diff --git a/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-182-rsa-encryption.md b/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-182-rsa-encryption.md index df95c1f9f0..7ffc6ae376 100644 --- a/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-182-rsa-encryption.md +++ b/curriculum/challenges/english/10-coding-interview-prep/project-euler/problem-182-rsa-encryption.md @@ -10,16 +10,59 @@ dashedName: problem-182-rsa-encryption The RSA encryption is based on the following procedure: -Generate two distinct primes p and q.Compute n=pq and φ=(p-1)(q-1). +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` -Find an integer e, 1<e<φ, such='' that='' gcd(e,φ)='1.' a='' message='' in='' this='' system='' is='' number='' the='' interval='' \[0,n-1].='' text='' to='' be='' encrypted='' then='' somehow='' converted='' messages='' (numbers='' \[0,n-1]).='' encrypt='' text,='' for='' each='' message,='' m,='' c='me' mod='' n='' calculated.='' decrypt='' following='' procedure='' needed:='' calculate='' d='' ed='1' φ,='' c,='' m='cd' n.='' there='' exist='' values='' of='' e='' and='' me='' call='' which='' unconcealed='' messages.='' an='' issue='' when='' choosing='' should='' not='' too='' many='' instance,='' let='' p='19' q='37.' φ='18\*36=648.' if='' we='' choose='' then,='' although='' gcd(181,648)='1' it='' turns='' out='' all='' possible='' messagesm='' (0≤m≤n-1)='' are='' calculating='' any='' valid='' choice='' some='' it's='' important='' at='' minimum.='' find='' sum='' e,='' 1<e<φ(1009,3643)='' so='' value='' <='' section=''></e<φ,> +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-- -`euler182()` should return 399788195976. +`RSAEncryption` should be a function. ```js -assert.strictEqual(euler182(), 399788195976); +assert(typeof RSAEncryption === 'function') +``` + +`RSAEncryption` should return a number. + +```js +assert.strictEqual(typeof RSAEncryption(19, 37), 'number'); +``` + +`RSAEncryption(19, 37)` should return `17766`. + +```js +assert.strictEqual(RSAEncryption(19, 37), 17766); +``` + +`RSAEncryption(283, 409)` should return `466196580`. + +```js +assert.strictEqual(RSAEncryption(283, 409), 466196580); +``` + +`RSAEncryption(1009, 3643)` should return `399788195976`. + +```js +assert.strictEqual(RSAEncryption(19, 37), 17766); ``` # --seed-- @@ -27,16 +70,44 @@ assert.strictEqual(euler182(), 399788195976); ## --seed-contents-- ```js -function euler182() { +function RSAEncryption(p, q) { return true; } -euler182(); +RSAEncryption(19, 37); ``` # --solutions-- ```js -// solution required +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; +} ```