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;
}
Diff: BigInt.cpp
- Revision:
- 12:a436f15b58b6
- Parent:
- 11:2f16a220ebbb
- Child:
- 13:d0b4d7cfeb98
--- a/BigInt.cpp Thu Mar 06 09:44:32 2014 +0000
+++ b/BigInt.cpp Thu Mar 06 10:58:30 2014 +0000
@@ -214,31 +214,33 @@
{
assert(a.isValid() && b.isValid());
- BigInt result;
-
// if a == 0 or b == 0 then result = 0
if(!a || !b)
- return result = 0;
+ return 0;
// if a == 1, then result = b
if(a == 1)
- return (result = b);
+ return b;
// if b == 1, then result = a
if(b == 1)
- return (result = a);
-
+ return a;
+
+ BigInt result;
result.size = a.size + b.size;
result.bits = new uint32_t[num(result.size)];
- memset(result.bits, 0, result.size);
+ memset(result.bits, 0, sizeof(uint32_t)*num(result.size));
for(int i = 0; i < num(a.size); ++i)
{
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;
+ uint32_t t = result.bits[i+j];
result.bits[i+j] += tmp;
- carry = tmp >> 32;
+ carry = tmp >> 32;
+ if(t > result.bits[i+j])
+ ++carry;
}
if(carry != 0)
result.bits[i+num(b.size)] += carry;
@@ -265,12 +267,10 @@
if(a == b)
return 1;
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 = b << m;
-
+ BigInt tmp = b;
+ tmp <<= m;
for(int j = m; j >= 0; --j)
{
if(tmp <= u)
@@ -278,7 +278,7 @@
u -= tmp;
BigInt tmp2 = 1;
tmp2 <<= j;
- q += tmp2;
+ q += tmp2;
}
tmp >>= 1;
}
@@ -295,13 +295,12 @@
{
assert(a.isValid());
+ if(m == 0)
+ return a;
+ if(m/8 >= a.size)
+ return 0;
+
BigInt result;
-
- if(m == 0)
- return result = a;
- if(m/8 >= a.size)
- return result = 0;
-
result.size = a.size - m/8;
result.bits = new uint32_t[num(result.size)];
uint8_t s = m%32;
@@ -334,7 +333,7 @@
result.size = m/8 + a.size;
uint32_t h = a.bits[num(a.size)-1];
- if((h << (m%32)) < h)
+ if((m%32)%8 != 0)
++result.size;
uint32_t l = num(result.size);
result.bits = new uint32_t[l];
@@ -347,9 +346,10 @@
else
result.bits[m/32+i] = (a.bits[i] << s) | (a.bits[i-1] >> (32-s));
}
- if(a.bits[num(a.size)-1] << s < a.bits[num(a.size)-1])
- result.bits[num(result.size)-1] = a.bits[num(a.size)-1] >> (32-s);
+ if(a.bits[num(a.size)-1] && s != 0)
+ result.bits[m/32+num(result.size)-1] |= a.bits[num(a.size)-1] >> (32-s);
+ result.trim();
return result;
}
@@ -362,19 +362,7 @@
BigInt operator%(const BigInt &a, const BigInt &b)
{
assert(a.isValid() && b.isValid() && b > 0);
-
- BigInt i = 1, result;
-
- while(a >= b*i && a < b*(i+1))
- {
- ++i;
- }
- --i;
- result = a - b*i;
-
- result.trim();
-
- return result;
+ return a - (a/b)*b;
}
BigInt& BigInt::operator%=(const BigInt &a)
@@ -411,8 +399,12 @@
return false;
uint32_t l = num(a.size);
for(int i = l-1; i >= 0; --i)
+ {
if(a.bits[i] < b.bits[i])
return true;
+ else if(a.bits[i] > b.bits[i])
+ return false;
+ }
return false;
}
@@ -431,8 +423,12 @@
return false;
uint32_t l = num(a.size);
for(int i = l-1; i >= 0; --i)
+ {
if(a.bits[i] > b.bits[i])
return true;
+ else if(a.bits[i] < b.bits[i])
+ return false;
+ }
return false;
}
@@ -571,7 +567,7 @@
BigInt tmp2 = 1;
tmp2 = 1 << r;
tmp2 *= a;
- BigInt montA = tmp2 - tmp2/modulus;
+ BigInt montA = tmp2%modulus;
montA.print();
BigInt tmp = montA;
BigInt tmpE = expn;
@@ -621,9 +617,7 @@
uint32_t BigInt::numBits() const
{
assert(isValid());
-
- print();
-
+
uint32_t n = (size-1)*8;
uint8_t a = bits[num(size)-1] >> ((size-1)%4)*8;
uint8_t tmp2 = 8;