Analysis of Giouan Battista Bellaso’s cipher challenges of 1555

ABSTRACT

In his book, Noui et singolari modi di cifrare, Giouan Battista Bellaso presented to his readers three challenge ciphertexts. Two of them are based on a polyalphabetic cipher with a mixed primary reciprocal alphabet and a long key, whereas the third one uses four independent homophonic substitution alphabets that change after each word. A successful approach to solving these ciphers, to which Bellaso gave the name “terzo modo” and “quarto modo,” performs a simulated annealing attack directly on the primary alphabet, respectively the four homophonic alphabets at once. This article focuses on the various applied fitness functions. The solutions for the ciphertexts are given and finally discussed, with a view toward the possibility of a hidden message within the plaintexts.

KEYWORDS:

• Countersign
• fitness function
• Giouan Battista Bellaso
• homophonic substitution
• index of coincidence
• kappa test
• polyalphabetic substitution
• reciprocal alphabet
• simulated annealing

Acknowledgments

The author would like to thank Thomas Bosbach and Byron Knutson for their invaluable help in editing this paper.

Notes

¹The minuscules j and v were also to be found in an Italian type case at that time, but at least in this context, they may be considered as mere typographic variations of i and u. In Bellaso’s first publication of 1553, for instance, we find “vno” and “versetto” in coexistence with “uno” and “uersetto.”

²This point is most important in terms of pattern search and n-gram statistics. Yet, strictly speaking, Bellaso mentioned the double letter reduction only in the section of the quarto modo cipher, where he declared it as mandatory for the latter cipher, but merely advised so for terzo modo: for the sake of brevity – “per più breuità” (Bellaso 1555, 3v). However, it turned out that he made largely use of the double letter reduction in his challenge ciphers of 1564 (e.g., “lefeto” = l’effetto), without even addressing this subject in that book. Hence, it was supposed that all challenges of 1555 were composed in line with this principle. A statistical analysis supported the assumption; see chapter 4.1.

³That is, the beginning of Virgil’s Aeneid.

⁴“per contrasegno uno Salmo, ouero uno Sonetto, ouero dieci uersi latini o uolgari, & più quanti à te piacerà” (Bellaso 1555, 2r).

⁵Greek numbering, i.e., Ps.125 in Hebrew numbering. Bellaso uses only a little more than one verse for his example text. Here, three verses are given to obtain comprehensible samples for both variants of enciphering.

⁶“Le Trè sottoscritte lettere in cifra, sono scritte con gli precetti disopra dimostrati, la intelligenza de lequali si discoprirà à chi le uorrà sapere passato un’anno” (Bellaso 1555, 4r).

⁷For example, http://ciphermysteries.com/other-ciphers/bellaso-ciphers (accessed November 13, 2017).

⁸The IC values in this article are non-normalised.

⁹That is, double letter reduction and insertion of “y” as a word divider, cf. chapter 2.2.

¹⁰Also, the abbreviation marks of both ciphertexts were ignored and only considered during the final deciphering.

¹¹Apart from eliminating spaces between words and punctuation, the texts for this corpus remained unmodified (particularly, no word divider was inserted). Only exception: The rarely occurring Latin letter k was replaced by c to suit the character set (cf. chap. 2.1).

¹²A “temperature” variable controls the probability of accepting a modification with a lower fitness. This variable is diminished continuously. The approach emulates annealing in metallurgy, hence the name.

¹³This is not a guarantee of finding the global maximum, but it increases chances.

¹⁴For an in-depth introduction, see “Optimization by Simulated Annealing” (Kirkpatrick, Gelatt, and Vecchi 1983).

¹⁵In conjunction with the actual way of modifying and, of course, general parameters like the number of iterations, the “cooling scheme,” etc.

¹⁶To the author’s knowledge, first described in “Automated Cryptanalysis of Substitution Ciphers” (Forsyth and Safavi-Naini 1993).

¹⁷The described fitness function is adopted from the website, “Practical Cryptography” (Lyons 2012). To the author’s knowledge, the approach was first suggested by Sam Hasinoff (2003, chap. 2.2).

¹⁸The rare groups of five letters (e.g., the last group “tgdaz”) were treated as two quadgrams each (“tgda” and “gdaz”).

¹⁹It should not be impossible in general: The cipher shows several weak points (the key stems from a natural language, the primary alphabet is keyword-derived and has fixed positions for the vowels, which reduces the key space dramatically, and there are only 11 alphabets). However, ciphertext 2 seemed too short for a successful attack assuming an “infinite” key.

²⁰Analogous to “gratia” and its alternative spelling “gracia,” either of which was pronounced like “grazia” in modern Italian (McGee, Rigg, and Klausner 1996, 191).

²¹Quoted from http://www.thelatinlibrary.com/vergil/ec3.shtml (accessed November 17, 2017).

²²Greek numbering, i.e., Ps. 51 in Hebrew numbering.

²³ę (“e caudata”) was used as an alternative representation of æ. The reduction of the vowel a to a mere diacritic mark here might serve as an explanation of the fact that Bellaso’s key has a single e for every instance of æ (e.g., letitia instead of laetitia).

²⁴Bellaso erroneously skipped one letter of the key.

²⁵The ciphertext has “utfa.” “utsa” (which yields a sense-making plaintext) may be assumed because f and ſ (long s) are prone to be mixed up by a typesetter.

²⁶Vincenzo Maggi[o] (1498–1564) was a writer and philosopher (“Vincenzo Maggi” 2016). Bellaso also mentions him in the preface of his first publication as “eccellentissimo filosofo, il Sig. Vincenzo Maggio” (Bellaso 1553).

²⁷Stemo male: dialect form of “stiamo male” (we are in bad condition).

²⁸Literally, argeca, which could not be found in any dictionary. Arghena is a Brescian word meaning crane: “machina per tirar pesi” (Pellizzari 1759, 10).

²⁹The original line break is five letters later here and has been shifted for layout reasons.

³⁰The punctuation follows Erasmus of Rotterdam (Erasmus 1534, 217, 527), cf. chapter 6.3.

³¹Bellaso ciphered “Avogaro,” an alternative spelling of the surname.

³²“Magistratura municipale giudiziaria civile che decideva nei concorsi di creditori, addizioni di eredità, pagamenti di dazi o fallimenti.” (Tedoldi 1999, 25).

³³Mηχανικά. Bellaso refers to the Latin name Quaestiones Mechanicae. The authorship of this work is being discussed, cf. the Wikipedia entry “Mechanics (Aristotle).”

³⁴He could also refer to the Latin translation by Nicolò Leonico Tomeo (Thomaeus 1525), as well as Aldo Manuzio’s first Greek printed edition of the end of the 15th century, “which circulated widely throughout the 16th (…) century” (Gullino 2013). However, Piccolomini’s paraphrase appears more plausible in this context because Piccolomini and Maggi[o], the addressee of the letter, were friends (“Vincenzo Maggi” 2016).

³⁶“Quæstio Decimatertia. Cap. XVIII.” (Piccolomini 1547, 35r).

³⁷“Haec succula dum versatur, funis quem ductarium dicunt, dum in ipsa conuoluitur, magna auxilio trochlearum, tollit pondera” (Piccolomini 1547, 36r): When this winch is turned, the rope, being called a ductarium, and which is wound inside it, lifts, with the help of pulleys, heavy weights.

³⁹Admittedly, this only applies if the ligature æ and the e caudata (ę), both denoting “ae,” are counted as two letters each, but the abbreviature ũ (denoting “um”) as one letter.

⁴⁰Both aphorisms originate in Diogenes Laërtius’ Lives and Opinions of Eminent Philosophers, which at Bellaso’s time circulated in many different editions and translations, which makes it hard to decide who copied from whom. However, it seems that the section “propterea (…) debent eruditi” has been added by Erasmus (Ter Meer 1969, 279) which means that Bellaso copied from him at least indirectly.

⁴¹(Knott and Fantham 2014, vol. 37, p.327).

⁴²(Knott and Fantham 2014, vol. 38, pp.824–825).

Additional information

Notes on contributors

Norbert Biermann

Norbert Biermann studied piano and conducting. He worked for more than twenty years as a musical coach and conductor at several German Opera houses. In 2014 Mr. Biermann was appointed Professor of Repertoire at the Berlin University of the Arts. About two years ago, he discovered cryptography for himself and ever since likes to solve diverse historical ciphers.