Ë
|
¬`iÜ- ã óx d Z ddlmZ ddlmZ ddlmZ dZdZddZ d Z
|
dd l m Z ee dd
|
« Zdd Zd Zd Zy)zHFunctions to create and test prime numbers.
|
|
:undocumented: __package__
|
é )ÚRandom)ÚInteger)Ú
|
iter_rangeé Nc ó t | t « s t | « } | dv rt S | j « rt S t d« }t | dz
|
« }| t j « j }t |« }d}|j « r|dz }|dz }|j « rt |« D ] }d}|||fv r5t j d| dz
|
|¬« }d|cxk r | dz
|
k sJ J |||fv r5t ||| « } | ||fv rTt d|« D ]% }
|
t | d| « } | |k( r w| |k( s t c c S t c S t S )a: Perform a Miller-Rabin primality test on an integer.
|
|
The test is specified in Section C.3.1 of `FIPS PUB 186-4`__.
|
|
:Parameters:
|
candidate : integer
|
The number to test for primality.
|
iterations : integer
|
The maximum number of iterations to perform before
|
declaring a candidate a probable prime.
|
randfunc : callable
|
An RNG function where bases are taken from.
|
|
:Returns:
|
``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
|
|
.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
|
©r é é é r r r )Ú min_inclusiveÚ max_inclusiveÚrandfunc) Ú
|
isinstancer ÚPROBABLY_PRIMEÚis_evenÚ COMPOSITEr ÚnewÚreadr Ú random_rangeÚpow) Ú candidateÚ
|
iterationsr ÚoneÚ minus_oneÚmÚaÚiÚbaseÚzÚjs õh H:\项ç®\archive\æµè¯ç»\èæ¬\Change_password\venv_build\Lib\site-packages\Crypto/Math/Primality.pyÚmiller_rabin_testr" - s~ ô( i¤Ô )ÜIÓ& àLÑ Ü Ð à×ÑÔÜÐä
|
!*CÜ A Ó&IàÐÜ::<×$Ñ$ô ÓAØ AØ
|
))+Ø aØ Qð ))+ô
|
Ó #ò ð ØsIÐ&Ñ&Ü×'Ñ'°aØ"+¨a¡-Ø %ô'Dð Ô- ¨A¡ Ò-Ð .Ñ-Ð .Ð-ð sIÐ&Ò&ô aÓ #Ø iÐ Ñ Ø ô AqÓ!ò AÜAq)Ó$AØI~ÙØCxÜ Ô ð ô Ò ð/ ô4 Ðó c ó t | t « s t | « } | dv rt S | j « s| j « rt
|
S d } |« D ]4 }| || fv r t j || « }|dk( rt
|
c S |dk( s4 n | dz }|j « dz
|
}t d« }t d« }t d« }t d« } t |dz
|
dd« D ] }
|
|j |« ||z }|| z }| j |« | |z } | z } | j ||« | j « r| | z } | dz } | | z } |j |
|
« rx|j |« || z }|j « r|| z }|dz }|| z }|j | « |j ||« |j « r|| z }|dz }|| z }ô|j |« |j | « |dk( rt S t
|
S )a_ Perform a Lucas primality test on an integer.
|
|
The test is specified in Section C.3.3 of `FIPS PUB 186-4`__.
|
|
:Parameters:
|
candidate : integer
|
The number to test for primality.
|
|
:Returns:
|
``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
|
|
.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
|
r c 3 ó@ K d} | | dkD r| dz } n| dz } | } w)Nr r r © )Úvalues r! Ú alternatez lucas_test.<locals>.alternate s8 è ø ØØØKØqyØ
|
à
|
ØFEð ùs r éÿÿÿÿr ) r r r r Úis_perfect_squarer Ú jacobi_symbolÚ size_in_bitsr ÚsetÚmultiply_accumulateÚis_oddÚget_bit) r r( ÚDÚjsÚKÚrÚU_iÚV_iÚU_tempÚV_tempr s r! Ú
|
lucas_testr9 w s ô i¤Ô )ÜIÓ& ð LÑ Ü Ð Ø×ÑÔi× 9Ñ 9Ô ;ÜÐòñ [ò Ø QBÑ Ø Ü × "Ñ " 1 iÓ 0Ø 7Ü Ò Ø 8Ù ðð A Aà Ó1Ñ Aô !*CÜ
|
!*CÜ QZFÜ QZFä Ar 2Ó &ó ! ð
|
|
3Ø# Ø)Ñà
|
|
3Ø# Ø! Ø×"Ñ" 3¨Ô,Ø ==?Ø iÑ FØ1 Ø)Ñà 99Q<à GGFOØ 6MCØzz|ØyÑ Ø AICØ 9Ñ Cà GGFOØ × #Ñ # F¨AÔ .Øzz|ØyÑ Ø AICØ 9Ñ Cà GGFOØ GGFOðC! ðF axÜ Ð Ü Ðr# )Ú
|
sieve_baseéd c óþ | t j « j }t | t « s t | « } t | « t v rt S t | j t « d}| j « t t fd|« « d d }t! | ||¬« t k( rt S t# | « t k( rt S t S # t $ r t cY S w xY w# t $ r d}Y [w xY w)að Test if a number is prime.
|
|
A number is qualified as prime if it passes a certain
|
number of Miller-Rabin tests (dependent on the size
|
of the number, but such that probability of a false
|
positive is less than 10^-30) and a single Lucas test.
|
|
For instance, a 1024-bit candidate will need to pass
|
4 Miller-Rabin tests.
|
|
:Parameters:
|
candidate : integer
|
The number to test for primality.
|
randfunc : callable
|
The routine to draw random bytes from to select Miller-Rabin bases.
|
:Returns:
|
``PROBABLE_PRIME`` if the number if prime with very high probability.
|
``COMPOSITE`` if the number is a composite.
|
For efficiency reasons, ``COMPOSITE`` is also returned for small primes.
|
)
|
)éÜ é )i é )i é )i é
|
)il é )iä é )iz r )i° é )i¤ r
|
)it r c ó | d k S )Nr r& )ÚxÚbit_sizes r! ú<lambda>z%test_probable_prime.<locals>.<lambda> s ø ¨h¸¸1¹©o r# r r ©r )r r r r r ÚintÚ _sieve_baser ÚmapÚfail_if_divisible_byÚ
|
ValueErrorr r, ÚlistÚfilterÚ
|
IndexErrorr" r9 )r r Ú mr_rangesÚ mr_iterationsrG s @r! Útest_probable_primerT Þ s ø ð, ÐÜ::<×$Ñ$ä i¤Ô )ÜIÓ& ô 9~Ñ$Ü Ð ðÜ I× *Ñ *¬KÔ8ð'Ið ×%Ñ%Ó'HðÜ VÓ$=Ø$-ó /ó 0Ø01ñ3Ø34ñ6 ô
|
MØ"*ô,Ü/8ò9äÐÜ)Ó ¤ Ò)ÜÐÜ Ðøô- ò ÜÒðûô ò Ø ðús$ ÁC  C. ÃC+Ã*C+Ã. C<Ã;C<c óº | j dd« }| j dd« }| j dd « }| r t d| j « z « | t d« |dk r t d « | t j « j
|
}t }|t k( r9t j ||¬
|
« d z } ||« s,t ||« }|t k( r9S ) ax Generate a random probable prime.
|
|
The prime will not have any specific properties
|
(e.g. it will not be a *strong* prime).
|
|
Random numbers are evaluated for primality until one
|
passes all tests, consisting of a certain number of
|
Miller-Rabin tests with random bases followed by
|
a single Lucas test.
|
|
The number of Miller-Rabin iterations is chosen such that
|
the probability that the output number is a non-prime is
|
less than 1E-30 (roughly 2^{-100}).
|
|
This approach is compliant to `FIPS PUB 186-4`__.
|
|
:Keywords:
|
exact_bits : integer
|
The desired size in bits of the probable prime.
|
It must be at least 160.
|
randfunc : callable
|
An RNG function where candidate primes are taken from.
|
prime_filter : callable
|
A function that takes an Integer as parameter and returns
|
True if the number can be passed to further primality tests,
|
False if it should be immediately discarded.
|
|
:Return:
|
A probable prime in the range 2^exact_bits > p > 2^(exact_bits-1).
|
|
.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
|
Ú
|
exact_bitsNr Ú prime_filterc ó y)NTr& )rF s r! rH z)generate_probable_prime.<locals>.<lambda>< s r# úUnknown parameters: z Missing exact_bits parameteré z Prime number is not big enough.©rV r r )
|
ÚpoprN Úkeysr r r r r ÚrandomrT )ÚkwargsrV r rW Úresultr s r! Úgenerate_probable_primera sÝ ðD L¨$Ó/JØzz* dÓ+HØ::n©nÓ=LÙ ÜÐ/°&·+±+³-Ñ?Ó@Ð@àÐÜÐ7Ó8Ð8ØCÒÜÐ:Ó;Ð;àÐÜ::<×$Ñ$ä FØ
|
IÒ
|
ÜNN¨jØ,4ô6Ø89ñ: áIÔ&Ø Ü$ Y°Ó9ð IÓ
|
ð Ðr# c ón | j dd« }| j dd« }| r t d| j « z « | t j « j
|
}t }|t k( rCt |dz
|
|¬« }|dz dz }|j « |k7 r5t ||¬« }|t k( rCS ) a Generate a random, probable safe prime.
|
|
Note this operation is much slower than generating a simple prime.
|
|
:Keywords:
|
exact_bits : integer
|
The desired size in bits of the probable safe prime.
|
randfunc : callable
|
An RNG function where candidate primes are taken from.
|
|
:Return:
|
A probable safe prime in the range
|
2^exact_bits > p > 2^(exact_bits-1).
|
rV Nr rY r r[ r rI )
|
r\ rN r] r r r r ra r, rT )r_ rV r r` Úqr s r! Ú generate_probable_safe_primerd R s± ð L¨$Ó/JØzz* dÓ+HÙ ÜÐ/°&·+±+³-Ñ?Ó@Ð@àÐÜ::<×$Ñ$ä FØ
|
IÒ
|
Ü #¨z¸A©~ÈÔ QØEAI Ø × !Ñ !Ó # zÒ 1Ø Ü$ Y¸ÔBð IÓ
|
ð Ðr# )N)Ú__doc__ÚCryptor ÚCrypto.Math.Numbersr ÚCrypto.Util.py3compatr r r r" r9 ÚCrypto.Util.numberr: Ú_sieve_base_larger- rK rT ra rd r& r# r! ú<module>rk sW ðñ>õ
|
Ý 'å ,à ØóGòT^õB ?ñ Ð# D SÐ)Ó* ó7òt7ót r#
|
