#include <iostream>
#include <cstring>
#include <ctime>
#include <cstdlib>
using namespace std;
#define WORD_SIZE 32
#define NUM_WORDS 64 // 2048位 = 64 * 32位
#define MONTGOMERY_R (1ULL << (WORD_SIZE * NUM_WORDS)) // R = 2^(2048)
struct BigNum
{
uint32_t words[NUM_WORDS];
BigNum()
{
memset(words, 0, sizeof(words));
}
BigNum(uint64_t val)
{
memset(words, 0, sizeof(words));
words[0] = val & 0xFFFFFFFF;
if (NUM_WORDS > 1)
{
words[1] = val >> 32;
}
}
void print() const
{
for (int i = NUM_WORDS - 1; i >= 0; i--)
{
printf("%08x", words[i]);
}
cout << endl;
}
};
struct ExtendedBigNum
{
uint32_t words[2 * NUM_WORDS];
ExtendedBigNum()
{
memset(words, 0, sizeof(words));
}
ExtendedBigNum(const BigNum& a)
{
memset(words, 0, sizeof(words));
for (int i = 0; i < NUM_WORDS; i++)
{
words[i] = a.words[i];
}
}
BigNum to_bignum() const
{
BigNum result;
for (int i = 0; i < NUM_WORDS; i++)
{
result.words[i] = words[i];
}
return result;
}
};
inline uint32_t ct_select_word(uint32_t a, uint32_t b, uint32_t selector)
{
return (a & ~selector) | (b & selector);
}
BigNum ct_select_bignum(const BigNum& a, const BigNum& b, uint32_t selector)
{
BigNum result;
uint32_t mask = selector;
for (int i = 0; i < NUM_WORDS; i++)
{
result.words[i] = ct_select_word(a.words[i], b.words[i], mask);
}
return result;
}
uint32_t ct_compare(const BigNum& a, const BigNum& b)
{
uint32_t gt = 0, lt = 0;
for (int i = NUM_WORDS - 1; i >= 0; i--)
{
uint32_t a_val = a.words[i];
uint32_t b_val = b.words[i];
gt |= ((a_val > b_val) & ~lt) & ~gt;
lt |= ((a_val < b_val) & ~gt) & ~lt;
}
return ct_select_word(0, ct_select_word(1, 2, gt), lt | gt);
}
BigNum ct_add(const BigNum& a, const BigNum& b)
{
BigNum result;
uint64_t carry = 0;
for (int i = 0; i < NUM_WORDS; i++)
{
uint64_t sum = (uint64_t)a.words[i] + b.words[i] + carry;
result.words[i] = sum & 0xFFFFFFFF;
carry = sum >> WORD_SIZE;
}
return result;
}
BigNum ct_sub(const BigNum& a, const BigNum& b)
{
BigNum result;
uint64_t borrow = 0;
for (int i = 0; i < NUM_WORDS; i++)
{
uint64_t diff = (uint64_t)a.words[i] - b.words[i] - borrow;
result.words[i] = diff & 0xFFFFFFFF;
borrow = (diff >> WORD_SIZE) & 1;
}
return result;
}
BigNum multiply(const BigNum& a, const BigNum& b)
{
BigNum result;
uint64_t temp[2 * NUM_WORDS] = { 0 };
for (int i = 0; i < NUM_WORDS; i++)
{
uint64_t carry = 0;
for (int j = 0; j < NUM_WORDS; j++)
{
uint64_t product = (uint64_t)a.words[i] * b.words[j] + temp[i + j] + carry;
temp[i + j] = product & 0xFFFFFFFF;
carry = product >> WORD_SIZE;
}
if (i + NUM_WORDS < 2 * NUM_WORDS)
{
temp[i + NUM_WORDS] = carry;
}
}
for (int i = 0; i < NUM_WORDS; i++)
{
result.words[i] = temp[i];
}
return result;
}
ExtendedBigNum multiply_extended(const BigNum& a, const BigNum& b)
{
ExtendedBigNum result;
for (int i = 0; i < NUM_WORDS; i++)
{
uint64_t carry = 0;
for (int j = 0; j < NUM_WORDS; j++)
{
uint64_t product = (uint64_t)a.words[i] * b.words[j] + result.words[i + j] + carry;
result.words[i + j] = product & 0xFFFFFFFF;
carry = product >> WORD_SIZE;
}
if (i + NUM_WORDS < 2 * NUM_WORDS)
{
result.words[i + NUM_WORDS] = carry;
}
}
return result;
}
BigNum barrett_precompute(const BigNum& modulus)
{
BigNum mu;
BigNum two_pow_2k;
two_pow_2k.words[2 * NUM_WORDS / WORD_SIZE - 1] = 1ULL << (2 * NUM_WORDS * WORD_SIZE % WORD_SIZE);
// 简化：这里我们使用一个近似值
// 实际实现需要完整的除法算法
mu = BigNum(1ULL << 32); // 简化实现
return mu;
}
BigNum barrett_reduce(const BigNum& a, const BigNum& modulus, const BigNum& mu)
{
// 简化实现，实际需要完整Barrett算法
BigNum result = a;
while (ct_compare(result, modulus) == 1)
{
result = ct_sub(result, modulus);
}
return result;
}
// 蒙哥马利乘法预计算参数
BigNum montgomery_precompute(const BigNum& modulus)
{
BigNum n_prime;
// 简化实现，实际需要扩展欧几里得算法
n_prime = BigNum(1);
return n_prime;
}
// 蒙哥马利乘法
BigNum montgomery_multiply(const BigNum& a, const BigNum& b, const BigNum& modulus, const BigNum& n_prime)
{
ExtendedBigNum t_ext = multiply_extended(a, b);
BigNum t = t_ext.to_bignum();
ExtendedBigNum m_ext = multiply_extended(t, n_prime);
BigNum m;
for (int i = 0; i < NUM_WORDS; i++)
{
m.words[i] = m_ext.words[i];
}
ExtendedBigNum m_times_n_ext = multiply_extended(m, modulus);
ExtendedBigNum result_ext;
uint64_t carry = 0;
for (int i = 0; i < 2 * NUM_WORDS; i++)
{
uint64_t sum = (uint64_t)t_ext.words[i] + m_times_n_ext.words[i] + carry;
result_ext.words[i] = sum & 0xFFFFFFFF;
carry = sum >> WORD_SIZE;
}
BigNum result;
for (int i = 0; i < NUM_WORDS; i++)
{
result.words[i] = result_ext.words[i + NUM_WORDS];
}
uint32_t cmp = ct_compare(result, modulus);
uint32_t selector = (cmp == 1) ? 0xFFFFFFFF : 0;
result = ct_select_bignum(result, ct_sub(result, modulus), selector);
return result;
}
// 转蒙哥马利形式
BigNum to_montgomery(const BigNum& a, const BigNum& modulus, const BigNum& n_prime)
{
BigNum r_squared;
r_squared.words[2 * NUM_WORDS / WORD_SIZE - 1] = 1ULL << (2 * NUM_WORDS * WORD_SIZE % WORD_SIZE);
r_squared = barrett_reduce(r_squared, modulus, barrett_precompute(modulus));
return montgomery_multiply(a, r_squared, modulus, n_prime);
}
// 转普通形式
BigNum from_montgomery(const BigNum& a_mont, const BigNum& modulus, const BigNum& n_prime)
{
// 乘以1，即计算 a_mont * 1 * R^{-1} mod N
BigNum one;
one.words[0] = 1;
return montgomery_multiply(a_mont, one, modulus, n_prime);
}
// 蒙哥马利幂运算
BigNum montgomery_power(const BigNum& base, const BigNum& exponent, const BigNum& modulus, const BigNum& n_prime)
{
BigNum base_mont = to_montgomery(base, modulus, n_prime);
BigNum r0, r1;
r0 = to_montgomery(BigNum(1), modulus, n_prime);
r1 = base_mont;
// 从最高位开始处理指数
for (int i = NUM_WORDS * WORD_SIZE - 1; i >= 0; i--)
{
int word_idx = i / WORD_SIZE;
int bit_idx = i % WORD_SIZE;
uint32_t bit = (exponent.words[word_idx] >> bit_idx) & 1;
uint32_t selector = bit ? 0xFFFFFFFF : 0;
BigNum temp_r0 = montgomery_multiply(r0, r1, modulus, n_prime);
BigNum temp_r1 = montgomery_multiply(r1, r1, modulus, n_prime);
BigNum r0_squared = montgomery_multiply(r0, r0, modulus, n_prime);
r0 = ct_select_bignum(r0_squared, temp_r0, selector);
r1 = ct_select_bignum(temp_r0, temp_r1, selector);
}
return from_montgomery(r0, modulus, n_prime);
}
void rsa_key_gen(BigNum& n, BigNum& e, BigNum& d)
{
// 在实际实现中，这里需要生成大素数p和q
// 简化：使用预定义的值
BigNum p(61);
BigNum q(53);
n = multiply(p, q);
BigNum phi_n = multiply(BigNum(60), BigNum(52));
e = BigNum(17);/
d = BigNum(2753);
}
// RSA加密
BigNum rsa_encrypt(const BigNum& message, const BigNum& e, const BigNum& n)
{
BigNum n_prime = montgomery_precompute(n);
return montgomery_power(message, e, n, n_prime);
}
// RSA解密
BigNum rsa_decrypt(const BigNum& ciphertext, const BigNum& d, const BigNum& n)
{
BigNum n_prime = montgomery_precompute(n);
return montgomery_power(ciphertext, d, n, n_prime);
}
int main()
{
BigNum n, e, d;
rsa_key_gen(n, e, d);
cout << "n: "; n.print();
cout << "e: "; e.print();
cout << "d: "; d.print();
// 明文
BigNum message(42);
cout << "明文："; message.print();
// 加密
BigNum ciphertext = rsa_encrypt(message, e, n);
cout << "密文："; ciphertext.print();
// 解密
BigNum decrypted = rsa_decrypt(ciphertext, d, n);
cout << "解密："; decrypted.print();
return 0;
}
上面的代码尝试完成使用Montgomery’s ladder算法的RSA实现，请修改正确、补充完整，要求：大数要至少2048位，用数组表示，不要使用openssl、NTL等库，乘法和模逆运算使用蒙哥马利乘法和Barrett归约法
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