Some applications of high-speed computers to the case 𝑛=2 of Algebraic Cryptography
Jack Levine · Mathematics of Computation · 1961
Introduction. In 1929 Hill [1] proposed the use of simultaneous linear congruences as a method of encipherment (see also [2], [3]).If the number, n, of such congruences be 5 or more this results in a cryptographic system of unusual security.In this article it is shown that high-speed computers can be used in the problem of the decipherment of the simplest case n = 2.We give first a brief description of the system using this value of n.The 26 letters of the alphabet are assigned numerical values according to some arrangement of the numbers 0, 1, 2, •• -, 24, 25.For example: ABCDEFGHI JKLMNOPQ 19 2 21 0 4 7 6 9 17 24 11 15 14 13 12 16 18 (1.1)RSTUVWXYZ 1 25 20 3 22 5 8 23 10 A 2 x 2 involutory matrix, mod 26, is selected to form the congruences (L2) C2 = cPl + dP2 mod 26 where the matrix is as) *-[: i].*•-'-[!?]■ mod26-As an illustration we use (1.4) H = [l ¿].A given plain-text, say CRYPTOGRAPHY, is then divided into two-letter groups, CR YP TO GR AP HY; each pair of letters is selected as the PiP2 of (1.2), and C'i, C2 calculated, using the numerical equivalents of (1.1).Thus, as C R = Pi P2 = 21 1, we find, using (1.4), d =" 4(21) + 7(1) m 13 = N, C% =-9(21) + 22(1) m 3 = U, so CR is enciphered by NU.The converse is also true, since matrix H is involutory.The complete encipherment becomes CR YP TO GR AP HY (1.5) NU VN XB WJ GU LLThe decipherment, knowing the matrix, is performed in an identical manner.