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/*
* Number Theory Functions
* (C) 1999-2011,2016,2018 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#include <botan/numthry.h>
#include <botan/pow_mod.h>
#include <botan/reducer.h>
#include <botan/monty.h>
#include <botan/rng.h>
#include <botan/internal/bit_ops.h>
#include <botan/internal/mp_core.h>
#include <botan/internal/ct_utils.h>
#include <botan/internal/monty_exp.h>
#include <botan/internal/primality.h>
#include <algorithm>
namespace Botan {
/*
* Return the number of 0 bits at the end of n
*/
size_t low_zero_bits(const BigInt& n)
{
size_t low_zero = 0;
if(n.is_positive() && n.is_nonzero())
{
for(size_t i = 0; i != n.size(); ++i)
{
const word x = n.word_at(i);
if(x)
{
low_zero += ctz(x);
break;
}
else
low_zero += BOTAN_MP_WORD_BITS;
}
}
return low_zero;
}
/*
* Calculate the GCD
*/
BigInt gcd(const BigInt& a, const BigInt& b)
{
if(a.is_zero() || b.is_zero())
return 0;
if(a == 1 || b == 1)
return 1;
BigInt X[2] = { a, b };
X[0].set_sign(BigInt::Positive);
X[1].set_sign(BigInt::Positive);
const size_t shift = std::min(low_zero_bits(X[0]), low_zero_bits(X[1]));
X[0] >>= shift;
X[1] >>= shift;
while(X[0].is_nonzero())
{
X[0] >>= low_zero_bits(X[0]);
X[1] >>= low_zero_bits(X[1]);
const uint8_t sel = static_cast<uint8_t>(X[0] >= X[1]);
X[sel^1] -= X[sel];
X[sel^1] >>= 1;
}
return (X[1] << shift);
}
/*
* Calculate the LCM
*/
BigInt lcm(const BigInt& a, const BigInt& b)
{
return ((a * b) / gcd(a, b));
}
/*
Sets result to a^-1 * 2^k mod a
with n <= k <= 2n
Returns k
"The Montgomery Modular Inverse - Revisited" Çetin Koç, E. Savas
https://citeseerx.ist.psu.edu/viewdoc/citations?doi=10.1.1.75.8377
A const time implementation of this algorithm is described in
"Constant Time Modular Inversion" Joppe W. Bos
http://www.joppebos.com/files/CTInversion.pdf
*/
size_t almost_montgomery_inverse(BigInt& result,
const BigInt& a,
const BigInt& p)
{
size_t k = 0;
BigInt u = p, v = a, r = 0, s = 1;
while(v > 0)
{
if(u.is_even())
{
u >>= 1;
s <<= 1;
}
else if(v.is_even())
{
v >>= 1;
r <<= 1;
}
else if(u > v)
{
u -= v;
u >>= 1;
r += s;
s <<= 1;
}
else
{
v -= u;
v >>= 1;
s += r;
r <<= 1;
}
++k;
}
if(r >= p)
{
r -= p;
}
result = p - r;
return k;
}
BigInt normalized_montgomery_inverse(const BigInt& a, const BigInt& p)
{
BigInt r;
size_t k = almost_montgomery_inverse(r, a, p);
for(size_t i = 0; i != k; ++i)
{
if(r.is_odd())
r += p;
r >>= 1;
}
return r;
}
BigInt ct_inverse_mod_odd_modulus(const BigInt& n, const BigInt& mod)
{
if(n.is_negative() || mod.is_negative())
throw Invalid_Argument("ct_inverse_mod_odd_modulus: arguments must be non-negative");
if(mod < 3 || mod.is_even())
throw Invalid_Argument("Bad modulus to ct_inverse_mod_odd_modulus");
if(n >= mod)
throw Invalid_Argument("ct_inverse_mod_odd_modulus n >= mod not supported");
/*
This uses a modular inversion algorithm designed by Niels Möller
and implemented in Nettle. The same algorithm was later also
adapted to GMP in mpn_sec_invert.
It can be easily implemented in a way that does not depend on
secret branches or memory lookups, providing resistance against
some forms of side channel attack.
There is also a description of the algorithm in Appendix 5 of "Fast
Software Polynomial Multiplication on ARM Processors using the NEON Engine"
by Danilo Câmara, Conrado P. L. Gouvêa, Julio López, and Ricardo
Dahab in LNCS 8182
https://conradoplg.cryptoland.net/files/2010/12/mocrysen13.pdf
Thanks to Niels for creating the algorithm, explaining some things
about it, and the reference to the paper.
*/
// todo allow this to be pre-calculated and passed in as arg
BigInt mp1o2 = (mod + 1) >> 1;
const size_t mod_words = mod.sig_words();
BOTAN_ASSERT(mod_words > 0, "Not empty");
BigInt a = n;
BigInt b = mod;
BigInt u = 1, v = 0;
a.grow_to(mod_words);
u.grow_to(mod_words);
v.grow_to(mod_words);
mp1o2.grow_to(mod_words);
secure_vector<word>& a_w = a.get_word_vector();
secure_vector<word>& b_w = b.get_word_vector();
secure_vector<word>& u_w = u.get_word_vector();
secure_vector<word>& v_w = v.get_word_vector();
CT::poison(a_w.data(), a_w.size());
CT::poison(b_w.data(), b_w.size());
CT::poison(u_w.data(), u_w.size());
CT::poison(v_w.data(), v_w.size());
// Only n.bits() + mod.bits() iterations are required, but avoid leaking the size of n
size_t bits = 2 * mod.bits();
while(bits--)
{
/*
const word odd = a.is_odd();
a -= odd * b;
const word underflow = a.is_negative();
b += a * underflow;
a.set_sign(BigInt::Positive);
a >>= 1;
if(underflow)
{
std::swap(u, v);
}
u -= odd * v;
u += u.is_negative() * mod;
const word odd_u = u.is_odd();
u >>= 1;
u += mp1o2 * odd_u;
*/
const word odd_a = a_w[0] & 1;
//if(odd_a) a -= b
word underflow = bigint_cnd_sub(odd_a, a_w.data(), b_w.data(), mod_words);
//if(underflow) { b -= a; a = abs(a); swap(u, v); }
bigint_cnd_add(underflow, b_w.data(), a_w.data(), mod_words);
bigint_cnd_abs(underflow, a_w.data(), mod_words);
bigint_cnd_swap(underflow, u_w.data(), v_w.data(), mod_words);
// a >>= 1
bigint_shr1(a_w.data(), mod_words, 0, 1);
//if(odd_a) u -= v;
word borrow = bigint_cnd_sub(odd_a, u_w.data(), v_w.data(), mod_words);
// if(borrow) u += p
bigint_cnd_add(borrow, u_w.data(), mod.data(), mod_words);
const word odd_u = u_w[0] & 1;
// u >>= 1
bigint_shr1(u_w.data(), mod_words, 0, 1);
//if(odd_u) u += mp1o2;
bigint_cnd_add(odd_u, u_w.data(), mp1o2.data(), mod_words);
}
CT::unpoison(a_w.data(), a_w.size());
CT::unpoison(b_w.data(), b_w.size());
CT::unpoison(u_w.data(), u_w.size());
CT::unpoison(v_w.data(), v_w.size());
BOTAN_ASSERT(a.is_zero(), "A is zero");
if(b != 1)
return 0;
return v;
}
/*
* Find the Modular Inverse
*/
BigInt inverse_mod(const BigInt& n, const BigInt& mod)
{
if(mod.is_zero())
throw BigInt::DivideByZero();
if(mod.is_negative() || n.is_negative())
throw Invalid_Argument("inverse_mod: arguments must be non-negative");
if(n.is_zero() || (n.is_even() && mod.is_even()))
return 0; // fast fail checks
if(mod.is_odd() && n < mod)
return ct_inverse_mod_odd_modulus(n, mod);
return inverse_euclid(n, mod);
}
BigInt inverse_euclid(const BigInt& n, const BigInt& mod)
{
if(mod.is_zero())
throw BigInt::DivideByZero();
if(mod.is_negative() || n.is_negative())
throw Invalid_Argument("inverse_mod: arguments must be non-negative");
if(n.is_zero() || (n.is_even() && mod.is_even()))
return 0; // fast fail checks
BigInt u = mod, v = n;
BigInt A = 1, B = 0, C = 0, D = 1;
while(u.is_nonzero())
{
const size_t u_zero_bits = low_zero_bits(u);
u >>= u_zero_bits;
for(size_t i = 0; i != u_zero_bits; ++i)
{
if(A.is_odd() || B.is_odd())
{ A += n; B -= mod; }
A >>= 1; B >>= 1;
}
const size_t v_zero_bits = low_zero_bits(v);
v >>= v_zero_bits;
for(size_t i = 0; i != v_zero_bits; ++i)
{
if(C.is_odd() || D.is_odd())
{ C += n; D -= mod; }
C >>= 1; D >>= 1;
}
if(u >= v) { u -= v; A -= C; B -= D; }
else { v -= u; C -= A; D -= B; }
}
if(v != 1)
return 0; // no modular inverse
while(D.is_negative()) D += mod;
while(D >= mod) D -= mod;
return D;
}
word monty_inverse(word input)
{
if(input == 0)
throw Exception("monty_inverse: divide by zero");
word b = input;
word x2 = 1, x1 = 0, y2 = 0, y1 = 1;
// First iteration, a = n+1
word q = bigint_divop(1, 0, b);
word r = (MP_WORD_MAX - q*b) + 1;
word x = x2 - q*x1;
word y = y2 - q*y1;
word a = b;
b = r;
x2 = x1;
x1 = x;
y2 = y1;
y1 = y;
while(b > 0)
{
q = a / b;
r = a - q*b;
x = x2 - q*x1;
y = y2 - q*y1;
a = b;
b = r;
x2 = x1;
x1 = x;
y2 = y1;
y1 = y;
}
const word check = y2 * input;
BOTAN_ASSERT_EQUAL(check, 1, "monty_inverse result is inverse of input");
// Now invert in addition space
y2 = (MP_WORD_MAX - y2) + 1;
return y2;
}
/*
* Modular Exponentiation
*/
BigInt power_mod(const BigInt& base, const BigInt& exp, const BigInt& mod)
{
if(mod.is_negative() || mod == 1)
{
return 0;
}
if(base.is_zero() || mod.is_zero())
{
if(exp.is_zero())
return 1;
return 0;
}
Power_Mod pow_mod(mod);
/*
* Calling set_base before set_exponent means we end up using a
* minimal window. This makes sense given that here we know that any
* precomputation is wasted.
*/
if(base.is_negative())
{
pow_mod.set_base(-base);
pow_mod.set_exponent(exp);
if(exp.is_even())
return pow_mod.execute();
else
return (mod - pow_mod.execute());
}
else
{
pow_mod.set_base(base);
pow_mod.set_exponent(exp);
return pow_mod.execute();
}
}
BigInt is_perfect_square(const BigInt& C)
{
if(C < 1)
throw Invalid_Argument("is_perfect_square requires C >= 1");
if(C == 1)
return 1;
const size_t n = C.bits();
const size_t m = (n + 1) / 2;
const BigInt B = C + BigInt::power_of_2(m);
BigInt X = BigInt::power_of_2(m) - 1;
BigInt X2 = (X*X);
for(;;)
{
X = (X2 + C) / (2*X);
X2 = (X*X);
if(X2 < B)
break;
}
if(X2 == C)
return X;
else
return 0;
}
/*
* Test for primality using Miller-Rabin
*/
bool is_prime(const BigInt& n,
RandomNumberGenerator& rng,
size_t prob,
bool is_random)
{
if(n == 2)
return true;
if(n <= 1 || n.is_even())
return false;
const size_t n_bits = n.bits();
// Fast path testing for small numbers (<= 65521)
if(n_bits <= 16)
{
const uint16_t num = static_cast<uint16_t>(n.word_at(0));
return std::binary_search(PRIMES, PRIMES + PRIME_TABLE_SIZE, num);
}
const size_t t = miller_rabin_test_iterations(n_bits, prob, is_random);
Modular_Reducer mod_n(n);
if(rng.is_seeded())
{
if(is_miller_rabin_probable_prime(n, mod_n, rng, t) == false)
return false;
return is_lucas_probable_prime(n, mod_n);
}
else
{
return is_bailie_psw_probable_prime(n, mod_n);
}
}
}
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