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 11:2f16a220ebbb, committed 2014-03-06
- Comitter:
- feb11
- Date:
- Thu Mar 06 09:44:32 2014 +0000
- Parent:
- 10:116e201f7d89
- Child:
- 12:a436f15b58b6
- Commit message:
- fixed multiplication algorithm
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 Wed Mar 05 19:36:22 2014 +0000
+++ b/BigInt.cpp Thu Mar 06 09:44:32 2014 +0000
@@ -227,24 +227,21 @@
// if b == 1, then result = a
if(b == 1)
return (result = a);
-
-
+
result.size = a.size + b.size;
result.bits = new uint32_t[num(result.size)];
- memset(result.bits, 0, sizeof(uint32_t)*num(result.size));
- uint32_t carry = 0;
- for(int i = 0; i < num(result.size); ++i)
+ memset(result.bits, 0, result.size);
+ for(int i = 0; i < num(a.size); ++i)
{
- uint32_t tmpA = 0, tmpB = 0;
- if(i < num(a.size))
- tmpA = a.bits[i];
-
- if(i < num(b.size))
- tmpB = b.bits[i];
-
- uint64_t tmp = (uint64_t)tmpA * (uint64_t)tmpB + (uint64_t)carry;
- result.bits[i] = tmp;
- carry = tmp >> 32;
+ uint64_t carry = 0;
+ for(int j = 0; j < num(b.size); ++j)
+ {
+ uint64_t tmp = (uint64_t)a.bits[i] * (uint64_t)b.bits[j] + carry;
+ result.bits[i+j] += tmp;
+ carry = tmp >> 32;
+ }
+ if(carry != 0)
+ result.bits[i+num(b.size)] += carry;
}
result.trim();
@@ -258,32 +255,35 @@
}
-BigInt operator/(const BigInt &a, const BigInt& 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();
+ BigInt u = a;
+ printf("a.bits = %d\n", a.numBits());
+ int m = a.numBits() - b.numBits();
printf("m=%d\n", m);
BigInt q = 0;
- BigInt tmp = 1;
- tmp <<= m;
+ BigInt tmp = b << m;
+
for(int j = m; j >= 0; --j)
{
- if(v*tmp <= u)
+ if(tmp <= u)
{
- u -= v*tmp;
- q += tmp;
+ u -= tmp;
+ BigInt tmp2 = 1;
+ tmp2 <<= j;
+ q += tmp2;
}
tmp >>= 1;
}
- return q;
+ q.trim();
+ return q;
}
BigInt& BigInt::operator/=(const BigInt &b)
@@ -540,22 +540,6 @@
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;
@@ -584,7 +568,10 @@
uint32_t r = 8*modulus.size;
// convert a in montgomery world
- BigInt montA = montgomeryStep2(a,modulus,r);
+ BigInt tmp2 = 1;
+ tmp2 = 1 << r;
+ tmp2 *= a;
+ BigInt montA = tmp2 - tmp2/modulus;
montA.print();
BigInt tmp = montA;
BigInt tmpE = expn;
@@ -603,8 +590,10 @@
return size != 0 && bits != 0;
}
-void BigInt::print()
+void BigInt::print() const
{
+ assert(isValid());
+
printf("size: %d bytes\n", size);
uint32_t n = num(size);
for(int i = n-1; i >= 0; --i)
@@ -629,17 +618,18 @@
size = newSize;
}
-uint32_t BigInt::numBits()
+uint32_t BigInt::numBits() const
{
assert(isValid());
+ print();
+
uint32_t n = (size-1)*8;
- uint8_t a = bits[size/4] >> ((size-1)%4)*8;
- uint8_t tmp = 0x80;
+ uint8_t a = bits[num(size)-1] >> ((size-1)%4)*8;
uint8_t tmp2 = 8;
- while(!(a & tmp))
+ while(!(a & 0x80))
{
- tmp >>= 1;
+ a <<= 1;
--tmp2;
}
n += tmp2;
--- a/BigInt.h Wed Mar 05 19:36:22 2014 +0000
+++ b/BigInt.h Thu Mar 06 09:44:32 2014 +0000
@@ -16,20 +16,20 @@
void import(uint8_t *data, uint32_t length);
// Arithmetic operations
- friend BigInt operator+(const BigInt &a, const BigInt& b);
+ friend BigInt operator+(const BigInt &a, const BigInt &b);
BigInt& operator+=(const BigInt &b);
BigInt& operator++();
BigInt operator++(int);
- friend BigInt operator-(const BigInt &a, const BigInt& b);
+ friend BigInt operator-(const BigInt &a, const BigInt &b);
BigInt& operator-=(const BigInt &b);
BigInt& operator--();
BigInt operator--(int);
- friend BigInt operator*(const BigInt &a, const BigInt& b);
+ friend BigInt operator*(const BigInt &a, const BigInt &b);
BigInt& operator*=(const BigInt &b);
- friend BigInt operator/(const BigInt &a, 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);
@@ -63,12 +63,12 @@
bool isValid() const;
// debug
- void print();
+ void print() const;
private :
void trim();
- uint32_t numBits();
+ uint32_t numBits() const;
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);