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  "body": "\n## Example\n\nThe RSA cryptosystem is an example of [[public-key cryptography|01E9MNJYEX]].\nTo make the keys, choose two distinct primes $p$ and $q$, and put $n = pq$ and $\\varphi(n) = (p - 1)(q - 1)$, the value of [[Euler's totient function|0VV04M41AN]].\nChoose an integer $e$ with $1 \u003c e \u003c \\varphi(n)$ and $\\gcd(e, \\varphi(n)) = 1$. Let $d$ be the [[inverse|00GXTQMY8Z]] of $e$ modulo $\\varphi(n)$.\nThe public key is $(n, e)$, and the private key is $d$.\nTo encrypt a message $m$ with $0 \\le m \u003c n$, compute $c = m^e \\bmod n$. To decrypt, compute $c^d \\bmod n$, which is equal to $m$.\nA small case: $p = 5$, $q = 11$, $n = 55$, $\\varphi(n) = 40$, $e = 3$ and $d = 27$, since $3 \\cdot 27 = 81 \\equiv 1 \\pmod{40}$.\nThe message $m = 2$ gives $c = 2^3 = 8$, and $8^{27} \\equiv 2 \\pmod{55}$.\n\n## Remarks\n\nDecryption works by [[Euler's theorem|0VC4DVE7DQ]]: $ed = 1 + k \\varphi(n)$, so $m^{ed} \\equiv m \\pmod n$ when $\\gcd(m, n) = 1$. The general case follows from the [[Chinese remainder theorem|0FNN9B4W64]].\nThe security rests on the belief that it is hard to factor $n$ when $p$ and $q$ are large.\nRon Rivest, Adi Shamir and Leonard Adleman described the system publicly in 1977.\n\n## Explanations\n\n### L1 human\n\nAnyone can raise a message to the power $e$ modulo $n$. To undo this, you need $d$, and to find $d$ you need the factors of $n$.\nMultiplying two large primes is easy, but recovering them from the product is believed to be very hard.\n",
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