RSA codes a Read the procedure for encryption and decryption

RSA codes:

a) Read the procedure for encryption and decryption of RSA codes and Briefly explain

i) why they select two prime numbers p and q and then form the number N = pq; why not just one prime p = N, or more than one primes, or even powers of one or more primes, to make the number N even larger?

ii) What is the significance of k, why do we look for k to be a positive number?

b) A person who wants to use RSA coding to receive messages from others announces N = 143 and encoder E = 7. This person receives an encrypted message 48. Find the decoder and use it to compute the original message.

c) You announce to the world that if they need to write you a secret message they should convert it to numbers and then use the encryptor N = 2021 and the public exponent E = 5 to convert their messages, and then send their converted messages to you. Try to go through this process step by step:

i) first pretend you are the sender: convert the message he and send it to the receiver (this is the encrypted R that is sent to the you)

ii) then pretend you are the receiver of the message, determine the values D and k, decipher the encrypted message, R, and make sure you recover the original message, that is \"he\".

Solution

a).

The reason is as follows,

If you don\'t guarantee it , the number of factors of the number will be huge, to break the algorithm you don\'t need to factorize into the exact factor they was used by the private key, but any factor other than 1*n will suffice, which means since most of the large numbers have many factors, it is easier. So the hardness depends on the smallest factor (which is in any case a prime)

Every number is nothing but a product of many primes, as the number of factors increase, it is easier to factorize the number. so essentially you want the largest \"factorizable\" number without exposing many factors, and hence the best number is a product of two primes.