<Text-field layout="Heading 1" style="Heading 1">Monalphabetic substitution ciphers</Text-field>

Cryptography

with(StringTools):

toNumbers:=message-> map(x->x-96,map(Ord,(Explode(Select(IsLower,LowerCase(message)))))):

toNumbers("Affine cipher");

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modpositive:=(m,n)->modp(m-1,n)+1:

map(x->modpositive(9*x+18,26),[1, 6, 6, 9, 14, 5, 3, 9, 16, 8, 5, 18]);

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toLetters:=numbers->UpperCase(Join(map(Char,map(x->x+96,numbers)),"")):

toLetters([1, 20, 20, 21, 14, 11, 19, 21, 6, 12, 11, 24]);

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LowerCase(toLetters(map(x->modpositive(3*(x-18),26),toNumbers("ATTUNKSUFLKX"))));

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<Text-field layout="Heading 1" style="Heading 1">The Pohlig-Hellman cipher and the discrete logarithm problem</Text-field>

toNumbers("polygraphic");

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n := 123456789: for i from n while not(isprime(n)) do n:=n+1: end do: print(n);

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gcd(123,123456790);

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map(x->modp(x&^123,123456791),[16151225, 07180116, 080903]);

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msolve({123*rprime=1},123456790);

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map(x->modp(x&^50185687,123456791),[85925373, 3137134, 6901637]);

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toLetters([16,15,12,25,7,18,1,16,8,9,3]);

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<Text-field layout="Heading 1" style="Heading 1">The Massey-Omura exchange</Text-field>

a:=123456789:

p:=41889443039:

s:=85479892289:

gcd(s,p-1);

NiMiIiI=

a1:=modp(a&^s,p);

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r:=46828919:

gcd(r,p-1);

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a2:=modp(a1&^r,p);

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msolve({s*sprime=1},p-1);

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a3:=modp(a2&^32914471425,p);

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msolve({r*rprime=1},p-1);

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modp(a3&^32579883087,p);

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<Text-field layout="Heading 1" style="Heading 1">The RSA algorithm</Text-field>

p:=115518123229:

q:=8445806041:

n:=p*q;

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m:=(p-1)*(q-1);

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r:=875189387279:

gcd(m,r);

NiMiIiI=

modp(123456789&^r,n);

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msolve({r*rprime=1},m);

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modp(164186964938094304597&^305536865450871050159,n);

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<Text-field layout="Heading 1" style="Heading 1">Notes</Text-field>

<Text-field layout="Heading 2" style="Heading 2">Computing powers mod <Font italic="true">p</Font></Text-field>

ourpowermod:=proc(a,k,n) local b,c,ktemp,r; b:=1; c:=a; ktemp:=k; while ktemp>0 do r:=modp(ktemp,2); if r=1 then b:=modp(b*c,n) end if; c:=modp(c^2,n); ktemp:=(ktemp-r)/2; end do; return(b); end proc:

ourpowermod(123456789, 85479892289, 41889443039);

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