About RSA Algorithms

About RSA Algorithms

--Remembering the "Eternal Blue" event

RSA encryption and decryption are all done within the integer ring $z_n$.

Set plaintext $x$ and ciphertext $y$? is the element within the $z_n$, which is encrypted using the public key as:

• Given the public key $ (n,e) $ and plaintext $x$, then ciphertext $y=x^e (mod n) $, where $x,y \in z_n$.

Decryption with the private key can be expressed as:

• Given the private key $ (n,d) $ and ciphertext $y$, the plaintext $x=y^d (mod n) $, where $x,y \in z_n$.

Typically, $x $, $y $, $n $ and $d$ are very large numbers. $e $ is sometimes referred to as the public index, $d $ is called the secrecy index.

The following is the procedure for calculating the public key $ (n,e) $ and the private key $ (n,d) $ in the RSA Cryptography system:

1. Select two large integers $p$ and $q$.
2. Calculates $n=p\times q$.
3. Calculates $\varphi (n) = (p-1) \times (q-1) $.
4. Random selection satisfies the public exponent of $ (E,\varphi (n)) =1$ $e \in \{1,2,..., \varphi (n) -1\}$.
5. The calculation satisfies the confidentiality index $d$ of $e \times d \equiv 1 (mod n) $.

Prove the feasibility of the RSA scheme:

The condition $ (e,\varphi (n)) =1$ guarantees that there is an inverse of $e$ in $z_{\varphi (n)}$, that is, the secret index $d$ must exist.

Set $e \times d=k \times \varphi (n) +1$, according to the encryption formula and decryption formula and Euler theorem,

There are $x \equiv y^d \equiv (x^e) ^d \equiv x^{ed} \equiv x^{k\varphi (n) +1} \equiv X^{k\varphi (n)} \times x \equiv x (mod n) $. Certificate of Completion.

About RSA Algorithms