Consider an RSA key set with p=17, q=23, N=391, and e=3 (as...

Key Concepts

Modular Exponentiation

Modular exponentiation is the process of raising a number to an exponent while taking the remainder with respect to a modulus. This concept is central to RSA encryption and decryption, as both involve computing exponentiations modulo a composite number. It ensures that the operations yield results within the confined set of residues.

Modular Multiplicative Inverse

The modular multiplicative inverse of an integer e modulo ?(n) is the integer d such that e * d is congruent to 1 modulo ?(n). In RSA, finding this inverse is essential for determining the decryption key. The existence of d ensures that the encryption and decryption operations will inverse each other.

Euler Totient Function

The Euler totient function, typically denoted as ?(n), counts the number of integers up to n that are relatively prime to n. In the context of RSA, it is used to determine the order of the multiplicative group modulo n, and it is calculated for a product of two distinct primes as ?(n) = (p - 1)(q - 1). This value is critical for determining the private key.

RSA Algorithm

The RSA algorithm is a public-key cryptosystem that relies on the difficulty of factoring large composite numbers. It involves generating two keys: a public key for encryption, and a private key for decryption. The system operates by choosing two primes, computing their product, and then using modular arithmetic to enable secure information exchange.

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