o j{,@s|dZddlmZddlmZddlmZdZdZdddZ d d Z dd l m Z ee dd Zdd dZddZddZdS)zHFunctions to create and test prime numbers. :undocumented: __package__ )Random)Integer) iter_rangeNc Cs<t|ts t|}|dvrtS|rtStd}t|d}|dur(tj}t|}d}|r>|dL}|d7}|s2t|D]Y}d}|||fvrltj d|d|d}d|krc|dksfJJ|||fvsLt |||} | ||fvryqBtd|D]} t | d|} | |krn| |krtSq~tSqBtS)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 rrNrr) min_inclusive max_inclusiverandfunc) isinstancerPROBABLY_PRIMEis_even COMPOSITErnewreadr random_rangepow) candidate iterationsr one minus_onemaibasezjr/root/aizidognhua/tmp/workspace/projects/ec89d86c-575f-41c9-af57-ac45cbdbf775/venv/lib/python3.10/site-packages/Cryptodome/Math/Primality.pymiller_rabin_test-sL          r!c Cst|ts t|}|dvrtS|s|rtSdd}|D]}||| fvr*q t||}|dkr8tS|dkr>nq |d}|d}td}td}td}td} t|dddD]v} | |||9}||;}| || |9} | |9} | ||| r| |7} | dL} | |;} | | r| ||| 7}| r||7}|dL}||;}| | | ||| r||7}|dL}||;}qa| || | qa|dkrtStS)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 rcss0d} |V|dkr|d7}n|d8}| }q)Nr Trrr)valuerrr alternates zlucas_test..alternaterr) r rrris_perfect_squarer jacobi_symbol size_in_bitsrsetmultiply_accumulateis_oddget_bit) rr#DjsKrU_iV_iU_tempV_temprrrr lucas_testwsh              r4) sieve_basedcs|dur tj}t|tst|}t|tvrtSzt|j tWn t y-t YSwd}| zt tfdd|dd}Wn tyPd}Ynwt|||dt kr\t St|t krdt StS)aTest 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. N) ))i)i)i )il)i)izr )i)ir)itrcs |dkS)Nrrxbit_sizerr  s z%test_probable_prime..rrr )rrrr rint _sieve_basermapfail_if_divisible_by ValueErrorrr'listfilter IndexErrorr!r4)rr mr_ranges mr_iterationsrrAr test_probable_primesB      rOcKs|dd}|dd}|ddd}|rtd||dur&td|d kr.td |dur7tj}t}|tkrTtj||d d B}||sKq9t ||}|tks=|S) axGenerate 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 exact_bitsNr prime_filtercSsdS)NTrr?rrr rC<sz)generate_probable_prime..Unknown parameters: zMissing exact_bits parameterzPrime number is not big enough.rPr r) poprIkeysrrrrrrandomrO)kwargsrPr rQresultrrrr generate_probable_primes. "   rZcKs|dd}|dd}|rtd||durtj}t}|tkrDt|d|d}|dd}||kr:q!t ||d}|tks%|S) aGenerate 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). rPNr rRrrTrrD) rUrIrVrrrrrZr'rO)rXrPr rYqrrrr generate_probable_safe_primeRs      r\)N)__doc__ CryptodomerCryptodome.Math.NumbersrCryptodome.Util.py3compatrrrr!r4Cryptodome.Util.numberr5_sieve_base_larger(rFrOrZr\rrrr s    J a : :