RSA calls for Alice to pick two equal size large primes: P, and Q, then:
compute n=P*Q
compute Z = (p1)*(q1)
pick a secret encryption key e from the range: 0,1,2.....(n1) such that it is coprime with Z: gcd(Z,e)=1
find a corresponding decryption key, d to satisfy: e*d = 1 mod Z
define encryption over any plaintext number P, from the series 0,1,2....(n1), as C = P^{e}
decrypt C back to P by computing: P = C^{d}
So any number from 0,1,...(n1) can be encrypted to another number from the same set, and decrypted back to the original using two different keys: e and d.
To deduce e from d or vice versa one has to know the value of Z. To find Z one needs to find p and q, and for that one needs to factor n=pq, since n is in the open to practice both encryption and decryption.
To properly use RSA one needs to pick p and q sufficiently large to insure that the adversarial computing power will not be able to factor n in a timely manner.
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Fredrick Gauss and his followers managed to estimate the frequency of primes, and accordingly, we are assured of a sufficiently large supply for RSA keys...
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In 1903, Professor Cole at Columbia University was met with standing ovations when he managed a feat: finding the prime ingredients for the largest number that was ever decomposed to its two large primes: 2^{67}1:
2^{67}1 = 193,707,721 x 761,838,257,287 
