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 9:3c191fa04f6e, committed 2013-10-05
- Comitter:
- feb11
- Date:
- Sat Oct 05 15:26:03 2013 +0000
- Parent:
- 8:2a361f1b41da
- Child:
- 10:116e201f7d89
- Commit message:
- refactored code + implemented modular exponentiation
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 Fri Oct 04 15:16:06 2013 +0000
+++ b/BigInt.cpp Sat Oct 05 15:26:03 2013 +0000
@@ -4,7 +4,7 @@
#include <stdlib.h>
#include <iostream>
#include <climits>
-
+#include <cassert>
static uint32_t max(const uint32_t a, const uint32_t b)
{
@@ -49,7 +49,10 @@
BigInt::~BigInt()
{
if(bits)
+ {
+ printf("deleting %d bytes\n", size);
delete[] bits;
+ }
bits = 0;
}
@@ -81,8 +84,10 @@
BigInt operator+(const BigInt &a, const BigInt& b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result;
-
+
result.size = a.size > b.size ? a.size : b.size;
size_t l = result.size/4;
if(result.size % 4 != 0)
@@ -141,6 +146,8 @@
// No negative number allowed
BigInt operator-(const BigInt &a, const BigInt& b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result;
if(b >= a)
@@ -185,15 +192,7 @@
}
}
- // trim result
- uint8_t *tmp = (uint8_t*)result.bits;
- uint32_t newSize = result.size;
- while(tmp[newSize-1] == 0 && newSize > 0)
- newSize--;
- result.size = newSize;
- uint32_t tmpL = num(result.size);
- if(tmpL < l)
- result.bits = (uint32_t*)realloc(result.bits, sizeof(uint32_t)*tmpL);
+ result.trim();
return result;
}
@@ -218,6 +217,8 @@
BigInt operator*(const BigInt &a, const BigInt& b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result;
// if a == 0 or b == 0 then result = 0
@@ -251,14 +252,7 @@
carry = tmp >> 32;
}
- // trim result
- uint8_t *tmp = (uint8_t*)result.bits;
- uint32_t newSize = result.size;
- while(tmp[newSize-1] == 0 && newSize > 0)
- newSize--;
- if(num(newSize) < num(result.size))
- result.bits = (uint32_t*)realloc(result.bits, sizeof(uint32_t)*num(newSize));
- result.size = newSize;
+ result.trim();
return result;
}
@@ -270,7 +264,10 @@
BigInt operator>>(const BigInt &a, const uint32_t m)
{
+ assert(a.isValid());
+
BigInt result;
+
if(m == 0)
return result = a;
if(m/8 >= a.size)
@@ -298,8 +295,10 @@
BigInt operator<<(const BigInt &a, const uint32_t m)
{
+ assert(a.isValid());
+
BigInt result;
-
+
if(m == 0)
return result = a;
@@ -332,18 +331,15 @@
BigInt operator%(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result = a;
+
+
while(result > b)
result -= b;
- // trim result
- uint8_t *tmp = (uint8_t*)result.bits;
- uint32_t newSize = result.size;
- while(tmp[newSize-1] == 0 && newSize > 0)
- newSize--;
- if(num(newSize) < num(result.size))
- result.bits = (uint32_t*)realloc(result.bits, sizeof(uint32_t)*num(newSize));
- result.size = newSize;
+ result.trim();
return result;
}
@@ -355,6 +351,8 @@
bool operator==(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
if(a.size != b.size)
return false;
@@ -372,6 +370,8 @@
bool operator<(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
if(a.size < b.size)
return true;
if(a.size > b.size)
@@ -385,11 +385,13 @@
bool operator<=(const BigInt &a, const BigInt &b)
{
- return (a < b) || (a == b);
+ return !(a > b);
}
bool operator>(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
if(a.size > b.size)
return true;
if(a.size < b.size)
@@ -403,11 +405,13 @@
bool operator>=(const BigInt &a, const BigInt &b)
{
- return (a > b) || (a == b);
+ return !(a < b);
}
bool operator!(const BigInt &a)
{
+ assert(a.isValid());
+
if(a.size != 1)
return false;
return a.bits[0] == 0;
@@ -416,6 +420,8 @@
BigInt operator&(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result;
result.size = a.size < b.size ? a.size : b.size;
@@ -426,15 +432,7 @@
for(uint32_t i = 0; i < l; ++i)
result.bits[i] = a.bits[i] & b.bits[i];
-
- // trim result
- uint8_t *tmp = (uint8_t*)result.bits;
- uint32_t newSize = result.size;
- while(tmp[newSize-1] == 0 && newSize > 0)
- newSize--;
- if(num(newSize) < num(result.size))
- result.bits = (uint32_t*)realloc(result.bits, sizeof(uint32_t)*num(newSize));
- result.size = newSize;
+ result.trim();
return result;
}
@@ -446,6 +444,8 @@
BigInt operator|(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
BigInt result;
uint32_t na = num(a.size);
@@ -475,6 +475,9 @@
BigInt operator^(const BigInt &a, const BigInt &b)
{
+ assert(a.isValid() && b.isValid());
+
+
BigInt result;
uint32_t na = num(a.size);
@@ -494,6 +497,8 @@
result.bits[i] = b.bits[i];
}
+ result.trim();
+
return result;
}
@@ -501,7 +506,30 @@
{
return (*this = *this ^ a);
}
-
+
+BigInt modPow(const BigInt &a, const BigInt &expn, const BigInt &modulus)
+{
+ assert(a.isValid() && expn.isValid() && modulus.isValid());
+
+ BigInt result = 1;
+ BigInt tmpE = expn;
+ BigInt base = a;
+ while(tmpE > 0)
+ {
+ if(tmpE % 2 == 1)
+ result = (result * base) % modulus;
+ tmpE >>= 1;
+ base = (base * base) % modulus;
+ }
+
+ return result;
+}
+
+bool BigInt::isValid() const
+{
+ return size != 0 && bits != 0;
+}
+
void BigInt::print()
{
printf("size: %d bytes\n", size);
@@ -512,3 +540,15 @@
}
printf("\n");
}
+
+void BigInt::trim()
+{
+ uint8_t *tmp = (uint8_t*)bits;
+ uint32_t newSize = size;
+ while(tmp[newSize-1] == 0 && newSize > 0)
+ newSize--;
+ if(num(newSize) < num(size))
+ bits = (uint32_t*)realloc(bits, sizeof(uint32_t)*num(newSize));
+ size = newSize;
+}
+
--- a/BigInt.h Fri Oct 04 15:16:06 2013 +0000
+++ b/BigInt.h Sat Oct 05 15:26:03 2013 +0000
@@ -50,15 +50,22 @@
friend BigInt operator&(const BigInt &a, const BigInt &b);
BigInt& operator&=(const BigInt &a);
friend BigInt operator|(const BigInt &a, const BigInt &b);
- BigInt& operator|=(const BigInt &a);
+ BigInt& operator|=(const BigInt &a);
friend BigInt operator^(const BigInt &a, const BigInt &b);
- BigInt& operator^=(const BigInt &a);
-
+ BigInt& operator^=(const BigInt &a);
+
+ // 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
uint32_t *bits;
};