Let p=5 and q=7 so that m=35, and let e=11. Find d= e^-1 φ(m). Then let x=22 and compute y=x^e 3

Step 1: Calculate $u005Cvarphi(m)$ where $m u003D pq u003D 35$ (since $p u003D 5$ and $q u003D 7$). The function $u005Cv

Let p=5 and q=7 so that m=35, and let e=11. Find d= e^-1...

Key Concepts

RSA Algorithm

The RSA algorithm is a widely used public-key cryptosystem that relies on the difficulty of factoring large prime numbers. It involves generating a public key and a private key using prime numbers, computing Euler’s Totient, and using concepts like modular multiplicative inverses and modular exponentiation to encrypt and decrypt messages securely.

Modular Exponentiation

Modular exponentiation is the process of finding the remainder when an integer x raised to an exponent e is divided by a modulus m. This operation is efficient and fundamental in many cryptographic algorithms, including RSA, where large powers must be computed under a modular constraint without dealing with excessively large numbers.

Euler’s Totient Function

Euler’s Totient Function, often denoted as ?(m), measures the number of integers less than m that are coprime to m. It is crucial in number theory and is particularly used in cryptography, as seen in RSA, for determining properties of modular arithmetic groups and for computing the modular multiplicative inverse of the encryption exponent.

Modular Multiplicative Inverse

The modular multiplicative inverse of a number e modulo ?(m) is an integer d such that the product e·d is congruent to 1 modulo ?(m). This concept is central to RSA cryptography because it allows the decryption key (d) to be computed from the encryption key (e) ensuring that the encryption and decryption operations are reversible.

Let $p=5$ and $q=7$ so that $m=35$, and let $e=11$. Find $d=$ $e^{-1} \bmod \varphi(m)$. Then let $x=22$ and compute $y=x^e \bmod 35$ and $z=$ $y^d \bmod m$.

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