In Part1, we have shown you the proof of Euler’s Theorem and Fermat’s little Theorem. And in Part2, I would show you how they would work in cryptography and ssh(secure shell protocol).
RSA
RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym RSA comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent system was developed secretly, in 1973 at GCHQ (the British signals intelligence agency), by the English mathematician Clifford Cocks. That system was declassified in 1997.
Encryption
Firstly, I wanna show you Encryption Process of RSA.
- Select a pair of prime numbers $(p,q)$, which are usually pretty large prime numbers ($\geq 10^{16}$).
- Let $n = p \cdot q$.
- Select a number $e$, which meets condition $\gcd(e,(p-1)\cdot (q-1)) = 1$.
- Based on above manipulation, we have generated Public Key $(e, n)$.
- Select a number $d$, which meets condition $d\cdot e\equiv1(\mod (p-1)\cdot (q-1))$.
- Such that we have also generated Private Key $(d, n)$.
- And Encryption Process could be denoted as:
$$m’ = rem(m^e, n)$$
Above Notations: $m’$ means encyption message, and $m$ means original message.
Decryption
The Decryption Process could be denoted as:
$$m = rem((m’)^{d}, n)$$
The Decryption and Encryption seems so simple that make us not help to consider : “Is RSA really secure enough?”.
The answer of this question is obvious, because almost all entities in our world utilize RSA to encrypt their data, even including Bank and Government. However, just knowing the answer of this question seems not adequate, so I would show you why RSA is reliable in mathematics.
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