In addition to ciphers operating on binary bits and ciphers operating on letters in the conventional 26-letter alphabet, ciphers operating on base-3 and base-5 strings of symbols can be applied to the encryption of both bits and letters, but in different ways.
125, which is 5^3, is very close to 128, which is 2^7. And the difference between the two numbers is 3. On the other hand, 32, which is 2^5, minus 27, or 3^3, is 5. So, if a string of binary bits is broken into groups of seven and five bits, one can produce base-3 and base-5 symbols from it; the seven-bit chunks will favor base-5 symbols, and the five-bit chunks will favor, less strongly, base-3 symbols.
On the other hand, 25 is 5^2, and 27 is 3^3, and they bracket 26 on either side. These modified alphabets have long been used in manual fractionation systems. An interesting possibility for a computerized system would be to use the fact that 25*27 is 675, one less than 676, or 26 squared. Except, therefore, for choosing one digraph to be ignored, a string of letters could be converted into strings of base-5 and base-3 digits.
We have just seen an efficient coding which allows 47 binary bits to be represented as 10 letters. Since both bits and letters can be converted to symbols from an alphabet of 3 and from an alphabet of 5 for fractionation, one could divide a binary message into blocks of 47 bits, and convert half of them before fractionation and half after based on a pseudorandom sequence of bits.
The blocks of 47 bits which are still in binary form would be grouped for fractionation in a number of ways, from 5+5+5+5+5+5+5+5+7 to 7+7+7+7+7+7+5. (Unfortunately, more equal divisions are not available. However, this can be dealt with by considering the binary blocks in pairs whenever possible, even if they are not contiguous. A pair could then be handled as nine groups of five bits and seven groups of seven bits.) As noted above, the groups of seven bits usually become three base-5 digits, but occasionally become one base-3 digit, and the groups of five bits usually become three base-3 digits, but occasionally become one base-5 digit.
The blocks that are in alphabetic form are divided into pairs of letters, and all but one value would become two base-5 digits and three base-3 digits.
Then, after performing a polygraphic substitution or a transposition on the base-3 and base-5 digits we have obtained, we reconstitute the alphabetic or binary blocks. The base-3 digits and base-5 digits are allocated to the groups of 7 or 5 bits, and the groups of two letters, such that each is of exactly the same type as we started with. Thus, the information on which groups became the most common combinations of base-3 or base-5 digits, and which became the less common combinations, is preserved, allowing reversing the encryption process on those digits to enable the full reversal of the whole complex encryption step. (Symbol type can be considered to be a separate type of information as well, also subject to encipherment, but only by transposition, so that the other information may be placed within the output. Note that on decipherment, symbol type must be recalculated by repeating the encipherment step, not by attempting to perform the decipherment form of the transposition, since information on whether a bit belongs to a group of 5 bits or 7 bits is not present in the message.)
The following is an attempt at an illustration of the procedure.
First, we convert part of the binary message to alphabetic form.
10110100110001001110100101000101111010101111011 00000111010001011111010011010111010111011101101 0aaaa(14 bits/3 lt)(14 bits/3 lt)(14 bits/3 lt) A11101aaaabbbbo1010iiijjjcccc1011iiibbbbkkk B HZ T S O V N V 10110001111010001101101110011000001111010111011
Then, we take the message in its mixed form, and convert it to base-3 and base-5 symbols, using the different appropriate methods for its binary and alphabetic parts.
10110 1001100 01001 11010 01010 0010111 1010101 11101 1 CBA 312 BAA ACB CAC 423 153 AAC AB HZ TS OV NV 24CBC 13AAB 42CCA 51BAC 12ACB 1011 0001111 01000 1101101 1100110 0000111 10101 11011 ACC 451 5 345 122 553 BBB ACC
The base-3 and base-5 components of the message are then separated out, and enciphered independently. One effective method for doing so is to transpose, substitute, and then transpose again. The substitution can be done from a table; perhaps one with 625 entries for enciphering base-5 symbols, and one with 243 entries for enciphering base-3 symbols.
CONTINENTAL ----------- CBABAAACBCA CAACCBCAABC CABACACBACC BBBACC CBC CCCB ACC ACCC ACC AABB ABAC CAB BAAB BCAA BAA CBCCC CBACC ACCCA CCAAB BABAC CABBA ABBCA ABAA ABACB BACCA CBACB ABACC CABAB ACBAC BABAC ABAA IMPERIAL -------- ABACBBAC CACBACBA BACCCABA BACBACBA BACABAA ABBBA CBCBA ACBBB BCACA CAAA BAAAA ACCCC BACAB
DELUXE ------ 312423 153241 342511 245153 451225 53 313245 154453 31135 23251 42512 24152 3132 4515 4453 3113 5232 5142 5122 4152 1211 3154 2213 3512 4133 2133 1145 3421 PROPINQUITY ----------- 12113154221 33512413321 3311453421 324 232 145 151 133 111 513 233 221 434 11
Then, we complete the fractionation process by reconstituting a mixed binary and alphabetic message in the same form as the one with which we began, from the enciphered base-3 and base-5 symbols.
ABB 324 BAC BCB AAC 232 145 BBB 01110 0110100 11001 00011 11101 1001011 1110010 10101 0 15BCA 11CAC 33AAA 11BAA 15AAA LS TU VR QM NP CCC 132 3 322 143 411 CBA CAB 0110 1101001 11110 0100100 0011000 1011010 10110 01101Finally, we convert the remaining binary blocks to alphabetic, thus completing the armoring step of converting an initially all-binary message to an all-alphabetic form for transmission. Such a method should truly blur the boundaries between bits and letters.
A Table of Powers, useful for devising other elaborate methods of fractionation involving symbols from alphabets of different lengths.
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