Cryptography

Substitution Method

with(StringTools):

alphabet:="abcdefghijklmnopqrstuvwxyz "; cibet:="NOPQKGUVHIDEFMZXYSLJTWCRAB ";

UTxhYmNkZWZnaGlqa2xtbm9wcXJzdHV2d3h5en42Ig==

UTxOT1BRS0dVVkhJREVGTVpYWVNMSlRXQ1JBQn42Ig==

plai:="Cryptography is the science of writing in secret code and is an ancient art; the first documented use of cryptography in writing dates back to circa 1900 B.C. when an Egyptian scribe used non-standard hieroglyphs in an inscription. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications ranging from diplomatic missives to war-time battle plans. It is no surprise, then, that new forms of cryptography came soon after the widespread development of computer communications. In data and telecommunications, cryptography is necessary when communicating over any untrusted medium, which includes just about any network, particularly the Internet.";

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

plai:=LowerCase(plai);

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s1:=Explode(cibet); s2:=sort(s1);

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CharacterMap(alphabet,cibet,plai);

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encode:=plaintext->CharacterMap(alphabet,cibet,plaintext);

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u:=encode(plai); decode(%);

Exercise: Create a function that deciphers a given message.

decode:=citext->CharacterMap(cibet,alphabet,citext);

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Explode(Sort((u)));

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What are the weaknesses of substitution method?

Caesar cipher

ASCII and the convert command.

convert("mat331",bytes);

NygiJDQiIiMoKiIkOyIiI15GJiIjXA==

convert([100,110,123],bytes);

USRkbnxmcjYi

Exercise: Using Matrix, create a maple command that displays a matrix of all ASCII characters equivalent to numbers 0 to 127,

matrix([[seq(i,i=0..9)],seq([10*j,seq(convert([10*j+i],bytes),i=0..9)],j=0..12)]);

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?Matrix

Exercise: Create a function that takes a list of integers and returns a list of the cubes of those integers.

f:=pts->[seq(pts[i]^3,i=1..nops(pts))];

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f([2,4,5,60,2]);

NyciIikiI2siJEQiIicrZ0BGIw==

map(x->x^3,[2,4,90,0,23]);

NyciIikiI2siJyshSCgiIiEiJm5AIg==

g:=pts->map(x->x^3,pts);

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g([1,2,35]);

NyUiIiIiIikiJnZHJQ==

Make a procedure that takes a plain message and

* converts each of its characters into the ASCII equivalent,

* adds a number n to each of the number of the characters

* converts the list of numbers into the ASCII equivalent characters.

How do we decipher the ciphertext?

This is not a good cryptosystem, find the flaws.

message:="this is our message";

UTR0aGlzfmlzfm91cn5tZXNzYWdlNiI=

mylist:=convert(message,bytes);

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n:=5;

IiIm

mySecondList:=map(x->x+5,mylist);

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convert(mySecondList,bytes);

UTR5bW54JW54JXR6dyVyanh4ZmxqNiI=

FIRST TRIAL (but it is not a correct encoding function)

enc:=proc(plaintext,n) convert(plaintext,bytes); map(x->x+n,%); convert(%,bytes); end proc;

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enc("hi z",20);

USd8Z3J8aHI0fGp5fGp2fGh2NiI=

Review: Modular arithmetic and the division algorithm

Exercise: Describe all the solutions of the following congruences:

1. xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEoJmVxdWl2O0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDRi8=23 (mod 7) 2. 4xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEoJmVxdWl2O0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZDRi8=23 (mod 11)

(optional ) 3. 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