On Leibniz: Expanded Edition

17. Leibniz and Cryptography

313 1. Cryptography’s Place in Leibniz’s Polymathic Project It is unquestionably an exaggeration to say, with Voltaire, that men use speech only to conceal their thoughts from the view of others. But it is certainly the case that they sometimes do so. The symbolic encoding of information and its concealment and revelation was of paramount interest to Leibniz throughout his entire career from beginning to end, and it was a topic that stimulated his mind in many directions . And cryptography, so Leibniz tells John Bernoulli, is a part of this project that is well deserving of the attention of a mathematician.1 The “art of decipherment is something virtually mathematical” (art de dechifrer . . . est une matiere encor demy-mathematique),2 and finding the key to a cryptogram is akin to finding the solution of equations in algebra.3 Thus a brief 1674 sketch of the art of innovation (ars inveniendi) states that this includes cryptography (ars explicandi crytophemata) and, like the latter, admits of pursuit via appropriate general rules.4 And in a long letter to E. W. von Tschirnhaus of May 1678 Leibniz describes cryptography (ars deciphra­ toria) as an integral part of the universal science (scientia generalis),5 which has close connections with algebra and constitutes a key component of the combinatories (ars combinatorial).6 Despite the power of the analytic method, it proves insufficient in cryptography, where a more extensive (longior) procedure of synthesis will prove necessary.7 Moreover, encoding transforms the foundation of a body of information from one format to another—much as with the representation of geometric figures from diagramatic to algebraic  17 Leibniz and Cryptography 314 Leibniz and cryptography representation in Cartesian geometry. His note of 1678 on the ars inveniendi remarks that the art of decipherment (ars deciphrandi) represents a sector of the field where analysis alone will not suffice for discovery and observes that, while analysis is generally more difficult, synthesis is more laborious.8 As to the type of synthetic reasoning involved, Leibniz likens the type of reasoning invoked in decipherment to finding good moves at playing chess.9 That everything can be said by the use of numbers is a key thesis of Leibniz ’s universal characteristic.10 And in a way, the object of cryptanalysis is the inverse of the Leibnizian characteristic: the latter seeks to make language more perspicuous and transparent, the former more difficult to access. Coding and decoding of information in symbolic systems are, after all, inverse procedures, and the steps that can make these processes simpler can be reversed to render them more complex and obscure. And Leibniz insisted that in this way advances in cryptography can serve to convey instructive insights into the ways of scientific inquiry. For as Leibniz saw it, cryptanalysis is something of a paradigm for scientific method, the ars faciendi hypotheses.11 Thus he observes that the investigation of causes is easier when different phenomena exhibit a commonality, even as “it is far easier to solve an encipherment when several encrypted letters are enciphered using the same key word” (facilius est cryptographemata solvere, si plures literas occultando sensu secundum eandem clavem scriptas).12 In the Nouveaux Essais Leibniz writes that “the art of discovering the cause of phenomena or finding the true hypotheses is akin to the art of deciphering” (l’art de decouvrir les causes des phenomenes, ou les hypotheses veritables, est comme l’Art de dechifrer).13 For in scientific explanation “a hypothesis is like the key to a cryptograph, and the simpler it is, and the greater the number of events that can be explained by it, the more probable it is.”14 Leibniz accordingly endorsed fully the idea—already found in Bacon’s Novum Organon and in the 1586 Traicté des Chiffres of the French algebraist and diplomatist Blaise de Vignière15 —that science aims to decode the secrets of nature. In just this way he claimed in relation to the conservation of force that “I have every reason to believe that I have here deciphered a part of nature ’s mysteries” (j’ay toutes les raisons de croire que j’ay dechifré une partie de ce mystere de la nature).16 Leibniz saw what he called “the method of hypotheses” as a key tool of scientific inquiry, and the deciphering of a cryptogram was his favorite illus- Leibniz and cryptography 315 tration of the workings of this method of hypothesis...