Multiple-precision Integers (Advanced topics)

The xmpz type

gmpy2 provides access to an experimental integer type called xmpz. The xmpz type is a mutable integer type. In-place operations (+=, //=, etc.) modify the original object and do not create a new object. Instances of xmpz cannot be used as dictionary keys.

>>> import gmpy2
>>> from gmpy2 import xmpz
>>> a = xmpz(123)
>>> b = a
>>> a += 1
>>> a
xmpz(124)
>>> b
xmpz(124)

The ability to change an xmpz object in-place allows for efficient and rapid bit manipulation.

Individual bits can be set or cleared:

>>> a[10]=1
>>> a
xmpz(1148)

Slice notation is supported. The bits referenced by a slice can be either ‘read from’ or ‘written to’. To clear a slice of bits, use a source value of 0. In 2s-complement format, 0 is represented by an arbitrary number of 0-bits. To set a slice of bits, use a source value of ~0. The tilde operator inverts, or complements the bits in an integer. (~0 is -1 so you can also use -1.) In 2s-complement format, -1 is represented by an arbitrary number of 1-bits.

If a value for stop is specified in a slice assignment and the actual bit-length of the xmpz is less than stop, then the destination xmpz is logically padded with 0-bits to length stop.

>>> a=xmpz(0)
>>> a[8:16] = ~0
>>> bin(a)
'0b1111111100000000'
>>> a[4:12] = ~a[4:12]
>>> bin(a)
'0b1111000011110000'

Bits can be reversed:

>>> bin(a)
'0b10001111100'
>>> a[::] = a[::-1]
>>> bin(a)
'0b111110001'

The iter_bits() method returns a generator that returns True or False for each bit position. The methods iter_clear(), and iter_set() return generators that return the bit positions that are 1 or 0. The methods support arguments start and stop that define the beginning and ending bit positions that are used. To mimic the behavior of slices. the bit positions checked include start but the last position checked is stop - 1.

>>> a=xmpz(117)
>>> bin(a)
'0b1110101'
>>> list(a.iter_bits())
[True, False, True, False, True, True, True]
>>> list(a.iter_clear())
[1, 3]
>>> list(a.iter_set())
[0, 2, 4, 5, 6]
>>> list(a.iter_bits(stop=12))
[True, False, True, False, True, True, True, False, False, False, False, False]

The following program uses the Sieve of Eratosthenes to generate a list of prime numbers.

from __future__ import print_function
import time
import gmpy2

def sieve(limit=1000000):
    '''Returns a generator that yields the prime numbers up to limit.'''

    # Increment by 1 to account for the fact that slices  do not include
    # the last index value but we do want to include the last value for
    # calculating a list of primes.
    sieve_limit = gmpy2.isqrt(limit) + 1
    limit += 1

    # Mark bit positions 0 and 1 as not prime.
    bitmap = gmpy2.xmpz(3)

    # Process 2 separately. This allows us to use p+p for the step size
    # when sieving the remaining primes.
    bitmap[4 : limit : 2] = -1

    # Sieve the remaining primes.
    for p in bitmap.iter_clear(3, sieve_limit):
        bitmap[p*p : limit : p+p] = -1

    return bitmap.iter_clear(2, limit)

if __name__ == "__main__":
    start = time.time()
    result = list(sieve())
    print(time.time() - start)
    print(len(result))

Advanced Number Theory Functions

The following functions are based on mpz_lucas.c and mpz_prp.c by David Cleaver.

A good reference for probable prime testing is http://www.pseudoprime.com/pseudo.html

is_bpsw_prp(…)
is_bpsw_prp(n) will return True if n is a Baillie-Pomerance-Selfridge-Wagstaff probable prime. A BPSW probable prime passes the is_strong_prp() test with base 2 and the is_selfridge_prp() test.
is_euler_prp(…)

is_euler_prp(n,a) will return True if n is an Euler (also known as Solovay-Strassen) probable prime to the base a.

Assuming:
gcd(n, a) == 1
n is odd

Then an Euler probable prime requires:
a**((n-1)/2) == 1 (mod n)
is_extra_strong_lucas_prp(…)

is_extra_strong_lucas_prp(n,p) will return True if n is an extra strong Lucas probable prime with parameters (p,1).

Assuming:
n is odd
D = p*p - 4, D != 0
gcd(n, 2*D) == 1
n = s*(2**r) + Jacobi(D,n), s odd

Then an extra strong Lucas probable prime requires:
lucasu(p,1,s) == 0 (mod n)
or
lucasv(p,1,s) == +/-2 (mod n)
or
lucasv(p,1,s*(2**t)) == 0 (mod n) for some t, 0 <= t < r
is_fermat_prp(…)

is_fermat_prp(n,a) will return True if n is a Fermat probable prime to the base a.

Assuming:
gcd(n,a) == 1

Then a Fermat probable prime requires:
a**(n-1) == 1 (mod n)
is_fibonacci_prp(…)

is_fibonacci_prp(n,p,q) will return True if n is a Fibonacci probable prime with parameters (p,q).

Assuming:
n is odd
p > 0, q = +/-1
p*p - 4*q != 0

Then a Fibonacci probable prime requires:
lucasv(p,q,n) == p (mod n).
is_lucas_prp(…)

is_lucas_prp(n,p,q) will return True if n is a Lucas probable prime with parameters (p,q).

Assuming:
n is odd
D = p*p - 4*q, D != 0
gcd(n, 2*q*D) == 1

Then a Lucas probable prime requires:
lucasu(p,q,n - Jacobi(D,n)) == 0 (mod n)
is_selfridge_prp(…)
is_selfridge_prp(n) will return True if n is a Lucas probable prime with Selfidge parameters (p,q). The Selfridge parameters are chosen by finding the first element D in the sequence {5, -7, 9, -11, 13, …} such that Jacobi(D,n) == -1. Let p=1 and q = (1-D)/4 and then perform a Lucas probable prime test.
is_strong_bpsw_prp(…)
is_strong_bpsw_prp(n) will return True if n is a strong Baillie-Pomerance-Selfridge-Wagstaff probable prime. A strong BPSW probable prime passes the is_strong_prp() test with base 2 and the is_strongselfridge_prp() test.
is_strong_lucas_prp(…)

is_strong_lucas_prp(n,p,q) will return True if n is a strong Lucas probable prime with parameters (p,q).

Assuming:
n is odd
D = p*p - 4*q, D != 0
gcd(n, 2*q*D) == 1
n = s*(2**r) + Jacobi(D,n), s odd

Then a strong Lucas probable prime requires:
lucasu(p,q,s) == 0 (mod n)
or
lucasv(p,q,s*(2**t)) == 0 (mod n) for some t, 0 <= t < r
is_strong_prp(…)

is_strong_prp(n,a) will return True if n is a strong (also known as Miller-Rabin) probable prime to the base a.

Assuming:
gcd(n,a) == 1
n is odd
n = s*(2**r) + 1, with s odd

Then a strong probable prime requires one of the following is true:
a**s == 1 (mod n)
or
a**(s*(2**t)) == -1 (mod n) for some t, 0 <= t < r.
is_strong_selfridge_prp(…)
is_strong_selfridge_prp(n) will return True if n is a strong Lucas probable prime with Selfidge parameters (p,q). The Selfridge parameters are chosen by finding the first element D in the sequence {5, -7, 9, -11, 13, …} such that Jacobi(D,n) == -1. Let p=1 and q = (1-D)/4 and then perform a strong Lucas probable prime test.
lucasu(…)
lucasu(p,q,k) will return the k-th element of the Lucas U sequence defined by p,q. p*p - 4*q must not equal 0; k must be greater than or equal to 0.
lucasu_mod(…)
lucasu_mod(p,q,k,n) will return the k-th element of the Lucas U sequence defined by p,q (mod n). p*p - 4*q must not equal 0; k must be greater than or equal to 0; n must be greater than 0.
lucasv(…)
lucasv(p,q,k) will return the k-th element of the Lucas V sequence defined by parameters (p,q). p*p - 4*q must not equal 0; k must be greater than or equal to 0.
lucasv_mod(…)
lucasv_mod(p,q,k,n) will return the k-th element of the Lucas V sequence defined by parameters (p,q) (mod n). p*p - 4*q must not equal 0; k must be greater than or equal to 0; n must be greater than 0.