Determine which of the following statements are True or Fals

Determine which of the following statements are True or False If a, n elementof Z, and gcd(a, n) = 1, and a^n mod n = a then n is prime. There are 216 numbers less than 247: that are relatively prime to 247. If a^x mod n =1 where gcd(a, n) = 1 and n is prime then x is the order of a In RSA algorithm, it is possible that the public key = private key.

Solution

GCD:

GCD (Greatest Common Divisor) or HCF (Highest Common Factor) of two numbers is the largest number that divides both of them.

Relatively Prime:

Two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the only common positive factor of the two numbers is 1.

Two numbers for which the only common factor is 1. In other words, relatively prime numbers have a greatest common factor (gcf) of 1.

There are 216 numbers less than 247, that are relatively prime to 247:

False: 247 is not a prime number. The numbers which are less than 247, their gcf is not 1.

RSA algorithm:

RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and differs from the decryption key which is kept secret.

RSA is a cryptosystem for public-key encryption, and is widely used for securing sensitive data, particularly when being sent over an insecure network such as the Internet.

In RSA algorithm,it is possible that public key= private key

True

Public-key cryptography, also known as asymmetric cryptography, uses two different but mathematically linked keys, one public and one private. The public key can be shared with everyone, whereas the private key must be kept secret. In RSA cryptography, both the public and the private keys can encrypt a message; the opposite key from the one used to encrypt a message is used to decrypt it.