with(linalg):with(StringTools):
<Text-field style="Heading 1" layout="Heading 1">Cryptanalysis tools</Text-field>Commands are given for counting letter frequencies, computing the index of coincidence, decimating, or selecting "columns," of polyalphabetic crypts for frequency or I.C. analysis, and computing sums of products of frequencies for matching alphabets.squeeze: a procedure that squeezes out all characters but the letters in an input text.squeeze := proc(msg)
local alph, ans, i, c;
alph:="abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";
ans := "";
for i from 1 to length(msg) do
c := substring(msg, i..i);
if searchtext(c,alph)>0 then ans := cat(op(ans),c) fi;
od;
RETURN(ans);
end:
The freqlist procedure produces a list of the frequencies of the letters in the input message in standard order.freqlist := proc(msg)
local ALPH, ms, i, j, c, freq;
ALPH := "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
ms := UpperCase(squeeze(msg));
freq := [];
for j from 1 to 26 do freq := [op(freq), 0]; od;
for i from 1 to length(ms) do
c := substring(ms, i..i);
j := searchtext(c, ALPH);
freq[j] := freq[j]+1;
od;
RETURN(freq);
end:
The freqdisplay procedure displays the table of frequencies of letters in the input message.freqdisplay := proc(msg)
local ALPH, ans, j, f;
ALPH := `ABCDEFGHIJKLMNOPQRSTUVWXYZ`; ans := array(1..2,1..26); f := freqlist(msg);
for j from 1 to 26 do
ans[1,j] := substring(ALPH, j..j);
ans[2,j] := f[j];
od;
RETURN(evalm(ans));
end:
IC calculates the index of coincidence of the input text.IC := proc(msg)
local f, N, num, i;
f := freqlist(msg);
N := 0; num := 0;
for i from 1 to 26 do
N := N + f[i];
num := num + f[i]*(f[i]-1);
od;
RETURN(evalf(num/(N*(N-1))));
end:
`indexofcoincidence` calculates the index of coincidence of the input text.indexofcoincidence := proc(msg)
local f, N, num, i;
f := freqlist(msg);
N := 0; num := 0;
for i from 1 to 26 do
N := N + f[i];
num := num + f[i]*(f[i]-1);
od;
RETURN(evalf(num/(N*(N-1))));
end:
The decimate procedure takes the input text and produces a list of k decimations of the letters of the text at interval k, starting at the first, second, ..., and k'th letters.For example, the command colum := decimate(msg, 5)yields five lists (columns for polyalphabetic decryption),colum[1], colum[2], colum[3], colum[4], and colum[5].decimate := proc(msg, k)
local ms, dec, i, j, L, subs;
ms := squeeze(msg); L := length(msg); subs := [];
for i from 1 to k do
dec := "";
for j from i to L by k do
dec := cat(dec, substring(ms,j..j));
od;
subs := [op(subs), dec];
od;
RETURN(subs);
end:
The ICs procedure calculates the indexes of coincidence for each of the k columns of the input text when that text is arranged in k columns.ICs := proc(msg, k)
local cols, i, ans;
cols := decimate(msg,k);
ans := [];
for i from 1 to k do
ans := [op(ans), IC(cols[i])];
od;
RETURN(ans);
end:
`sumf1f2` calculates the sum of products of corresponding frequencies in two input frequency lists of length 26.sumf1f2 := proc(freq1,freq2)
local s, i;
s := 0;
for i from 1 to 26 do
s := s + freq1[i]*freq2[i];
od;
RETURN(s);
end:
The sumf1f2all procedure calculates all 26 sums of products of frequencies for one frequency distribution shifted against another.See p. 85 of Sinkov's text.sumf1f2all := proc(freq1, freq2) local i, j, f1, f2, ans; ans := array(1..2,1..26); for j from 1 to 26 do ans[1,j] := j-1; ans[2,j] := 0; for i from 1 to 26 do ans[2,j] := ans[2,j] + freq1[i]*freq2[(i-1+j-1 mod 26) +1]; od; od; RETURN(evalm(ans));end:
<Text-field style="Text" layout="Heading 1">Example: Vigenere crypt from Barr's example 2.5.3</Text-field>
<Text-field style="Heading 1" layout="Heading 1"></Text-field># The Vigenere crypt of Barr example 2.5.3 is considered.barrex := "CTMYR DOIBS RESRR RIJYR EBYLD IYMLC CYQXS RRMLQ FSDXF OWFKT CYJRR IQZSM X";freqdisplay(barrex);ICs(barrex, 1);ICs(barrex, 2);ICs(barrex, 3);colum := decimate(barrex, 3);freq1 := freqlist(colum[1]);freq2 := freqlist(colum[2]);freq3 := freqlist(colum[3]);sumf1f2all(freq1, freq2);sumf1f2all(freq1, freq3);sumf1f2all(freq2, freq3);