ElGamal Public Key Cryptosystem

Use the Galois field array function, gf, to implement an ElGamal public key cryptosystem.

Key Generation

Define the polynomial degree, m.

m = 15;
q = 2^m;

Find a primitive polynomial and group generator. Set the random number generator seed to produce a repeatable result.

poly = primpoly(m,'nodisplay');

primeFactors = unique(factor(2^m-1));
rng(123456);
while 1
g = gf(randi([1,q-1]),m,poly);
isPrimitive = true;
for i = 1:length(primeFactors)
if g^((q-1)/primeFactors(i)) == gf(1,m,poly)
isPrimitive = false;
break;
end
end
if isPrimitive
break;
end
end

Construct private and public keys.

privateKey = 12;
publicKey = {g,g^privateKey,poly};

Encryption

Create and display the original message.

text = ['The Fourier transform decomposes a function of time (a signal)' newline ...
'into the frequencies that make it up, in a way similar to how a' newline ...
'musical chord can be expressed as the amplitude (or loudness) of' newline ...
'its constituent notes.'];
disp(text);
The Fourier transform decomposes a function of time (a signal)
into the frequencies that make it up, in a way similar to how a
musical chord can be expressed as the amplitude (or loudness) of
its constituent notes.

Convert the message to binary and group them every m bits. The message uses ASCII characters. Since the ASCII table has 128 characters, seven bits per character is sufficient.

bitsPerChar = 7;
binMsg = int2bit(int8(text'),bitsPerChar);
end
textToEncrypt = bit2int(binMsg,m);

Encrypt the original message.

cipherText = gf(zeros(length(textToEncrypt),2),m,poly);

for i = 1:length(textToEncrypt)
k = randi([1 2^m-2]);
cipherText(i,:) = [publicKey{1}^k, ...
gf(textToEncrypt(i),m,poly)*publicKey{2}^k];
end

Display the encrypted message.

tmp = cipherText.x;
disp(de2char(tmp(:,2),bitsPerChar,m));
vTchba~*TzEC> *o_c;a5
vS>Do7]{B#wDc0`YCDVxV]:FC0|o|	],7
S)d/uW
]A=]T	sKX+q3K&q ex_>f=C_gZ"*slX=N&\$~*[Xh+R[<5:(NpO6
8+@/)9sBm	&is#Z<DN`Qo~?GaOFIzYA~a+Lygzv?l

Decryption

Decrypt the encrypted original message.

decipherText = gf(zeros(size(cipherText,1),1),m,poly);
for i = 1:size(cipherText,1)
decipherText(i) = cipherText(i,2) * cipherText(i,1)^(-privateKey);
end

Display the decrypted message.

disp(de2char(decipherText.x,bitsPerChar,m));
The Fourier transform decomposes a function of time (a signal)
into the frequencies that make it up, in a way similar to how a
musical chord can be expressed as the amplitude (or loudness) of
its constituent notes.

Supporting Function

de2char converts the bits to char messages.

function text = de2char(msg,bitsPerChar,m)
binDecipherText = int2bit(msg,m);
text = char(bit2int(binDecipherText(1:end-mod(numel(binDecipherText), ...
bitsPerChar)),bitsPerChar))';
end