Chapter 1, Part 3

One-Way Keys

In 1976, Whitfield Diffie and Martin E. Hellman of Stanford University forever changed the cryptographic landscape with their open publication of one-way keys. In conventional cryptosystems, a single key is used for both encryption and decryption. Such systems are called symmetric. The weakness of these systems is their requirement of protecting any exchange of such keys over a secure channel, which is inconvenient at best. (If a secure channel were available, why use encryption in the first place?)

The introduction of Diffie and Hellman to asymmetric keys made possible the concept of “public key cryptography,” which allows the participants to communicate without requiring a secret means of delivering the keys. It is possible to have a system in which one key is used for encryption and a different key for decipherment. One can publish the encryption key widely for those who would send a message. The encryption key is useless for decipherment. When the message is received by the intended recipient, his private complementary key is used for deciphering the message. This private key is available to no one.

Asymmetric cryptosystems are based on mathematical techniques that are easy to compute in one direction but excessively onerous and slow to solve in the reverse. The two main public-key algorithms are the Diffie-Helman (and its variants such as the Digital Signature Standard from the National Institute of Standards and Technology, ElGamal, and elliptic curve approaches) and RSA, developed at the Massachusetts Institute of Technology by Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman. A fairly advanced form of encryption technology is known as “PGP,” for “Pretty Good Protection,” and is readily available over the Internet for the diligent.16

These systems can also be adapted for authentication and electronic “signatures” for approving documents, contracts, and the like over email, etc.

The ability to communicate through the Internet, within which a gigantic and dynamically changing data base is simultaneously available to both a sender and a receiver, also makes practical, reasonably secure encryption widely available for industrial and private uses. Symmetric systems are still the most efficient, and the public key techniques, while involving more substantial computational loads, make the conveyance of the necessary keys secure. The ability to share extensive, dynamically changing keys, accompanied by necessary sophisticated software at both ends, makes practical protection readily available to anyone. The open availability of this technology leaves those who abhor privacy – especially governments and extreme “liberals” – very uncomfortable.

The computing power presently available on the author’s own desk exceeds the computing power which was available to him when he headed up the Computer Center of the Ford Motor Company in the 1960s The mathematical sophistication now available to an encryptor, and demanded by the would-be cryptanalyst, staggers the imagination.

It is interesting to review the development of mechanical aids to cryptographic techniques from a historical perspective.

Cipher Disks

Single-step monoalphabetic substitution ciphers generally prevailed until the 15th century. Then there emerged a series of advances that not only transformed cryptology, but laid the foundation for both advanced statistics and the computer. Among those advances was the appearance of the “cipher disk,” generally attributed to the first of the “Renaissance men,” Leon Battista Alberti (1404-1472), who became known as “the father of cryptology.”17 By taking two copper disks, a larger stationary one on the same axis as a smaller movable one, and inscribing each with a sequence of letters, it made simple substitution ciphers more convenient and easier to use.

Figure 1-4: Cipher Disks

Figure 1-4: Cipher Disks

The crucial addition was the ability to change the position of the index during the message, resulting in the modern polyalphabetic cipher which was vastly more complex than those which were previously in use. The Wheatstone Disk and Thomas Jefferson’s Cipher Wheels were also mechanical variations of such devices. Modern cipher machines produce polyalphabetic ciphers that can exploit millions of cipher alphabets.

(There are some scholars who speculate that Ezekiel’s “wheels within wheels” may be a hint of polyalphabetic applications hidden within the Biblical text.18)


Most ciphers employ a key, which specifies such things as the arrangement of the letters within a cipher alphabet, the pattern of shuffling in a transposition, or the settings on a cipher machine.

Blaise de Vigènere, in 1585, invented the “autokey” in which the encrypted message itself makes up the key. After the first letter (or word or phrase), known to both the sender and the recipient, the deciphered plaintext becomes the key for the subsequent encipherment of the remaining ciphertext. Using an Alberti cipher disk, for example, after a key word, the deciphered letters of the plaintext continue to tell the recipient of this polyalphabetic cipher how the inner wheel should be turned against the outer.

The Cardano Grille

Another of the remarkable “Renaissance men” was Girolamo Cardano of Pavia in the 16th century. A highly successful physician,19 Cardano was also an outstanding mathematician, publishing the earliest solutions for cubic and quadratic equations.20 One of Cardano’s contributions to cryptology reflected his background in kabbalistic encoding and decoding – skipping letters within an otherwise plausible ciphertext. (His method also anticipated a form of masking used in computer instruction processing.) His method consisted of a mask (“grille”) with precut holes. The encoder writes his plaintext in the holes, removes the mask, and then fills the remainder with blind text (nulls), preferably retaining the appearance of an innocuous message.

To decipher the message, the recipient must possess a mask (or “grille”) identical to the sender’s, or must know the spacing that created it.

Figure 1-5: Cardano Grille

Figure 1-5: Cardano Grille

(The “equidistant letter sequences” which have been discovered in the Bible are the equivalent of a simplified Cardano Grille and will be explored in Chapter 11.)

The primary difficulty with this method is, of course, that any awkwardness in the phrasing of the cover message may betray the existence of a hidden message.21 Such “awkwardness,” however, can also be deliberate to enlist the attention of the specially informed to look deeper. (Such a clue is called a remez, and will also be explored in Chapter 11.)

A close cousin of the Cardano Grille was the Turning Grille used by the Germans. A turning grille is usually a square sheet of cardboard divided into cells. One quarter of these are punched out in a pattern such that when the grille is rotated to its four ordinal positions, all the cells on the paper beneath will be exposed and none will be exposed more than once. A 6 x 6 grille and its application for a 36-letter message is shown in Fig. 1-6, which follows.

This is laid over a sheet of paper and the first nine letters are written through the apertures. Then it is turned 90°, the next nine letters written through the openings, and so on for two more turns. By then each of the 36 cells on the paper will have a letter inscribed in it. Then the letters can be taken off in any predetermined order.

Figure 1-6: The Turning Grille

Figure 1-6: The Turning Grille

Grille systems are particularly susceptible to multiple anagramming, which is the general solution for transposition systems. They are not very effective for purposes of secrecy; the German use of grilles against the French only lasted four months.22 (However, such grilles can serve as a subtle means of authentication. We will explore this application of anagrams, acrostics, and related techniques in subsequent sections of this book.)

Mathematical Statistics

The pursuit of cracking an unknown code increasingly relies on the techniques of mathematical statistics. While the development of mathematics has many very ancient roots, the field of mathematical statistics is of a much more recent vintage. It wasn’t until the 17th century that Sir John Wallis, England’s preeminent cryptologist, driven by the advent of polyalphabetic ciphers, laid the foundations for calculus and the binomial theorem, a crucial element in the science of statistics.

But it was a French contemporary of Wallis, Blaise Pascal, who was to lay the foundation for modern statistical science and who is also venerated by many as the “father of the computer.” The Pascal computer language was named after him.23 Having completed the equivalent of a doctoral education in the humanities, arts, and sciences by the age of 12, he was breaking new ground in calculus and conic sections24 by the age of 16. But as a mathematician, Pascal is best known for having laid the groundwork for the theory of probability, the cornerstone of the field of statistics.25

It should be understood that mathematics is generally deterministic: two times two always equals four. Engineers and scientists are the beneficiaries of Western civilization which is largely deterministic in its physics, relying on a conceptual model of “cause-and-effect.” It takes a special insight and training to deal with stochastic (probabilistic) variables, and the skills and talent to successfully deal with statistical models are relatively rare. (The paradoxes which have emerged from quantum physics, and from the departure from “cause and effect” determinism, will be explored in Chapter 23.)

« Back to Part 2Continue to Part 4 »


  1. Phillip R. Zimmermann, “Cryptography on the Internet,” Scientific American, October 1998, pp.110-115. An excellent tutorial on contemporary cryptographic techniques in the public domain.
  2. Kahn, p. 127. Alberti was a monk who was also an architect, athlete, mathematician, moralist, musician, painter, poet, sculptor, etc.
  3. Ezekiel 1:15, 16;10:2ff.
  4. Credited with an accurate clinical description of typhus, developed a treatment for syphilis, and invented a Braille-like system to permit the blind to read by touch.
  5. A century before Pascal and Fermat, he published systematic computations for probabilities and statistics.
  6. Kahn, p.144.
  7. Kahn, p. 309.
  8. When the author was CEO of Western Digital Corporation, one of its innovative developments was a microprogrammed microchip which directly executed Pascal (and, thus, ADA, a Department of Defense derivative).
  9. Conic sections are the family of curves which result when a cone is intersected at various angles by a single plane – i.e., circles, ellipses, parabolas, and hyperbolas. The elegant relationship between three-dimensional geometry and two-dimensional algebra has fascinated mathematical minds for centuries and continues to do so today. These concepts can be extended to hyperspaces (spaces of more than three dimensions) through the application of metric tensors and Riemann geometry.
  10. Pascal is even more widely known for his religious writings, especially Pensées (Musings), his preliminary sketches for a comprehensive defense of the Christian faith.