Francois Berder / BigInt

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:
10:116e201f7d89
Parent:
9:3c191fa04f6e
Child:
11:2f16a220ebbb
--- 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;
+}