20283964805

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Build # 20283964805

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Pull #5180

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Merge 9310065bb into 3d96b675e

Pull Request Pull Request #5180: Avoid integer modulo operations when sieving for primes

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/src/lib/math/numbertheory/make_prm.cpp

/*
* Prime Generation
* (C) 1999-2007,2018,2019 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/

#include <botan/internal/primality.h>

#include <botan/numthry.h>
#include <botan/rng.h>
#include <botan/internal/barrett.h>
#include <botan/internal/bit_ops.h>
#include <botan/internal/ct_utils.h>
#include <botan/internal/divide.h>
#include <botan/internal/loadstor.h>
#include <algorithm>

namespace Botan {

namespace {

class Prime_Sieve final {
   public:
      Prime_Sieve(const BigInt& init_value, size_t sieve_size, word step, bool check_2p1) :
            m_sieve(std::min(sieve_size, PRIME_TABLE_SIZE)), m_step(step), m_check_2p1(check_2p1) {
         for(size_t i = 0; i != m_sieve.size(); ++i) {
            m_sieve[i] = ct_mod_word(init_value, PRIMES[i]);
         }
      }

      size_t sieve_size() const { return m_sieve.size(); }

      bool check_2p1() const { return m_check_2p1; }

      bool next() {
         auto passes = CT::Mask<word>::set();
         for(size_t i = 0; i != m_sieve.size(); ++i) {
            m_sieve[i] = sieve_step_incr(m_sieve[i], m_step, PRIMES[i]);

            // If m_sieve[i] == 0 then val % p == 0 -> not prime
            passes &= CT::Mask<word>::expand(m_sieve[i]);

            if(this->check_2p1()) {
               /*
               If v % p == (p-1)/2 then 2*v+1 == 0 (mod p)

               So if potentially generating a safe prime, we want to
               avoid this value because 2*v+1 will certainly not be prime.

               See "Safe Prime Generation with a Combined Sieve" M. Wiener
               https://eprint.iacr.org/2003/186.pdf
               */
               passes &= ~CT::Mask<word>::is_equal(m_sieve[i], (PRIMES[i] - 1) / 2);
            }
         }

         return passes.as_bool();
      }

   private:
      // Return (v + step) % mod
      //
      // This assumes v is already < mod, which is an invariant of the sieve
      static constexpr word sieve_step_incr(word v, word step, word mod) {
         BOTAN_DEBUG_ASSERT(v < mod);
         // The sieve step and primes are public so this modulo is ok
         const word stepmod = (step >= mod) ? (step % mod) : step;

         // This sum is at most 2*(mod-1)
         const word next = (v + stepmod);
         return next - CT::Mask<word>::is_gte(next, mod).if_set_return(mod);
      }

      std::vector<word> m_sieve;
      const word m_step;
      const bool m_check_2p1;
};

#if defined(BOTAN_ENABLE_DEBUG_ASSERTS)

bool no_small_multiples(const BigInt& v, const Prime_Sieve& sieve) {
   const size_t sieve_size = sieve.sieve_size();
   const bool check_2p1 = sieve.check_2p1();

   if(v.is_even())
      return false;

   const BigInt v_x2_p1 = 2 * v + 1;

   for(size_t i = 0; i != sieve_size; ++i) {
      if((v % PRIMES[i]) == 0)
         return false;

      if(check_2p1) {
         if(v_x2_p1 % PRIMES[i] == 0)
            return false;
      }
   }

   return true;
}

#endif

}  // namespace

/*
* Generate a random prime
*/
BigInt random_prime(
   RandomNumberGenerator& rng, size_t bits, const BigInt& coprime, size_t equiv, size_t modulo, size_t prob) {
   if(bits <= 1) {
      throw Invalid_Argument("random_prime: Can't make a prime of " + std::to_string(bits) + " bits");
   }
   if(coprime.is_negative() || (!coprime.is_zero() && coprime.is_even()) || coprime.bits() >= bits) {
      throw Invalid_Argument("random_prime: invalid coprime");
   }
   // TODO(Botan4) reduce this to ~1000
   if(modulo == 0 || modulo >= 100000) {
      throw Invalid_Argument("random_prime: Invalid modulo value");
   }

   equiv %= modulo;

   if(equiv == 0) {
      throw Invalid_Argument("random_prime Invalid value for equiv/modulo");
   }

   // Handle small values:

   if(bits <= 16) {
      if(equiv != 1 || modulo != 2 || coprime != 0) {
         throw Not_Implemented("random_prime equiv/modulo/coprime options not usable for small primes");
      }

      if(bits == 2) {
         return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 2 : 3));
      } else if(bits == 3) {
         return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 5 : 7));
      } else if(bits == 4) {
         return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 11 : 13));
      } else {
         for(;;) {
            // This is slightly biased, but for small primes it does not seem to matter
            uint8_t b[4] = {0};
            rng.randomize(b, 4);
            const size_t idx = load_le<uint32_t>(b, 0) % PRIME_TABLE_SIZE;
            const uint16_t small_prime = PRIMES[idx];

            if(high_bit(small_prime) == bits) {
               return BigInt::from_word(small_prime);
            }
         }
      }
   }

   const size_t MAX_ATTEMPTS = 32 * 1024;

   const size_t mr_trials = miller_rabin_test_iterations(bits, prob, true);

   while(true) {
      BigInt p(rng, bits);

      // Force lowest and two top bits on
      p.set_bit(bits - 1);
      p.set_bit(bits - 2);
      p.set_bit(0);

      // Force p to be equal to equiv mod modulo
      p += (modulo - (p % modulo)) + equiv;

      Prime_Sieve sieve(p, bits, modulo, true);

      for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt) {
         p += modulo;

         if(!sieve.next()) {
            continue;
         }

         // here p can be even if modulo is odd, continue on in that case
         if(p.is_even()) {
            continue;
         }

         BOTAN_DEBUG_ASSERT(no_small_multiples(p, sieve));

         auto mod_p = Barrett_Reduction::for_secret_modulus(p);

         if(coprime > 1) {
            /*
            First do a single M-R iteration to quickly eliminate most non-primes,
            before doing the coprimality check which is expensive
            */
            if(!is_miller_rabin_probable_prime(p, mod_p, rng, 1)) {
               continue;
            }

            /*
            * Check if p - 1 and coprime are relatively prime, using gcd.
            * The gcd computation is const-time
            */
            if(gcd(p - 1, coprime) > 1) {
               continue;
            }
         }

         if(p.bits() > bits) {
            break;
         }

         if(!is_miller_rabin_probable_prime(p, mod_p, rng, mr_trials)) {
            continue;
         }

         if(prob > 32 && !is_lucas_probable_prime(p, mod_p)) {
            continue;
         }

         return p;
      }
   }
}

BigInt generate_rsa_prime(RandomNumberGenerator& keygen_rng,
                          RandomNumberGenerator& prime_test_rng,
                          size_t bits,
                          const BigInt& coprime,
                          size_t prob) {
   if(bits < 512) {
      throw Invalid_Argument("generate_rsa_prime bits too small");
   }

   /*
   * The restriction on coprime <= 64 bits is arbitrary but generally speaking
   * very large RSA public exponents are a bad idea both for performance and due
   * to attacks on small d.
   */
   if(coprime <= 1 || coprime.is_even() || coprime.bits() > 64) {
      throw Invalid_Argument("generate_rsa_prime coprime must be small odd positive integer");
   }

   const size_t MAX_ATTEMPTS = 32 * 1024;

   const size_t mr_trials = miller_rabin_test_iterations(bits, prob, true);

   while(true) {
      BigInt p(keygen_rng, bits);

      CT::poison(p);

      /*
      Force high two bits so multiplication always results in expected n bit integer

      Force the two low bits, and step by 4, so the generated prime is always == 3 (mod 4).
      This way when we perform the inversion modulo phi(n) it is always of the form 2*o
      with o odd, which allows a fastpath and avoids leaking any information about the
      structure of the prime.
      */
      p.set_bit(bits - 1);
      p.set_bit(bits - 2);
      p.set_bit(1);
      p.set_bit(0);

      const word step = 4;

      Prime_Sieve sieve(p, bits, step, false);

      for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt) {
         p += step;

         if(!sieve.next()) {
            continue;
         }

         BOTAN_DEBUG_ASSERT(no_small_multiples(p, sieve));

         auto mod_p = Barrett_Reduction::for_secret_modulus(p);

         /*
         * Do a single primality test first before checking coprimality, since
         * currently a single Miller-Rabin test is faster than computing gcd,
         * and this eliminates almost all wasted gcd computations.
         */
         if(!is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, 1)) {
            continue;
         }

         /*
         * Check if p - 1 and coprime are relatively prime.
         */
         if(gcd(p - 1, coprime) > 1) {
            continue;
         }

         if(p.bits() > bits) {
            break;
         }

         if(is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, mr_trials)) {
            CT::unpoison(p);
            return p;
         }
      }
   }
}

/*
* Generate a random safe prime
*/
BigInt random_safe_prime(RandomNumberGenerator& rng, size_t bits) {
   if(bits <= 64) {
      throw Invalid_Argument("random_safe_prime: Can't make a prime of " + std::to_string(bits) + " bits");
   }

   const size_t error_bound = 128;

   BigInt q;
   BigInt p;
   for(;;) {
      /*
      Generate q == 2 (mod 3), since otherwise [in the case of q == 1 (mod 3)],
      2*q+1 == 3 (mod 3) and so certainly not prime.
      */
      q = random_prime(rng, bits - 1, BigInt::zero(), 2, 3, error_bound);
      p = (q << 1) + 1;

      if(is_prime(p, rng, error_bound, true)) {
         return p;
      }
   }
}

}  // namespace Botan

1	/*
2	* Prime Generation
3	* (C) 1999-2007,2018,2019 Jack Lloyd
4	*
5	* Botan is released under the Simplified BSD License (see license.txt)
6	*/
7
8	#include <botan/internal/primality.h>
9
10	#include <botan/numthry.h>
11	#include <botan/rng.h>
12	#include <botan/internal/barrett.h>
13	#include <botan/internal/bit_ops.h>
14	#include <botan/internal/ct_utils.h>
15	#include <botan/internal/divide.h>
16	#include <botan/internal/loadstor.h>
17	#include <algorithm>
18
19	namespace Botan {
20
21	namespace {
22
23	class Prime_Sieve final {	×
24	public:
25	Prime_Sieve(const BigInt& init_value, size_t sieve_size, word step, bool check_2p1) :	926✔
26	m_sieve(std::min(sieve_size, PRIME_TABLE_SIZE)), m_step(step), m_check_2p1(check_2p1) {	926✔
27	for(size_t i = 0; i != m_sieve.size(); ++i) {	508,405✔
28	m_sieve[i] = ct_mod_word(init_value, PRIMES[i]);	507,479✔
29	}
30	}	926✔
31
32	size_t sieve_size() const { return m_sieve.size(); }
33
34	bool check_2p1() const { return m_check_2p1; }	1,147,853,322✔
35
36	bool next() {	1,345,804✔
37	auto passes = CT::Mask<word>::set();	1,345,804✔
38	for(size_t i = 0; i != m_sieve.size(); ++i) {	1,149,199,126✔
39	m_sieve[i] = sieve_step_incr(m_sieve[i], m_step, PRIMES[i]);	1,147,853,322✔
40
41	// If m_sieve[i] == 0 then val % p == 0 -> not prime
42	passes &= CT::Mask<word>::expand(m_sieve[i]);	1,147,853,322✔
43
44	if(this->check_2p1()) {	1,147,853,322✔
45	/*
46	If v % p == (p-1)/2 then 2*v+1 == 0 (mod p)
47
48	So if potentially generating a safe prime, we want to
49	avoid this value because 2*v+1 will certainly not be prime.
50
51	See "Safe Prime Generation with a Combined Sieve" M. Wiener
52	https://eprint.iacr.org/2003/186.pdf
53	*/
54	passes &= ~CT::Mask<word>::is_equal(m_sieve[i], (PRIMES[i] - 1) / 2);	1,120,514,186✔
55	}
56	}
57
58	return passes.as_bool();	1,345,804✔
59	}
60
61	private:
62	// Return (v + step) % mod
63	//
64	// This assumes v is already < mod, which is an invariant of the sieve
65	static constexpr word sieve_step_incr(word v, word step, word mod) {	1,147,853,322✔
66	BOTAN_DEBUG_ASSERT(v < mod);	1,147,853,322✔
67	// The sieve step and primes are public so this modulo is ok
68	const word stepmod = (step >= mod) ? (step % mod) : step;	1,147,853,322✔
69
70	// This sum is at most 2*(mod-1)
71	const word next = (v + stepmod);	1,147,853,322✔
72	return next - CT::Mask<word>::is_gte(next, mod).if_set_return(mod);	1,147,853,322✔
73	}
74
75	std::vector<word> m_sieve;
76	const word m_step;
77	const bool m_check_2p1;
78	};
79
80	#if defined(BOTAN_ENABLE_DEBUG_ASSERTS)
81
82	bool no_small_multiples(const BigInt& v, const Prime_Sieve& sieve) {
83	const size_t sieve_size = sieve.sieve_size();
84	const bool check_2p1 = sieve.check_2p1();
85
86	if(v.is_even())
87	return false;
88
89	const BigInt v_x2_p1 = 2 * v + 1;
90
91	for(size_t i = 0; i != sieve_size; ++i) {
92	if((v % PRIMES[i]) == 0)
93	return false;
94
95	if(check_2p1) {
96	if(v_x2_p1 % PRIMES[i] == 0)
97	return false;
98	}
99	}
100
101	return true;
102	}
103
104	#endif
105
106	} // namespace
107
108	/*
109	* Generate a random prime
110	*/
111	BigInt random_prime(	800✔
112	RandomNumberGenerator& rng, size_t bits, const BigInt& coprime, size_t equiv, size_t modulo, size_t prob) {
113	if(bits <= 1) {	800✔
114	throw Invalid_Argument("random_prime: Can't make a prime of " + std::to_string(bits) + " bits");	8✔
115	}
116	if(coprime.is_negative() \|\| (!coprime.is_zero() && coprime.is_even()) \|\| coprime.bits() >= bits) {	1,603✔
117	throw Invalid_Argument("random_prime: invalid coprime");	×
118	}
119	// TODO(Botan4) reduce this to ~1000
120	if(modulo == 0 \|\| modulo >= 100000) {	798✔
121	throw Invalid_Argument("random_prime: Invalid modulo value");	×
122	}
123
124	equiv %= modulo;	798✔
125
126	if(equiv == 0) {	798✔
127	throw Invalid_Argument("random_prime Invalid value for equiv/modulo");	1✔
128	}
129
130	// Handle small values:
131
132	if(bits <= 16) {	797✔
133	if(equiv != 1 \|\| modulo != 2 \|\| coprime != 0) {	38✔
134	throw Not_Implemented("random_prime equiv/modulo/coprime options not usable for small primes");	×
135	}
136
137	if(bits == 2) {	19✔
138	return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 2 : 3));	2✔
139	} else if(bits == 3) {	18✔
140	return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 5 : 7));	2✔
141	} else if(bits == 4) {	17✔
142	return BigInt::from_word(((rng.next_byte() % 2) == 0 ? 11 : 13));	1✔
143	} else {
144	for(;;) {	9,390✔
145	// This is slightly biased, but for small primes it does not seem to matter
146	uint8_t b[4] = {0};	4,703✔
147	rng.randomize(b, 4);	4,703✔
148	const size_t idx = load_le<uint32_t>(b, 0) % PRIME_TABLE_SIZE;	4,703✔
149	const uint16_t small_prime = PRIMES[idx];	4,703✔
150
151	if(high_bit(small_prime) == bits) {	9,406✔
152	return BigInt::from_word(small_prime);	16✔
153	}
154	}	4,687✔
155	}
156	}
157
158	const size_t MAX_ATTEMPTS = 32 * 1024;	778✔
159
160	const size_t mr_trials = miller_rabin_test_iterations(bits, prob, true);	778✔
161
162	while(true) {	×
163	BigInt p(rng, bits);	778✔
164
165	// Force lowest and two top bits on
166	p.set_bit(bits - 1);	778✔
167	p.set_bit(bits - 2);	778✔
168	p.set_bit(0);	778✔
169
170	// Force p to be equal to equiv mod modulo
171	p += (modulo - (p % modulo)) + equiv;	778✔
172
173	Prime_Sieve sieve(p, bits, modulo, true);	778✔
174
175	for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt) {	1,312,223✔
176	p += modulo;	1,312,223✔
177
178	if(!sieve.next()) {	1,312,223✔
179	continue;	1,311,445✔
180	}
181
182	// here p can be even if modulo is odd, continue on in that case
183	if(p.is_even()) {	61,162✔
184	continue;	11,317✔
185	}
186
187	BOTAN_DEBUG_ASSERT(no_small_multiples(p, sieve));	19,264✔
188
189	auto mod_p = Barrett_Reduction::for_secret_modulus(p);	19,264✔
190
191	if(coprime > 1) {	19,264✔
192	/*
193	First do a single M-R iteration to quickly eliminate most non-primes,
194	before doing the coprimality check which is expensive
195	*/
196	if(!is_miller_rabin_probable_prime(p, mod_p, rng, 1)) {	99✔
197	continue;	93✔
198	}
199
200	/*
201	* Check if p - 1 and coprime are relatively prime, using gcd.
202	* The gcd computation is const-time
203	*/
204	if(gcd(p - 1, coprime) > 1) {	12✔
205	continue;	×
206	}
207	}
208
209	if(p.bits() > bits) {	19,171✔
210	break;
211	}
212
213	if(!is_miller_rabin_probable_prime(p, mod_p, rng, mr_trials)) {	19,171✔
214	continue;	18,393✔
215	}
216
217	if(prob > 32 && !is_lucas_probable_prime(p, mod_p)) {	778✔
218	continue;	×
219	}
220
221	return p;	778✔
222	}	19,264✔
223	}	778✔
224	}
225
226	BigInt generate_rsa_prime(RandomNumberGenerator& keygen_rng,	148✔
227	RandomNumberGenerator& prime_test_rng,
228	size_t bits,
229	const BigInt& coprime,
230	size_t prob) {
231	if(bits < 512) {	148✔
232	throw Invalid_Argument("generate_rsa_prime bits too small");	×
233	}
234
235	/*
236	* The restriction on coprime <= 64 bits is arbitrary but generally speaking
237	* very large RSA public exponents are a bad idea both for performance and due
238	* to attacks on small d.
239	*/
240	if(coprime <= 1 \|\| coprime.is_even() \|\| coprime.bits() > 64) {	296✔
241	throw Invalid_Argument("generate_rsa_prime coprime must be small odd positive integer");	×
242	}
243
244	const size_t MAX_ATTEMPTS = 32 * 1024;	148✔
245
246	const size_t mr_trials = miller_rabin_test_iterations(bits, prob, true);	148✔
247
248	while(true) {	×
249	BigInt p(keygen_rng, bits);	148✔
250
251	CT::poison(p);	148✔
252
253	/*
254	Force high two bits so multiplication always results in expected n bit integer
255
256	Force the two low bits, and step by 4, so the generated prime is always == 3 (mod 4).
257	This way when we perform the inversion modulo phi(n) it is always of the form 2*o
258	with o odd, which allows a fastpath and avoids leaking any information about the
259	structure of the prime.
260	*/
261	p.set_bit(bits - 1);	148✔
262	p.set_bit(bits - 2);	148✔
263	p.set_bit(1);	148✔
264	p.set_bit(0);	148✔
265
266	const word step = 4;	148✔
267
268	Prime_Sieve sieve(p, bits, step, false);	148✔
269
270	for(size_t attempt = 0; attempt <= MAX_ATTEMPTS; ++attempt) {	33,581✔
271	p += step;	33,581✔
272
273	if(!sieve.next()) {	33,581✔
274	continue;	33,433✔
275	}
276
277	BOTAN_DEBUG_ASSERT(no_small_multiples(p, sieve));	4,478✔
278
279	auto mod_p = Barrett_Reduction::for_secret_modulus(p);	4,478✔
280
281	/*
282	* Do a single primality test first before checking coprimality, since
283	* currently a single Miller-Rabin test is faster than computing gcd,
284	* and this eliminates almost all wasted gcd computations.
285	*/
286	if(!is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, 1)) {	4,478✔
287	continue;	4,330✔
288	}
289
290	/*
291	* Check if p - 1 and coprime are relatively prime.
292	*/
293	if(gcd(p - 1, coprime) > 1) {	296✔
294	continue;	×
295	}
296
297	if(p.bits() > bits) {	148✔
298	break;
299	}
300
301	if(is_miller_rabin_probable_prime(p, mod_p, prime_test_rng, mr_trials)) {	148✔
302	CT::unpoison(p);	148✔
303	return p;	296✔
304	}
305	}	4,478✔
306	}	148✔
307	}
308
309	/*
310	* Generate a random safe prime
311	*/
312	BigInt random_safe_prime(RandomNumberGenerator& rng, size_t bits) {	8✔
313	if(bits <= 64) {	8✔
314	throw Invalid_Argument("random_safe_prime: Can't make a prime of " + std::to_string(bits) + " bits");	×
315	}
316
317	const size_t error_bound = 128;	8✔
318
319	BigInt q;	8✔
320	BigInt p;	8✔
321	for(;;) {	248✔
322	/*
323	Generate q == 2 (mod 3), since otherwise [in the case of q == 1 (mod 3)],
324	2*q+1 == 3 (mod 3) and so certainly not prime.
325	*/
326	q = random_prime(rng, bits - 1, BigInt::zero(), 2, 3, error_bound);	248✔
327	p = (q << 1) + 1;	496✔
328
329	if(is_prime(p, rng, error_bound, true)) {	248✔
330	return p;	8✔
331	}
332	}
333	}	8✔
334
335	} // namespace Botan

randombit / botan / 20283964805

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