p and g. In reality, however, there are mil- For each of these public keys, the

Question

p and g. In reality, however, there are mil- For each of these public keys, the primes p and g 2, e2) such that b. The RSA cryptosystem is secure given tacker to factor N in order to determine the primes public lions of public keys (N,e) on the Internet. Fo ncrators can be buand (N2,e2) Sucken are generated at random, and random-number gee(N1,e1 can find on the Internet many thousands of pairs of put Ni and N2 share use given N pg, it is computationally hard for an at- public kegys (wi.ems e bb a common factor. In this situation, both cryptosystems can be easily bro Suppose that N p and N2 p2, where p.q1,92 of digits long). Given Ni and N2, suggest an efficient nct large primes (each hundreds N2. sugest an eiche 9z are distinct large primes (each h integers m and n, with m > n, it is computationally easy to compute the re- n, even if both m and n are very large (hundreds of digits long). mainder of m modulo

Explanation / Answer

RSA algorithm is asymmetric cryptography algorithm, which simply means is works on two different keys, i.e. one Private key and one Public Key. The idea behind the algorithm is based on the fact that it is difficult to factorize a large number, for the problem mentioned:

N1=pq1 and N2=pq2

Given, p, q1, q2 are sufficiently large prime numbers, the best way to factor both N1 and N2 is as follows:

Step 1) Make a list between 1 and N and remove the common factors, shown

1, 2, 3, 4, 5, 6, ..., 573, ..., N1 ........ (1)

1, 2, 3, 4, 5, 6, ...., 579, ..., N2 ........(2)

Now, No common factors with N1 and N2 from equatiions (1) and (2), respectively

also, calculate a third variable length, L=(p-1)*(q-1), as

L1 = (p-1) * (q1-1) .........(3)

similarly,

L2 = (p-1) * (q2-1) ........(4)

Step 2) Now, pick up the encryption key, the simplest rules for picking up the encryption key is:

Coprime (L1, N1) and Coprime (L2, N2)

Thus, the keys obtained can be used as factors for the same, and for the large primes (hundreds or thousands of digits long), the common factor problem can be avoided.

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