The Flaw in the JN-25 Series of Ciphers, II

From 1939 to 1945 the Imperial Japanese Navy made heavy use of a series of additive cipher systems generically named JN-25 by the cryptanalytical unit of the United States Navy. Each of these consisted of a code-book assigning a five-digit 'group', always a multiple of three, to each word or phrase in a very long list and encrypting these by 'false' (non-carrying) addition of a five-digit group ('the additive') taken from a long table of essentially random such. The author's earlier paper explains how this use of multiples of three provided a route for relatively rapid recovery of the additive and thus the decryption of intercepts. Another quite different and rather surprising source of insecurity inherent in this use of multiples of three was noted only in 1943 and became the basis of a process code-named 'Mamba' needed in 1944. This note sets out a further consequence of the statistics underlying Mamba: the use of multiples of three in JN-25 codebooks betrays itself very quickly. This paper deals with codebooks assigning to each word or phrase in a (long) list a 5-digit 'group' which is a multiple of three or scannable in the jargon of WW2. Here the groups are allowed to have initial 0s: thus 12345 and 00078 are groups. Being a multiple of three (equivalently, being divisible by three) means that the corresponding number, 12345 or 78 in these examples, is a multiple by three or, equivalently, the sum of the five digits, 15 in both examples, is a multiple of three. The characteristic χ of such a group is defined to be the reduction modulo 10 of the sum of the five digits: thus χ(12345) = 5. One might well guess that of the 33334 scannable groups from 00000 to 99999 about 3333 have characteristic 0, about another 3333 have characteristic 1, etc. This is far from being the case. A simple electronic calculation checks that for 0 ≤ i ≤ 9 the number m(i) of scannable groups with characteristic i is as set out below: m(0) = 3247 m(1) = 5875 m(2) = 1780 m(3) = 1780 m(4) = 5875 m(5) = 3247 m(6) = 925 m(7) = 4840 m(8) = 4840 m(9) = 925. These Mamba numbers, were known to the American naval cryptanalysis unit Op-20-G in 1943 but the calculation must have been quite tedious. Working out that it was worth calculating at all …