study applies a clinical-research methodology—characterized by pre-specification and adversarial review—to a problem of historical cryptanalysis. The design is robust, utilizing a positive control (B2) to validate the scoring framework before attempting to foreclose hypotheses for B1 and B3. By exhaustively enumerating the Hill 2 × 2 matrix space and testing multiple transposition and substitution periods, the author minimizes the "search space" for remaining possibilities. A peer reviewer might flag the limited scope of the Hill matrix test (only 2 × 2), but given the historical context of the Beale ciphers, this is a proportionate starting point.
The evidence supports the conclusion that B1 and B3 are likely fabrications. The "alphabet-walking" model transforms a qualitative observation (the Gillogly anomaly) into a quantitative likelihood ratio, moving the argument from "this looks strange" to "this is statistically improbable as encryption." This extends the existing body of knowledge by replacing anecdotal "failure to decrypt" with a documented, reproducible negative result.
If these findings are accepted, the Beale ciphers move from the realm of "unsolved puzzles" to "demonstrated hoaxes," shifting the focus from cryptanalysis to the psychology of the original fraud. To further falsify the hoax claim, a researcher would need to identify a key-text or a non-standard composite cipher that the current pipeline failed to capture.
The cross-disciplinary application of evidence-based medicine to history suggests a growing trend: the migration of rigorous statistical auditing from the hard sciences into the humanities to resolve long-standing ambiguities.
Bridge Questions: Does the exclusion of Hill matrices larger than 2 × 2 leave a viable path for genuine encryption, or is the "hoax" probability high enough to render further search futile? How would the results change if the key-text corpus were expanded beyond the Declaration of Independence?