BigInt - This library provides a way to easily handle arbi…

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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;
}

BigInt.cpp

Committer:: feb11
Date:: 2014-03-06
Revision:: 12:a436f15b58b6
Parent:: 11:2f16a220ebbb
Child:: 13:d0b4d7cfeb98

File content as of revision 12:a436f15b58b6:

#include "BigInt.h"
#include <string.h>
#include <stdio.h>
#include <stdlib.h>
#include <iostream>
#include <climits>
#include <cassert>
#include <algorithm>

static uint32_t num(const uint32_t a)
{
    return a/4 + (a%4 ? 1:0); 
}


BigInt::BigInt():
size(0),
bits(0)
{
}

BigInt::BigInt(const uint32_t a)
{
    if(a >> 24)
        size = 4;
    else if(a >> 16)
        size = 3;
    else if(a >> 8)
        size = 2;
    else
        size = 1;
    bits = new uint32_t[1];
    bits[0] = a;
}

BigInt::BigInt(const BigInt &a):
size(a.size)
{
    uint32_t l = num(size);
    bits = new uint32_t[l];
    for(int i = 0; i < l; ++i)
        bits[i] = a.bits[i];
}


BigInt::~BigInt()
{
    if(size)
    {
        delete[] bits;
    }
}

BigInt& BigInt::operator=(const BigInt& a)
{
    size = a.size;
    uint32_t l = num(size);
    if(bits)
        delete[] bits;
    bits = new uint32_t[l];
    for(int i = 0; i < l; ++i)
        bits[i] = a.bits[i];    
    return *this;
}

void BigInt::import(uint8_t *data, uint32_t length)
{
    size = length;
    size_t l = size/4;
    if(size % 4 != 0)
        l++;
    if(bits)
        delete[] bits;
    bits = new uint32_t[l];
    memset(bits, 0, sizeof(uint32_t)*l);
    for(int i = length-1; i >=0; --i)
        bits[i/4] |= data[i] << ((i%4)*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;
    size_t l = result.size/4;
    if(result.size % 4 != 0)
        l++;
    result.bits = new uint32_t[l];
    memset(result.bits, 0, sizeof(uint32_t)*l);
    uint32_t al = num(a.size);
    uint32_t bl = num(b.size);
    uint32_t carry = 0;
    for(int i = 0; i < l; ++i)
    {
        uint32_t tmpA = 0, tmpB = 0;
        if(i < al)
            tmpA = a.bits[i];
        if(i < bl)
            tmpB = b.bits[i];
        result.bits[i] = tmpA + tmpB + carry;
        carry = result.bits[i] < std::max(tmpA, tmpB);
    }
    if(carry)
    {
        result.size++;
        if(result.size > l*4)
        { 
            l++;
            result.bits = (uint32_t*)realloc(result.bits, l * sizeof(uint32_t));
            result.bits[l-1] = 0x00000001;
        }
        else
        {
            result.bits[l-1] += 1 << (8 *((result.size-1)%4));
        }
    }
    
    return result;
}

BigInt& BigInt::operator+=(const BigInt &b)
{
    return (*this = (*this) + b);
}

BigInt& BigInt::operator++()
{
    return (*this += 1);
}

BigInt BigInt::operator++(int)
{
    BigInt t = *this;
    *this += 1;
    return t;
}

// a - b, if b >= a, returns 0
// No negative number allowed
BigInt operator-(const BigInt &a, const BigInt& b)
{
    assert(a.isValid() && b.isValid());

    BigInt result;

    if(b >= a)
    {
        return result = 0;
    }
    else
    {
        result.size = a.size;
        uint32_t l = num(a.size);
        result.bits = new uint32_t[l];
        memset(result.bits, 0, sizeof(uint32_t)*l);
        uint32_t bl = num(b.size);
        uint8_t borrow = 0;
        for(uint32_t i = 0; i < l; ++i)
        {
            uint32_t tmpA = a.bits[i], tmpB = 0;
            if(i < bl)
                tmpB = b.bits[i];
            if(borrow)
            {
                if(tmpA == 0)
                    tmpA = ULONG_MAX;
                else
                    --tmpA;
                    
                if(tmpA < tmpB)            
                    result.bits[i] = tmpA + 1 + (ULONG_MAX - tmpB);
                else
                    result.bits[i] = tmpA - tmpB;

                if(a.bits[i] != 0 && tmpA > tmpB)
                    borrow = 0;
            }
            else
            {
                if(tmpA < tmpB)            
                    result.bits[i] = tmpA + 1 + (ULONG_MAX - tmpB);
                else
                    result.bits[i] = tmpA - tmpB;
                borrow = tmpA < tmpB;
            }
        }
        
        result.trim();
        
        return result;
    }
}

BigInt& BigInt::operator-=(const BigInt &b)
{
    return (*this = (*this) - b);
}

BigInt& BigInt::operator--()
{
    return (*this -= 1);
}

BigInt BigInt::operator--(int)
{
    BigInt t = *this;
    *this -= 1;
    return t;
}

BigInt operator*(const BigInt &a, const BigInt& b)
{
    assert(a.isValid() && b.isValid());

    // if a == 0 or b == 0 then result = 0
    if(!a || !b)
        return 0;
    
    // if a == 1, then result = b
    if(a == 1)
        return b;
    
    // if b == 1, then result = a
    if(b == 1)
        return a;
    
    BigInt result;          
    result.size = a.size + b.size;
    result.bits = new uint32_t[num(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;     
            if(t > result.bits[i+j])
                ++carry;                    
        }
        if(carry != 0)
            result.bits[i+num(b.size)] += carry;
    }
    
    result.trim();
    
    return result;
}

BigInt& BigInt::operator*=(const BigInt &b)
{
    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; 
    int m = a.numBits() - b.numBits();
    BigInt q = 0;
    BigInt tmp = b;
    tmp <<= m;
    for(int j = m; j >= 0; --j)
    {
        if(tmp <= u)
        {
            u -= tmp;
            BigInt tmp2 = 1;
            tmp2 <<= j;
            q += tmp2;  
        }   
        tmp >>= 1;
    }
    q.trim();
    return q;
}

BigInt& BigInt::operator/=(const BigInt &b)
{
    return (*this = (*this) / b);
} 

BigInt operator>>(const BigInt &a, const uint32_t m)
{
    assert(a.isValid());

    if(m == 0)
        return a;
    if(m/8 >= a.size)
        return 0;
    
    BigInt result;
    result.size = a.size - m/8;
    result.bits = new uint32_t[num(result.size)];
    uint8_t s = m%32;
    for(uint32_t i = 0; i < num(result.size); ++i)
    {
        if(m/32+i+1 < num(a.size))
            result.bits[i] =  (a.bits[m/32+i+1] << (32-s)) |  (a.bits[m/32+i] >> s);
        else
            result.bits[i] = (a.bits[m/32+i] >> s);
    }
    
    result.trim();
        
    return result;
}

BigInt& BigInt::operator>>=(const uint32_t m)
{
    return *this = *this >> m;
}
 
BigInt operator<<(const BigInt &a, const uint32_t m)
{
    assert(a.isValid());

    BigInt result;
    
    if(m == 0)
        return result = a;

    result.size = m/8 + a.size;
    uint32_t h = a.bits[num(a.size)-1];
    if((m%32)%8 != 0)
        ++result.size;
    uint32_t l = num(result.size);
    result.bits = new uint32_t[l];
    memset(result.bits, 0, sizeof(uint32_t)*l);
    uint32_t s = m % 32;
    for(uint32_t i = 0; i < num(a.size); ++i)
    {
        if(i == 0)
            result.bits[m/32+i] = a.bits[i] << s;
        else
            result.bits[m/32+i] = (a.bits[i] << s) | (a.bits[i-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;
}

BigInt& BigInt::operator<<=(const uint32_t m)
{
    return (*this = *this << m);
}

BigInt operator%(const BigInt &a, const BigInt &b)
{
    assert(a.isValid() && b.isValid() && b > 0);
    return a - (a/b)*b;
}

BigInt& BigInt::operator%=(const BigInt &a)
{
    return (*this = *this % a);
}

bool operator==(const BigInt &a, const BigInt &b)
{
    assert(a.isValid() && b.isValid());

    if(a.size != b.size)
        return false;
        
    uint32_t l = num(a.size);
    for(int i = 0; i < l; ++i)
        if(a.bits[i] != b.bits[i])
            return false;
    return true;
}

bool operator!=(const BigInt &a, const BigInt &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)
        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;
}

bool operator<=(const BigInt &a, const BigInt &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)
        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;
}

bool operator>=(const BigInt &a, const BigInt &b)
{
    return !(a < b);
}

bool operator!(const BigInt &a)
{
    assert(a.isValid());

    if(a.size != 1)
        return false;
    return a.bits[0] == 0;
}
 
 
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;
    uint32_t l = num(result.size);
    result.bits = new uint32_t[l];
    memset(result.bits, 0, l);
    
    for(uint32_t i = 0; i < l; ++i)
        result.bits[i] = a.bits[i] & b.bits[i];

    result.trim();
    
    return result;
}

BigInt& BigInt::operator&=(const BigInt &a)
{
    return (*this = *this & a);
}

BigInt operator|(const BigInt &a, const BigInt &b)
{
    assert(a.isValid() && b.isValid());

    BigInt result;

    uint32_t na = num(a.size);
    uint32_t nb = num(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);
    
    for(uint32_t i = 0; i < l; ++i)
    {
        if(i < na && i < nb)
            result.bits[i] = a.bits[i] | b.bits[i];
        else if(i < na)
            result.bits[i] = a.bits[i];
        else
            result.bits[i] = b.bits[i];
    }
    
    return result;
}

BigInt& BigInt::operator|=(const BigInt &a)
{
    return (*this = *this | a);
}

BigInt operator^(const BigInt &a, const BigInt &b)
{
    assert(a.isValid() && b.isValid());


    BigInt result;

    uint32_t na = num(a.size);
    uint32_t nb = num(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);
    
    for(uint32_t i = 0; i < l; ++i)
    {
        if(i < na && i < nb)
            result.bits[i] = a.bits[i] ^ b.bits[i];
        else if(i < na)
            result.bits[i] = a.bits[i];
        else
            result.bits[i] = b.bits[i];
    }
    
    result.trim();
    
    return result;
}

BigInt& BigInt::operator^=(const BigInt &a)      
{
    return (*this = *this ^ a);
}

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());
    
    // precompute R and R^2
    uint32_t r = 8*modulus.size;

    // convert a in montgomery world
    BigInt tmp2 = 1;
    tmp2 = 1 << r;
    tmp2 *= a;
    BigInt montA = tmp2%modulus;
    montA.print();
    BigInt tmp = montA;
    BigInt tmpE = expn;
    while(tmpE > 1)
    {
        tmp = montgomeryStep(montA, tmp, modulus, r);
        --tmpE;
    }
    
    // convert a to normal world
    return montgomeryStep(tmp, 1, modulus, r);
}

bool BigInt::isValid() const
{
    return size != 0 && bits != 0;
}

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)
    {
        printf("%08x ", bits[i]);
    }
    printf("\n");
}

void BigInt::trim()
{
    uint8_t *tmp = (uint8_t*)bits;
    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() const
{
    assert(isValid());

    uint32_t n = (size-1)*8;
    uint8_t a = bits[num(size)-1] >> ((size-1)%4)*8;
    uint8_t tmp2 = 8;
    while(!(a & 0x80)) 
    {
        a <<= 1;
        --tmp2;
    }
    n += tmp2;

    return n;
}

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Created:	20 Sep 2013
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The code in this repository is Apache licensed.