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RSA

p * q = n
Coprime Phi is calculated either by Euler Totient or greatest common divisor via euclidean algorithm


1 < $\phi$ < n

There is also


$\phi$ = (p-1) * (q-1)

Encryption, public key e is a prime between 2 and phi


2 < e < $\phi$

possible_e = []
for i in range (2, phi):
    if gcd(n, i) == 1 and gcd(phi, i) == 1:
        possible_e.append()

Decryption, private key d


d * e mod $\phi$ = 1

possible_d = []
for i in range (phi + 1, phi + foo):
    if i * e mod phi == 1 :
       possible_d.append()

\( Cipher = msg ** d mod \phi \)
\( Cleartext = cipher ** e mod \phi )

Euklid

Just a short excourse:
A greatest common divisior out of an example a = 32 and b = 14 would be the groups of the following divisors

a = 32, b = 24
a = {1, 2, 4, 8, 16}
b = {1, 2, 3, 8, 12}
gcd(a,b) = 8

Greatest Common Divisor (GCD)

Two values are prime and have themselves and only 1 as a divisor are called coprime. To check if a and b have a greatest common divisor do the euclidean algorithm.

def gcd(a, b):
    if b == 0:
        return a
    return gcd(b, a % b)

Extended GCD

#TODO

Fermat's Little Theorem

If modulus p is a prime and and modulus n is not a prime, p defines a finite field (ring).


n \in F_{p} \{0,1,...,p-1\}

The field consists of elements n which have an inverse m resulting in n + m = 0 and n * m = 1.

So , n^p - n is a multiple of p then n^p \equiv n\ mod\ p and therefore n = n^p\ mod\ p. An example


4 = 4^{31}\ mod\ 31

Further, p while still a prime results in 1 = n^{p-1} mod\ p. An example


1 = 5^{11-1}\ mod\ 11

Modular Inverse

Coming back to the modular inverse n, it can be found in the following way n^{p-1} \equiv 1\ mod\ p
n^{p-1} * n^{-1} \equiv n^{-1}\ mod\ p
n^{p-2} * n * n^-1 \equiv n^{-1}\ mod\ p
n^{p-2} * 1 \equiv n^{-1}\ mod\ p
n^{p-2} \equiv n^{-1}\ mod\ p

Quadratic Residue

m is a quadratic residue when \pm n^2 = m\ mod\ p with two solutions. Otherwise it is a quadratic non residue.

So a porperty of quad res are, if Quadratic Residue QR = 1 and Quadratic NonResidue QN = -1


QR * QR = QR\\
QR * QN = QN\\
QN * QN = QR\\

Legendre


\frac{a}{p} =
\begin{cases}
1, & if\ a\ quadratic\ residue\ mod\ p\ and\ not\ a\ \equiv\ 0\ (mod\ p),\\
-1, & if\ a\ is\ a\ non\ residue\ mod\ p,\\
0, & if\ a\ \equiv 0\ (mod\ p)\\
\end{cases}


\frac{a}{p} \equiv a^{p-1/2}\ (mod\ p)\ and\ \frac{a}{p} \in \{-1,0,1\}

Legendre Symbol test via Python with

pow(a,(p-1)/2,p)

Finding the square root of integer a which is quadratic residue

Given p \equiv 3\ mod\ 4 the square root is calculated through

pow(a,((p+1)//4),p)

Tonelli-Shanks - Modular Square Root

Find elliptic curve co-ordinates
Precondition: modulus is not a prime
TBD

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RSA