+ '(jk,Rt^RIHt^RIHt^RIHt^t^tR Rlt Rt ^RI H t ]!] R,4tR R ltR tR tR#) zHFunctions to create and test prime numbers. :undocumented: __package__ )Random)Integer) iter_rangeNc\V\4'g \V4pVR9d\#VP4'd\#\^4p\V^, 4pVf \ P !4Pp\V4p^pVP4'dV^,pV^, pK*\V4Fp^pWV39d>\P!^V^, VR7p^Tu;8:dV^, 8:dK@QhQh\WV4p WV39dK`\^V4F)p \V ^V4p W8XdKW8XgK!\uu# \u# \#)aPerform 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 ) min_inclusive max_inclusiverandfunc) isinstancerPROBABLY_PRIMEis_even COMPOSITErnewreadr random_rangepow) candidate iterationsrone minus_onemaibasezjs &&& J/home/ubuntu/cf-venv/lib/python3.14/site-packages/Crypto/Math/Primality.pymiller_rabin_testr!-sX( i ) )I& L  !*C A &I::<$$  A A ))++ a Q  #I&&''a"+a-%'D- A - .- .-  # i Aq!AAq)$A~x  " /$4 c\V\4'g \V4pVR9d\#VP4'gVP 4'd\ #RpV!4F<pWV)39dK\P !W 4pV^8Xd \ u#VR8XgK<M V^,pVP4^, p\^4p\^4p\^4p\^4p \V^, RR4EF\p VPV4W,pW,pV PV4W,p V X,p V PWw4V P4'd W, p V ^,p W,p VPV 4'dVPV4Wi, pVP4'd W`, pV^,pW`,pVPV 4VPW4VP4'd Wp, pV^,pWp,pEK:VPV4VPV 4EK_ V^8Xd\#\ #)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 c3V"^pVxV^8d V^, pM V^,pV)pK$5i)r )values r alternatelucas_test..alternates2Kqy  FEs')r ) rrrris_perfect_squarer jacobi_symbol size_in_bitsrsetmultiply_accumulateis_oddget_bit) rr'DjsKrU_iV_iU_tempV_temprs & r lucas_testr9ws i ) )I& L i99;;[ QB    " "1 0 7  8  A A 1A !*C !*C QZF QZF Ar2 &  3  3 ! ""3, ==??  F1  99Q<< GGFO MCzz||  AIC  C GGFO  # #F .zz||  AIC  C GGFO GGFOC'F ax r") sieve_base:NdNc6aVf \P!4Pp\V\4'g \ V4p\ V4\ 9d\#\VP\ 4RpTP4o\\T3RlT44^,^,p\!YTR7\8Xd\#\#T4\8Xd\#\# \d \u#i;i \d^pLbi;i)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. c<SV^,8#)r%)xbit_sizes&r %test_probable_prime.. s h1or"r) ))i)i)i )il)i)izr )i)ir )itr )rrrrrint _sieve_basermapfail_if_divisible_by ValueErrorrr,listfilter IndexErrorr!r9)rr mr_ranges mr_iterationsr@s&& @r test_probable_primerVs,::<$$ i ) )I&  9~$ I * *K8'I%%'HV$=$-/0013346 "*,/89) ) -   s$!C1'D1DD DDc VPRR4pVPRR4pVPRR4pV'd!\RVP4,4hVf \R4hV^8d \R4hVf \P!4P p\ pV\ 8Xd=\P!VVR 7^,pV!V4'gK:\WR4pKGX#) aGenerate a random probable prime. The prime will not have any specific properties (e.g. it will not be a *strong* prime). Random numbers are evaluated for primality until one passes all tests, consisting of a certain number of Miller-Rabin tests with random bases followed by a single Lucas test. The number of Miller-Rabin iterations is chosen such that the probability that the output number is a non-prime is less than 1E-30 (roughly 2^{-100}). This approach is compliant to `FIPS PUB 186-4`__. :Keywords: exact_bits : integer The desired size in bits of the probable prime. It must be at least 160. randfunc : callable An RNG function where candidate primes are taken from. prime_filter : callable A function that takes an Integer as parameter and returns True if the number can be passed to further primality tests, False if it should be immediately discarded. :Return: A probable prime in the range 2^exact_bits > p > 2^(exact_bits-1). .. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf exact_bitsNr prime_filtercR#)Tr%)r?s&r rA)generate_probable_prime..<sr"Unknown parameters: zMissing exact_bits parameterzPrime number is not big enough.rXr) poprPkeysrrrrrrandomrV)kwargsrXrrYresultrs, r generate_probable_primercsDL$/Jzz*d+H::nn=L /&++-?@@788C:;;::<$$ F I NNj,4689: I&& $Y9 r"c VPRR4pVPRR4pV'd!\RVP4,4hVf \P!4P p\ pV\ 8XdJ\V^, VR7pV^,^,pVP4V8wdKF\WRR7pKTX#)asGenerate a random, probable safe prime. Note this operation is much slower than generating a simple prime. :Keywords: exact_bits : integer The desired size in bits of the probable safe prime. randfunc : callable An RNG function where candidate primes are taken from. :Return: A probable safe prime in the range 2^exact_bits > p > 2^(exact_bits-1). rXNrr\r]rC) r^rPr_rrrrrcr,rV)rarXrrbqrs, r generate_probable_safe_primerfRs L$/Jzz*d+H /&++-?@@::<$$ F I  #zA~ QEAI  ! ! #z 1 $YB r")N)__doc__CryptorCrypto.Math.NumbersrCrypto.Util.py3compatrrrr!r9Crypto.Util.numberr:_sieve_base_larger-rMrVrcrfr%r"r rmsU> ', GT^B?#D)* 7t7tr"