Merge remote-tracking branch 'origin/main'

Author: OssieLin

README.md

@@ -1,7 +1,13 @@
-This project is under development, and it's focusing on the comparative analysis of different primality testing algorithms.\
+This project is the comparative analysis of primality testing algorithms.\
+\
+Modern cryptography algorithms like RSA need primality tests to generate large random primes.\
+Specifically, RSA uses the product of two prime numbers to generate the public and private keys; if these primes are predictable, it makes it easier for an attacker to break the encryption scheme.\
 \
-Modern cryptography algorithms like RSA or ElGamal need primality tests to generate large random primes.\
-For instance, RSA uses the product of two prime numbers to generate the public and private keys; if these primes are predictable, it makes it easier for an attacker to break the encryption scheme. 
 Large random primes are constructed in practice by generating pseudorandom numbers and then doing a primality test such as the Fermat primality test or something more powerful such as the Miller–Rabin primality test.\
 These primality tests fail for certain numbers called **pseudoprimes**.\
-My research focuses on analyzing the class of Perrin-type primality tests- **Lucas-Type Sequence**. Dougherty-Bliss and Zeilberger looked at these recently. They find that certain recurrent sequences produce primality tests for which the smallest pseudoprime is very large.
+My research focuses on analyzing the class of Perrin-type primality tests- **Lucas-Type Sequence**: \
+\
+$$V_{n+2} = c_1 V_{n+1} + c_0 V_{n}, \quad V_{0} = 2, \quad V_{1} = c_1$$\
+If p is prime, then we have the congruence $$V_{p}\equiv V_{1}(mod\quad p)$$
+\
+Dougherty-Bliss and Zeilberger looked at these recently. They find that certain recurrent sequences produce primality tests for which the smallest pseudoprime is very large.