update RSA

Cryptography/RSA.md

@@ -82,6 +82,47 @@ $n^{p-2} \equiv n^{-1}\ mod\ p$
 $m$ 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 
+```python
+pow(a,(p-1)/2,p)
+```
+
+[Finding the square root of integer a which is quadratic residue](http://mathcenter.oxford.emory.edu/site/math125/findingSquareRoots/)
+
+* Given $p \equiv 3\ mod\ 4$ the square root is calculated through
+```python
+pow(a,((p+1)//4),p)
+```
+
+## Tonelli-Shanks - Modular Square Root
+
+* Find elliptic curve co-ordinates
+* Precondition: modulus is not a prime
+* TBD
+
 ## Links
 
 * [Encryption+Decryption](https://www.cs.drexel.edu/~jpopyack/Courses/CSP/Fa17/notes/10.1_Cryptography/RSA_Express_EncryptDecrypt_v2.html)