A Primorial Approach for Generating Primes Illinois Wesleyan Universtiy David Lopez | Yash Thacker

PPAGP Problem | Motivation Primorials and Computational Time Generalized Algorithm Comparison Primorials have intrigued Primorial Primes The PPAGP Algorithm generates mathematicians for centuries primes of a given bit size by using Standard method: because of their historical Schinzel’s conjecture to find A primorial is defined as PPAGP Algorithm: significance when used by Euler generalized primorial primes. It the product of the first n to prove that there are an infinite loops through primes : number of primes. Fascinated by Pseudocode: this, we decided to look into their 1. Generate nth multiple of a properties, and we discovered a primorial given the desired bit way of Generating prime size with the form : numbers of a given bit size. This 2k and call it “P” has important applications in Generalized primorial 2. while: “P” does not equal prime: many fields.. Prime numbers of a primes are defined as 1. add the (nth +1) prime or desired bit size , they are prime numbers that are a 1 if the number of the loop is essential for making prime away from a 1. cryptography keys for secure multiple of a primorial. 2. if “P” is prime: data transmission in return “P” cryptosystems like the RSA. else “n” = “n”+1

Schinzel’s Current Solution Conjecture Conclusions

To generate prime numbers given Schinzel predicts that: The PPAG algorithm generates primes ~90% faster a bit size, algorithms generate than the current current solution for this problem. random numbers within given Currently, it can be applied to key-exchange protocols bounds and test for primality until a if: that do not require randomness, like the number is found. Theory predicts Diffie-Hellman key exchange protocol. Moreover, that the number of primality calls is future work could include looking for methods to proportional to the bit size. or randomize the algorithm, by looking at modifying the k . coefficient 2 Pseudocode: while: “p” does not equal prime: References: 1. Generate random number such This predicts that we will be that “p” is “n” bits able to find a prime near a Stephen P, Richards, A Number For Your Thoughts, 1982, p. 2. if “p” is prime: primorial within a range that is 200. return “p” always one or a prime away S.W. Golomb Evidence of Fortunes conjecture. Mathematics Magazine, Vol. 54, No. 4 (Sep., 1981), pp. 209-210. from the primorial 1