<< problem 181 - Investigating in how many ways objects of two ... | Maximum product of parts - problem 183 >> |
Problem 182: RSA encryption
(see projecteuler.net/problem=182)
The RSA encryption is based on the following procedure:
Generate two distinct primes p and q.
Compute n = pq and phi = (p-1)(q-1).
Find an integer e, 1 < e < phi, such that gcd(e, phi)=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 = m^e mod n is calculated.
To decrypt the text, the following procedure is needed: calculate d such that ed = 1 mod phi, then for each encrypted message, c, calculate m = c^d mod n.
There exist values of e and m such that m^e mod n = m.
We call messages m for which m^e 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 phi = 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 m^e 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.
Choose p = 1009 and q = 3643.
Find the sum of all values of e, 1 < e < phi(1009, 3643) and gcd(e, phi)=1, so that the number of unconcealed messages for this value of e is at a minimum.
My Algorithm
This problem bugged me for a while. However, searching the internet for "rsa fixed points" brought up a link to a discussion on StackOverflow
(see math.stackexchange.com/questions/1298664/rsa-fixed-point) which contained a simple formula:
(1 + gcd(e-1, p-1)) * (1 + gcd(e-1, q-1))
All I have to do is to enumerate all encrypted message (called encrypted
in my code) in a for
-loop:
- if the formula returns a value I already had then add
encrypted
tosum
- else "reset"
sum
to the current value ofencrypted
Note
The programming part isn't very exciting. Due the huge number of calls to isCoprime
(and gcd
) I researched ways to speed up these functions.
isCoprime
becomes about 30% if I check the lowest bits of its parameters: if both are zero then the parameters are both even and can't be coprime- using special CPU instructions (
ctz
on Intel CPUs) givesgcd
a performance boost of about 30% as well
#ifdef GNUC
because Visual C++ has differed names and calling conventions for their intrinsicsI updated my toolbox accordingly.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "19 37" | ./182
Output:
Note: the original problem's input 1009 3643
cannot be entered
because just copying results is a soft skill reserved for idiots.
(this interactive test is still under development, computations will be aborted after one second)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#include <iostream>
#include <numeric>
// I optimized the following two routines which originated in my toolbox
// greatest common divisor
template <typename T>
T gcd(T a, T b)
{
// C++17 will have std::gcd(a, b) which is most likely as fast as my implementation
#ifdef __GNUC__
// adopted from https://lemire.me/blog/2013/12/26/fastest-way-to-compute-the-greatest-common-divisor/
// and https://github.com/lemire/Code-used-on-Daniel-Lemire-s-blog/blob/master/2013/12/26/gcd.cpp
if (a == 0)
return b;
if (b == 0)
return a;
// MSVC++: _BitScanForward intrinsic instead
auto shift = __builtin_ctz(a | b);
a >>= __builtin_ctz(a);
do
{
b >>= __builtin_ctz(b);
if (a > b)
std::swap(a, b);
b -= a;
} while (b != 0);
return a << shift;
#else
// standard GCD
while (a != 0)
{
T c = a;
a = b % a;
b = c;
}
return b;
#endif
}
// return true if a and b are coprime
template <typename T>
bool isCoprime(T a, T b)
{
// fast reject if both are even (=> gcd(a,b) >= 2)
if (((a|b) & 1) == 0)
return false;
return gcd(a, b) == 1;
}
int main()
{
// problem's constants
unsigned int p = 1009;
unsigned int q = 3643;
std::cin >> p >> q;
// compute phi according to problem statement
auto phi = (p - 1) * (q - 1);
// lowest number of unconcealed messages, start with highest possible value
unsigned int best = 0xFFFFFFFF;
// sum of all e when best is minimized
unsigned long long sum = 0;
// iterate over all messages
for (unsigned int encryption = 0; encryption < phi; encryption++)
{
// must be coprime
if (!isCoprime(encryption, phi))
continue;
auto badP = gcd(p - 1, encryption - 1) + 1;
auto badQ = gcd(q - 1, encryption - 1) + 1;
auto numPlaintext = badP * badQ;
// same number of unconcealed messages ? add all of them
if (best == numPlaintext)
sum += encryption;
// improved (=lower) number of unconcealed messages ? reset sum
else if (best > numPlaintext)
{
best = numPlaintext;
sum = encryption;
}
}
// display result
std::cout << sum << std::endl;
return 0;
}
This solution contains 16 empty lines, 18 comments and 5 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.20 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL
)
See here for a comparison of all solutions.
Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL
.
Changelog
September 3, 2017 submitted solution
September 3, 2017 added comments
Difficulty
Project Euler ranks this problem at 60% (out of 100%).
Links
projecteuler.net/thread=182 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p182.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/182.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/182.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/182.cpp (written by Stephen Macke)
C github.com/LaurentMazare/ProjectEuler/blob/master/e182.c (written by Laurent Mazare)
Java github.com/HaochenLiu/My-Project-Euler/blob/master/182.java (written by Haochen Liu)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p182.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem182.java (written by Magnus Solheim Thrap)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p182.hs (written by Nayuki)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
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