# Lester Hill’s “The checking of the accuracy …”, part 1

Backstory: Lester Saunders Hill wrote an unpublished notes, 40 pages, undated but probably written in mid- to late 1920s, titled “The checking of the accuracy of transmittal of telegraphic communications by means of operations in finite algebraic fields”. In the 1960s this was given to David Kahn by Hill’s widow. These notes were typewritten, but mathematical symbols, tables, insertions, and some footnotes were often handwritten. These notes are being LaTeXed “as we speak”. I thank Chris Christensen, the National Cryptologic Museum, and NSA’s David Kahn Collection, for their help in obtaining these notes. Many thanks also to Rene Stein of the NSA Cryptologic Museum library and David Kahn for permission to publish this transcription. Comments by transcriber will look his this: [This is a comment. – wdj]. I used Sage (www.sagemath.org) to generate the tables in LaTeX.

Here is just the introductory first section of his paper. I hope to post more later.

Section 1: The problem considered

The forms of code in current use for the economical transmittal of cable and radio messages employ, almost uniformly, the so-called ”pronounceable” combinations. Pronounceability is estimated in terms of letter groups which commonly occur in one or more of eight designated ”telegraphic” languages: English, French, Spanish, etc. International regulations have, in fact, prescribed a system, of charges according to which a secret five-letter word is regarded, in the word count, as one-half of a telegraphic word, or as a full telegraphic word, according to as it is, or is not, pronounceable.

There is now descernible a definite tendancy to abandon this quite arbitrary pronounceability rule, which has heretofore hampered the development of code and cipher communication. And it seems not unlikely that, at some early session of the Interbational Telegraph Congress, all secret five-letter combinations will be placed upon the same basis with respect to transmittal charges.

A mathematical problem of considerable scientific and practical interest arises in this connection. Messages written in pronounceable code or cipher contain within themselves certain checks upon the accuracy of their telegraphic transmittal. But sequences of unpronounceable groups will generally be quite arbitrary, and no such internal check will ordinarily be available. The outstanding requirement is some device capable of protecting totally unrestricted sequences, in a telegraphically economical way, against the hazards of faulty transmittal.

Let the finite set $C$ of signs (letters, numerals, etc) upon which a given system of communication is based – plain-language, code, or cipher system – be placed in correspondence with the elements of a finite algebraic field $F$. Then the problem of checking sequences of signs in $C$ becomes that of checking sequences of elements in $F$. As well-known, if we denote by $\Gamma$ the number of elements, including the zero and unit elements, of a finite field $F$, then $\Gamma$ is one of the numbers:

$2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, \dots ,$

that is to say, $\Gamma$ is of the form $\Gamma = p^r$, where $p$ denotes a prime positive integer greater than $1$ and $r$ denotes a positive integer. Conversely, if $\Gamma$ is an integer of this form, we can very easily set up finite algebraic fields with $\Gamma$ elements. In devising a scheme of checks for messages based upon the system $C$ of communications with $n$ distinct signs, we should normally work with a field $F$ the number, $\Gamma$, of whose elements differs as little as possible form $n$. But this is not essential.

Our problem, viewed form the standpoint of mathematics, is simply that of providing economical and rigorous checks, easily applied in a practical way, upon the accuracy of transmittal of arbitrary sequences of elements in a finite algebraic field.

The method suggested will be applicable to telegraphic messages of any type: plain-language, code, cipher, cipher-code. It will, however, undoubtedly, be found susceptible of fundamental improvement in many respects; and it will probably yield to other, and more definitely practical, procedures that may subsequently be constructed be upon a basis of number-theoretical operations. The present paper will have served its purpose if it succeeds in directing attention to certain hitherto neglected practical possibilites inherent in elementary algebraic manipulations.