2.8 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f4231000cf542c50ff35 | Problem 182: RSA encryption | 5 | 301818 | problem-182-rsa-encryption |
--description--
The RSA encryption is based on the following procedure:
Generate two distinct primes p
and q
.
Compute n=p*q
and φ=(p-1)(q-1)
.
Find an integer e
, 1 < e < φ
, such that gcd(e,φ) = 1
A message in this system is a number in the interval [0,n-1]
.
A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n-1]
).
To encrypt the text, for each message, m
, c=me mod n is calculated.
To decrypt the text, the following procedure is needed: calculate d
such that ed=1 mod φ
, then for each encrypted message, c
, calculate m=cd mod n.
There exist values of e
and m
such that me mod n = m.
We call messages m
for which me mod n=m unconcealed messages.
An issue when choosing e
is that there should not be too many unconcealed messages.
For instance, let p=19
and q=37
.
Then n=19*37=703
and φ=18*36=648
.
If we choose e=181
, then, although gcd(181,648)=1
it turns out that all possible messages
m (0≤m≤n-1)
are unconcealed when calculating me mod n.
For any valid choice of e
there exist some unconcealed messages.
It's important that the number of unconcealed messages is at a minimum.
For any given p
and q
, find the sum of all values of e
, 1 < e < φ(p,q)
and gcd(e,φ)=1
, so that the number of unconcealed messages for this value of e
is at a minimum.
--hints--
RSAEncryption
should be a function.
assert(typeof RSAEncryption === 'function')
RSAEncryption
should return a number.
assert.strictEqual(typeof RSAEncryption(19, 37), 'number');
RSAEncryption(19, 37)
should return 17766
.
assert.strictEqual(RSAEncryption(19, 37), 17766);
RSAEncryption(283, 409)
should return 466196580
.
assert.strictEqual(RSAEncryption(283, 409), 466196580);
RSAEncryption(1009, 3643)
should return 399788195976
.
assert.strictEqual(RSAEncryption(19, 37), 17766);
--seed--
--seed-contents--
function RSAEncryption(p, q) {
return true;
}
RSAEncryption(19, 37);
--solutions--
function gcd(a, b) {
if (b)
return gcd(b, a % b);
else
return a;
}
function RSAEncryption(p, q) {
let phi = (p - 1) * (q - 1);
let best = Number.MAX_SAFE_INTEGER;
let sum = 0;
for (let e = 0; e < phi; ++e) {
if (!(gcd(e, phi) == 1))
continue;
let msg = (gcd(p - 1, e - 1) + 1) * (gcd(q - 1, e - 1) + 1);
if (best == msg) {
sum += e;
} else if (best > msg) {
best = msg;
sum = e;
}
}
return sum;
}