Francois Berder
This library provides a way to easily handle arbitrary large integers.
This library provides the following operations :
- addition, substraction, multiplication, division and modulo
- bits operators (AND, OR, XOR, left and right shifts)
- boolean operators
- modular exponentiation (using montgomery algorithm)
- modular inverse
Example
In this example, we use a 1024 bits long RSA key to encrypt and decrypt a message. We first encrypt the value 0x41 (65 in decimal) and then decrypt it. At the end, m should be equal to 0x41. The encryption is fast (0, 4 second) while the decryption is really slow. This code will take between 30 seconds and 2 minutes to execute depending on the compiler and optimization flags.
main.cpp
#include "mbed.h"
#include "BigInt.h"
#include <stdlib.h>
#include <stdio.h>
uint8_t modbits[] = {
0xd9, 0x4d, 0x88, 0x9e, 0x88, 0x85, 0x3d, 0xd8, 0x97, 0x69, 0xa1, 0x80, 0x15, 0xa0, 0xa2, 0xe6,
0xbf, 0x82, 0xbf, 0x35, 0x6f, 0xe1, 0x4f, 0x25, 0x1f, 0xb4, 0xf5, 0xe2, 0xdf, 0x0d, 0x9f, 0x9a,
0x94, 0xa6, 0x8a, 0x30, 0xc4, 0x28, 0xb3, 0x9e, 0x33, 0x62, 0xfb, 0x37, 0x79, 0xa4, 0x97, 0xec,
0xea, 0xea, 0x37, 0x10, 0x0f, 0x26, 0x4d, 0x7f, 0xb9, 0xfb, 0x1a, 0x97, 0xfb, 0xf6, 0x21, 0x13,
0x3d, 0xe5, 0x5f, 0xdc, 0xb9, 0xb1, 0xad, 0x0d, 0x7a, 0x31, 0xb3, 0x79, 0x21, 0x6d, 0x79, 0x25,
0x2f, 0x5c, 0x52, 0x7b, 0x9b, 0xc6, 0x3d, 0x83, 0xd4, 0xec, 0xf4, 0xd1, 0xd4, 0x5c, 0xbf, 0x84,
0x3e, 0x84, 0x74, 0xba, 0xbc, 0x65, 0x5e, 0x9b, 0xb6, 0x79, 0x9c, 0xba, 0x77, 0xa4, 0x7e, 0xaf,
0xa8, 0x38, 0x29, 0x64, 0x74, 0xaf, 0xc2, 0x4b, 0xeb, 0x9c, 0x82, 0x5b, 0x73, 0xeb, 0xf5, 0x49
};
uint8_t dbits[] = {
0x04, 0x7b, 0x9c, 0xfd, 0xe8, 0x43, 0x17, 0x6b, 0x88, 0x74, 0x1d, 0x68, 0xcf, 0x09, 0x69, 0x52,
0xe9, 0x50, 0x81, 0x31, 0x51, 0x05, 0x8c, 0xe4, 0x6f, 0x2b, 0x04, 0x87, 0x91, 0xa2, 0x6e, 0x50,
0x7a, 0x10, 0x95, 0x79, 0x3c, 0x12, 0xba, 0xe1, 0xe0, 0x9d, 0x82, 0x21, 0x3a, 0xd9, 0x32, 0x69,
0x28, 0xcf, 0x7c, 0x23, 0x50, 0xac, 0xb1, 0x9c, 0x98, 0xf1, 0x9d, 0x32, 0xd5, 0x77, 0xd6, 0x66,
0xcd, 0x7b, 0xb8, 0xb2, 0xb5, 0xba, 0x62, 0x9d, 0x25, 0xcc, 0xf7, 0x2a, 0x5c, 0xeb, 0x8a, 0x8d,
0xa0, 0x38, 0x90, 0x6c, 0x84, 0xdc, 0xdb, 0x1f, 0xe6, 0x77, 0xdf, 0xfb, 0x2c, 0x02, 0x9f, 0xd8,
0x92, 0x63, 0x18, 0xee, 0xde, 0x1b, 0x58, 0x27, 0x2a, 0xf2, 0x2b, 0xda, 0x5c, 0x52, 0x32, 0xbe,
0x06, 0x68, 0x39, 0x39, 0x8e, 0x42, 0xf5, 0x35, 0x2d, 0xf5, 0x88, 0x48, 0xad, 0xad, 0x11, 0xa1
};
int main()
{
BigInt e = 65537, mod, d;
mod.importData(modbits, sizeof(modbits));
d.importData(dbits, sizeof(dbits));
BigInt c = modPow(0x41,e,mod);
c.print();
BigInt m = modPow(c,d,mod);
m.print();
printf("done\n");
return 0;
}
Revision 10:116e201f7d89, committed 2014-03-05
- Comitter:
- feb11
- Date:
- Wed Mar 05 19:36:22 2014 +0000
- Parent:
- 9:3c191fa04f6e
- Child:
- 11:2f16a220ebbb
- Commit message:
- division implemented
Changed in this revision
| BigInt.cpp | Show annotated file Show diff for this revision Revisions of this file |
| BigInt.h | Show annotated file Show diff for this revision Revisions of this file |
--- a/BigInt.cpp Sat Oct 05 15:26:03 2013 +0000
+++ b/BigInt.cpp Wed Mar 05 19:36:22 2014 +0000
@@ -5,17 +5,14 @@
#include <iostream>
#include <climits>
#include <cassert>
-
-static uint32_t max(const uint32_t a, const uint32_t b)
-{
- return a < b ? b : a;
-}
+#include <algorithm>
static uint32_t num(const uint32_t a)
{
return a/4 + (a%4 ? 1:0);
}
-
+
+
BigInt::BigInt():
size(0),
bits(0)
@@ -48,12 +45,10 @@
BigInt::~BigInt()
{
- if(bits)
+ if(size)
{
- printf("deleting %d bytes\n", size);
delete[] bits;
}
- bits = 0;
}
BigInt& BigInt::operator=(const BigInt& a)
@@ -105,7 +100,7 @@
if(i < bl)
tmpB = b.bits[i];
result.bits[i] = tmpA + tmpB + carry;
- carry = result.bits[i] < max(tmpA, tmpB);
+ carry = result.bits[i] < std::max(tmpA, tmpB);
}
if(carry)
{
@@ -262,6 +257,40 @@
return (*this = (*this) * b);
}
+
+BigInt operator/(const BigInt &a, const BigInt& b)
+{
+ assert(a.isValid() && b.isValid() && b != 0);
+
+ if(b == 1)
+ return a;
+ if(a < b)
+ return 0;
+ if(a == b)
+ return 1;
+ BigInt u = a, v = b;
+ int m = u.numBits() - v.numBits();
+ printf("m=%d\n", m);
+ BigInt q = 0;
+ BigInt tmp = 1;
+ tmp <<= m;
+ for(int j = m; j >= 0; --j)
+ {
+ if(v*tmp <= u)
+ {
+ u -= v*tmp;
+ q += tmp;
+ }
+ tmp >>= 1;
+ }
+ return q;
+}
+
+BigInt& BigInt::operator/=(const BigInt &b)
+{
+ return (*this = (*this) / b);
+}
+
BigInt operator>>(const BigInt &a, const uint32_t m)
{
assert(a.isValid());
@@ -284,7 +313,8 @@
result.bits[i] = (a.bits[m/32+i] >> s);
}
-
+ result.trim();
+
return result;
}
@@ -331,14 +361,17 @@
BigInt operator%(const BigInt &a, const BigInt &b)
{
- assert(a.isValid() && b.isValid());
+ assert(a.isValid() && b.isValid() && b > 0);
- BigInt result = a;
+ BigInt i = 1, result;
+
+ while(a >= b*i && a < b*(i+1))
+ {
+ ++i;
+ }
+ --i;
+ result = a - b*i;
-
- while(result > b)
- result -= b;
-
result.trim();
return result;
@@ -450,8 +483,8 @@
uint32_t na = num(a.size);
uint32_t nb = num(b.size);
- uint32_t l = max(na,nb);
- result.size = max(a.size, b.size);
+ uint32_t l = std::max(na,nb);
+ result.size = std::max(a.size, b.size);
result.bits = new uint32_t[l];
memset(result.bits, 0, l);
@@ -482,8 +515,8 @@
uint32_t na = num(a.size);
uint32_t nb = num(b.size);
- uint32_t l = max(na,nb);
- result.size = max(a.size, b.size);
+ uint32_t l = std::max(na,nb);
+ result.size = std::max(a.size, b.size);
result.bits = new uint32_t[l];
memset(result.bits, 0, l);
@@ -507,22 +540,62 @@
return (*this = *this ^ a);
}
+BigInt montgomeryStep2(const BigInt &a, const BigInt &m, uint32_t r)
+{
+ BigInt result = a;
+ uint32_t tmpR = r;
+ while(r > 0)
+ {
+ if(r == 1)
+ result += (2 << tmpR);
+
+ if(result.bits[0] & 0x01)
+ result += m;
+ --r;
+ }
+ return result;
+}
+
+BigInt montgomeryStep(const BigInt &a, const BigInt &b, const BigInt &m, uint32_t r)
+{
+ BigInt result = 0;
+ BigInt tmp = a;
+ while(r > 0)
+ {
+ if(tmp.bits[0] & 0x01)
+ result += b;
+
+ if(result.bits[0] & 0x01)
+ result += m;
+
+ --r;
+ result >>= 1;
+ tmp >>= 1;
+ }
+ return result;
+}
+
+// Implementation using Montgomery algorithm
BigInt modPow(const BigInt &a, const BigInt &expn, const BigInt &modulus)
{
assert(a.isValid() && expn.isValid() && modulus.isValid());
- BigInt result = 1;
+ // precompute R and R^2
+ uint32_t r = 8*modulus.size;
+
+ // convert a in montgomery world
+ BigInt montA = montgomeryStep2(a,modulus,r);
+ montA.print();
+ BigInt tmp = montA;
BigInt tmpE = expn;
- BigInt base = a;
- while(tmpE > 0)
+ while(tmpE > 1)
{
- if(tmpE % 2 == 1)
- result = (result * base) % modulus;
- tmpE >>= 1;
- base = (base * base) % modulus;
+ tmp = montgomeryStep(montA, tmp, modulus, r);
+ --tmpE;
}
- return result;
+ // convert a to normal world
+ return montgomeryStep(tmp, 1, modulus, r);
}
bool BigInt::isValid() const
@@ -547,8 +620,29 @@
uint32_t newSize = size;
while(tmp[newSize-1] == 0 && newSize > 0)
newSize--;
+ if(newSize == 0)
+ newSize = 1;
if(num(newSize) < num(size))
+ {
bits = (uint32_t*)realloc(bits, sizeof(uint32_t)*num(newSize));
+ }
size = newSize;
}
+uint32_t BigInt::numBits()
+{
+ assert(isValid());
+
+ uint32_t n = (size-1)*8;
+ uint8_t a = bits[size/4] >> ((size-1)%4)*8;
+ uint8_t tmp = 0x80;
+ uint8_t tmp2 = 8;
+ while(!(a & tmp))
+ {
+ tmp >>= 1;
+ --tmp2;
+ }
+ n += tmp2;
+
+ return n;
+}
--- a/BigInt.h Sat Oct 05 15:26:03 2013 +0000
+++ b/BigInt.h Wed Mar 05 19:36:22 2014 +0000
@@ -28,7 +28,10 @@
friend BigInt operator*(const BigInt &a, const BigInt& b);
BigInt& operator*=(const BigInt &b);
-
+
+ friend BigInt operator/(const BigInt &a, const BigInt& b);
+ BigInt& operator/=(const BigInt &b);
+
friend BigInt operator>>(const BigInt &a, const uint32_t m);
BigInt& operator>>=(const uint32_t m);
friend BigInt operator<<(const BigInt &a, const uint32_t m);
@@ -56,17 +59,21 @@
// modular exponentiation
friend BigInt modPow(const BigInt &a, const BigInt &expn, const BigInt &modulus);
-
+
bool isValid() const;
// debug
void print();
-
+
private :
- void trim();
-
- uint32_t size; // number of bytes
+ void trim();
+ uint32_t numBits();
+
+ friend BigInt montgomeryStep(const BigInt &a, const BigInt &b, const BigInt &m, uint32_t r);
+ friend BigInt montgomeryStep2(const BigInt &a, const BigInt &m, uint32_t r);
+
+ uint32_t size; // smaller number of bytes needed to represent integer
uint32_t *bits;
};