kg_prime.c

cryptlib安全工具包

/****************************************************************************
*																			*
*					cryptlib Prime Generation/Checking Routines				*
*						Copyright Peter Gutmann 1997-2007					*
*																			*
****************************************************************************/

/* The Usenet Oracle has pondered your question deeply.
   Your question was:

   > O Oracle Most Wise,
   >
   > What is the largest prime number?

   And in response, thus spake the Oracle:

   } This is a question which has stumped some of the best minds in
   } mathematics, but I will explain it so that even you can understand it.
   } The first prime is 2, and the binary representation of 2 is 10.
   } Consider the following series:
   }
   }	Prime	Decimal Representation	Representation in its own base
   }	1st		2						10
   }	2nd		3						10
   }	3rd		5						10
   }	4th		7						10
   }	5th		11						10
   }	6th		13						10
   }	7th		17						10
   }
   } From this demonstration you can see that there is only one prime, and
   } it is ten. Therefore, the largest prime is ten.
													-- The Usenet Oracle */

#define PKC_CONTEXT		/* Indicate that we're working with PKC contexts */
#if defined( INC_ALL )
  #include "crypt.h"
  #include "context.h"
  #include "keygen.h"
#else
  #include "crypt.h"
  #include "context/context.h"
  #include "context/keygen.h"
#endif /* Compiler-specific includes */

/****************************************************************************
*																			*
*								Fast Prime Sieve							*
*																			*
****************************************************************************/

/* #include 4k of EAY copyright */

/* The following define is necessary in memory-starved environments.  It
   controls the size of the table used for the sieving */

#if defined( CONFIG_CONSERVE_MEMORY )
  #define EIGHT_BIT
#endif /* CONFIG_CONSERVE_MEMORY */

/* Pull in the table of primes */

#if defined( INC_ALL )
  #include "bn_prime.h"
#else
  #include "bn/bn_prime.h"
#endif /* Compiler-specific includes */

/* The number of primes in the sieve (and their values) that result in a
   given number of candidates remaining from 40,000.  Even the first 100
   primes weed out 91% of all the candidates, and after 500 you're only
   removing a handful for each 100 extra primes.

	 Number		   Prime	Candidates left
				  Values	from 40,000
	--------	---------	---------------
	  0- 99		   0- 541		3564
	100-199		 541-1223		3175
	200-299		1223-1987		2969
	300-399		1987-2741		2845
	400-499		2741-3571		2755
	500-599		3571-4409		2688
	600-699		4409-5279		2629
	700-799		5279-6133		2593
	800-899		6133-6997		2555
	900-999		6997-7919		2521

  There is in fact an even faster prime tester due to Dan Piponi that uses
  C++ templates as a universal computer and performs the primality test at
  compile time, however this requires the use of a fairly advanced C++
  compiler and isn't amenable to generating different primes */

/* Enable the following to cross-check the Miller-Rabin test using a Fermat 
   test and an alternative form of the Miller-Rabin test that merges the
   test loop and the modexp at the start.  Note that this displays 
   diagnostic timing output and expects to use Pentium performance counters 
   for timing so it's only (optionally) enabled for Win32 debug */

#if defined( __WIN32__ ) && !defined( NDEBUG ) && 0
  #define CHECK_PRIMETEST
#endif /* Win32 debug */

/* The size of the sieve array, 1 memory page (on most CPUs) = 4K candidate
   values.  When changing this value the LFSR parameters need to be adjusted
   to match */

#define SIEVE_SIZE				4096

/* When we're doing a sieve of a singleton candidate we don't run through
   the whole range of sieve values since we run into the law of diminshing
   returns after a certain point.  The following value sieves with every
   prime under 1000 */

#if NUMPRIMES < ( 21 * 8 )
  #define FAST_SIEVE_NUMPRIMES	NUMPRIMES
#else
  #define FAST_SIEVE_NUMPRIMES	( 21 * 8 )
#endif /* Small prime table */

/* Set up the sieve array for the number.  Every position that contains
   a zero is non-divisible by all of the small primes */

STDC_NONNULL_ARG( ( 1, 3 ) ) \
static void initSieve( IN_ARRAY( sieveSize ) BOOLEAN *sieveArray, 
					   IN_LENGTH_FIXED( SIEVE_SIZE ) const int sieveSize,
					   const BIGNUM *candidate )
	{
	int i;

	assert( isWritePtr( sieveArray, sieveSize * sizeof( BOOLEAN ) ) );
	assert( isReadPtr( candidate, sizeof( BIGNUM ) ) );

	REQUIRES_V( sieveSize == SIEVE_SIZE );

	memset( sieveArray, 0, sieveSize * sizeof( BOOLEAN ) );

	/* Walk down the list of primes marking the appropriate position in the
	   array as divisible by the prime.  We start at index 1 because the
	   candidate will never be divisible by 2 (== primes[ 0 ]) */
	for( i = 1; i < NUMPRIMES; i++ )
		{
		unsigned int step = primes[ i ];
		BN_ULONG sieveIndex = BN_mod_word( candidate, step );

		/* Determine the correct start index for this value */
		if( sieveIndex & 1 )
			sieveIndex = ( step - sieveIndex ) / 2;
		else
			{
			if( sieveIndex > 0 )
				sieveIndex = ( ( step * 2 ) - sieveIndex ) / 2;
			}

		/* Mark each multiple of the divisor as being divisible */
		while( sieveIndex >= 0 && sieveIndex < sieveSize )
			{
			sieveArray[ sieveIndex ] = 1;
			sieveIndex += step;
			}
		}
	}

/* An LFSR to step through each entry in the sieve array.  This isn't a true 
   pseudorandom selection since all that it's really doing is going through 
   the numbers in a linear order with a different starting point, but it's 
   good enough as a randomiser */

#define LFSR_POLYNOMIAL		0x1053
#define LFSR_MASK			0x1000

CHECK_RETVAL \
static int nextEntry( IN_INT_SHORT int value )
	{
	assert( LFSR_MASK == SIEVE_SIZE );
	
	REQUIRES( value > 0 && value < SIEVE_SIZE );

	/* Get the next value: Multiply by x and reduce by the polynomial */
	value <<= 1;
	if( value & LFSR_MASK )
		value ^= LFSR_POLYNOMIAL;
	return( value );
	}

/* A one-off sieve check for when we're testing a singleton rather than
   running over a range of values */

CHECK_RETVAL_BOOL STDC_NONNULL_ARG( ( 1 ) ) \
BOOLEAN primeSieve( const BIGNUM *candidate )
	{
	int i;

	assert( isReadPtr( candidate, sizeof( BIGNUM ) ) );

	for( i = 1; i < FAST_SIEVE_NUMPRIMES; i++ )
		{
		if( BN_mod_word( candidate, primes[ i ] ) == 0 )
			return( FALSE );
		}

	return( TRUE );
	}

/****************************************************************************
*																			*
*							Generate a Prime Number							*
*																			*
****************************************************************************/

#ifdef CHECK_PRIMETEST

/* Witness function, modified from original BN code as found at a UFO crash 
   site.  This looks nothing like a standard Miller-Rabin test because it 
   merges the modexp that usually needs to be performed as the first 
   portion of the test process and the remainder of the checking.  Destroys 
   param6 + 7 */

CHECK_RETVAL STDC_NONNULL_ARG( ( 1, 2, 3, 4, 5, 6, 7 ) ) \
static int witnessOld( INOUT PKC_INFO *pkcInfo, INOUT BIGNUM *a, 
					   INOUT BIGNUM *n, INOUT BIGNUM *n1, 
					   INOUT BIGNUM *mont_n1, INOUT BIGNUM *mont_1, 
					   INOUT BN_MONT_CTX *montCTX_n )
	{
	BIGNUM *y = &pkcInfo->param6;
	BIGNUM *yPrime = &pkcInfo->param7;		/* Safe to destroy */
	BN_CTX *ctx = pkcInfo->bnCTX;
	BIGNUM *mont_a = &ctx->bn[ ctx->tos++ ];
	const int k = BN_num_bits( n1 );
	int i, bnStatus = BN_STATUS;

	assert( isWritePtr( pkcInfo, sizeof( PKC_INFO ) ) );
	assert( isWritePtr( a, sizeof( BIGNUM ) ) );
	assert( isWritePtr( n, sizeof( BIGNUM ) ) );
	assert( isWritePtr( n1, sizeof( BIGNUM ) ) );
	assert( isWritePtr( mont_n1, sizeof( BIGNUM ) ) );
	assert( isWritePtr( mont_1, sizeof( BIGNUM ) ) );
	assert( isWritePtr( montCTX_n, sizeof( BN_MONT_CTX ) ) );

	/* All values are manipulated in their Montgomery form so before we 
	   begin we have to convert a to this form as well */
	if( !BN_to_montgomery( mont_a, a, montCTX_n, pkcInfo->bnCTX ) )
		{
		ctx->tos--;
		return( CRYPT_ERROR_FAILED );
		}

	CKPTR( BN_copy( y, mont_1 ) );
	for ( i = k - 1; i >= 0; i-- )
		{
		/* Perform the y^2 mod n check.  yPrime = y^2 mod n, if yPrime == 1
		   it's composite (this condition is virtually never met) */
		CK( BN_mod_mul_montgomery( yPrime, y, y, montCTX_n, 
								   pkcInfo->bnCTX ) );
		if( bnStatusError( bnStatus ) || \
			( !BN_cmp( yPrime, mont_1 ) && \
			  BN_cmp( y, mont_1 ) && BN_cmp( y, mont_n1 ) ) )
			{
			ctx->tos--;
			return( TRUE );
			}

		/* Perform another step of the modexp */
		if( BN_is_bit_set( n1, i ) )
			{
			CK( BN_mod_mul_montgomery( y, yPrime, mont_a, montCTX_n, 
									   pkcInfo->bnCTX ) );
			}
		else
			{
			BIGNUM *tmp;

			/* Input and output to modmult can't be the same, so we have to
			   swap the pointers */
			tmp = y; y = yPrime; yPrime = tmp;
			}
		}
	ctx->tos--;

	/* Finally we have y = a^u mod n.  If y == 1 (mod n) it's prime,
	   otherwise it's composite */
	return( BN_cmp( y, mont_1 ) ? TRUE : FALSE );
	}

/* Perform noChecks iterations of the Miller-Rabin probabilistic primality 
   test.  Destroys param8, tmp1-3, mont1 */

CHECK_RETVAL STDC_NONNULL_ARG( ( 1, 2 ) ) \
static int primeProbableOld( INOUT PKC_INFO *pkcInfo, 
							 INOUT BIGNUM *candidate, 
							 IN_RANGE( 1, 100 ) const int noChecks )
	{
	BIGNUM *check = &pkcInfo->tmp1;
	BIGNUM *candidate_1 = &pkcInfo->tmp2;
	BIGNUM *mont_candidate_1 = &pkcInfo->tmp3;
	BIGNUM *mont_1 = &pkcInfo->param8;		/* Safe to destroy */
	BN_MONT_CTX *montCTX_candidate = &pkcInfo->montCTX1;
	int i, bnStatus = BN_STATUS, status;

	assert( isWritePtr( pkcInfo, sizeof( PKC_INFO ) ) );
	assert( isWritePtr( candidate, sizeof( BIGNUM ) ) );

	REQUIRES( noChecks >= 1 && noChecks <= 100 );

	/* Set up various values */
	CK( BN_MONT_CTX_set( montCTX_candidate, candidate, pkcInfo->bnCTX ) );
	if( bnStatusError( bnStatus ) )
		return( getBnStatus( bnStatus ) );
	CK( BN_to_montgomery( mont_1, BN_value_one(), montCTX_candidate, 
						  pkcInfo->bnCTX ) );
	CKPTR( BN_copy( candidate_1, candidate ) );
	CK( BN_sub_word( candidate_1, 1 ) );
	CK( BN_to_montgomery( mont_candidate_1, candidate_1, montCTX_candidate, 
						  pkcInfo->bnCTX ) );
	if( bnStatusError( bnStatus ) )
		return( getBnStatus( bnStatus ) );

	/* Perform n iterations of Miller-Rabin */
	for( i = 0; i < noChecks; i++ )
		{
		/* Instead of using a bignum for the Miller-Rabin check we use a
		   series of small primes.  The reason for this is that if bases a1
		   and a2 are strong liars for n then their product a1*a2 is also 
		   very likely to be a strong liar so using a composite base 
		   doesn't give us any great advantage.  In addition an initial test 
		   with a=2 is beneficial since most composite numbers will fail 
		   Miller-Rabin with a=2, and exponentiation with base 2 is faster 
		   than general-purpose exponentiation.  Finally, using small values 
		   instead of random bignums is both significantly more efficient 
		   and much easier on the RNG.   In theory in order to use the first 
		   noChecks small primes as the base instead of using random bignum 
		   bases we would have to assume that the extended Riemann 
		   hypothesis holds (without this, which allows us to use values 
		   1 < check < 2 * log( candidate )^2, we'd have to pick random 
		   check values as required for Monte Carlo algorithms), however the 
		   requirement for random bases assumes that the candidates could be 
		   chosen maliciously to be pseudoprime to any reasonable list of 
		   bases, thus requiring random bases to evade the problem.  
		   Obviously we're not going to do this so one base is as good as 
		   another, and small primes work well (even a single Fermat test 
		   has a failure probability of around 10e-44 for 512-bit primes if 
		   you're not trying to cook the primes, this is why Fermat works as 
		   a verification of the Miller-Rabin test in generatePrime()) */
		BN_set_word( check, primes[ i ] );
		status = witnessOld( pkcInfo, check, candidate, candidate_1, 
							 mont_candidate_1, mont_1, montCTX_candidate );
		if( cryptStatusError( status ) )
			return( status );
		if( status )
			return( FALSE );	/* It's not a prime */
		}

	/* It's prime */
	return( TRUE );
	}
#endif /* CHECK_PRIMETEST */

/* Less unconventional witness function, which follows the normal pattern:

	x(0) = a^u mod n
	if x(0) = 1 || x(0) = n - 1 
		return "probably-prime"

	for i = 1 to k
		x(i) = x(i-1)^2 mod n
		if x(i) = n - 1
			return "probably-prime"
		if x(i) = 1
			return "composite";
	return "composite"

   Since it's a yes-biased Monte Carlo algorithm this witness function can
   only answer "probably-prime" so we reduce the uncertainty by iterating
   for the Miller-Rabin test */

CHECK_RETVAL STDC_NONNULL_ARG( ( 1, 2, 3, 4, 5, 7 ) ) \
static int witness( INOUT PKC_INFO *pkcInfo, INOUT BIGNUM *a, 
					const BIGNUM *n, const BIGNUM *n_1, const BIGNUM *u, 
					IN_LENGTH_SHORT const int k,