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integer_types( t reduceNt Errorc B s e Z d Z RS( s) Base class for exceptions in this module.( t __name__t
__module__t __doc__( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyR s t SquareRootErrorc B s e Z RS( ( R R ( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyR s t NegativeExponentErrorc B s e Z RS( ( R R ( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyR s c C s/ | d k r t d | n t | | | S( s+ Raise base to exponent, reducing by modulusi s# Negative exponents (%d) not allowed( R t pow( t baset exponentt modulus( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt modular_exp s
c C s | d d k s t t | d k s. t x t | t | k r | d d k r xI t d t | d D]+ } | | | d | | | | | <qs Wn | d d !} q1 W| S( s Reduce poly by polymod, integer arithmetic modulo p.
Polynomials are represented as lists of coefficients
of increasing powers of x.ii i i ( t AssertionErrort lent range( t polyt polymodt pt i( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt polynomial_reduce_mod/ s
,c C s t | t | d d g } xa t t | D]M } xD t t | D]0 } | | | | | | | | | | | <qM Wq4 Wt | | | S( s Polynomial multiplication modulo a polynomial over ints mod p.
Polynomials are represented as lists of coefficients
of increasing powers of x.i i ( R R R ( t m1t m2R R t prodR t j( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt polynomial_multiply_modG s
!2c C s | | k s t | d k r% d g S| } | } | d d k rJ | } n d g } xW | d k r | d } t | | | | } | d d k rV t | | | | } qV qV W| S( s Polynomial exponentiation modulo a polynomial over ints mod p.
Polynomials are represented as lists of coefficients
of increasing powers of x.i i i ( R R ( R R R R t Gt kt s( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt polynomial_exp_mod_ s
c C s | d k s t | d d k s( t | | } | d k rB d S| d k rR d S| d } } x) | d d k r | d | d } } qb W| d d k s | d d k s | d d k r d } n d } | d k r | S| d d k r| d d k r| } n | t | | | S( s
Jacobi symboli i i i i i ii ( R t jacobi( t at nt a1t eR ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyR | s$
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c C s d | k o | k n s" t d | k s4 t | d k rD d S| d k rT | St | | } | d k r t d | | f n | d d k r t | | d d | S| d d k rEt | | d d | } | d k r t | | d d | S| | d k r6d | t d | | d d | | St d
n x t d | D]r } t | | d | | d k rU| | d f } t d
| d d | | } | d d k st | d SqUWt d d S( s) Modular square root of a, mod p, p prime.i i i is %d has no square root modulo %di i i i s Shouldn't get here.s No b found.N( i i ( R R R R t RuntimeErrorR R ( R! R t jact dt bt ft ff( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt square_root_mod_prime s4 " (! c C s | d k s | | k r% | | } n | | } } d \ } } } } x] | d k r t | | | f \ } } } | | | | | | | | f \ } } } } qG W| d k s t | d k r | S| | Sd S( s Inverse of a mod m.i i N( i i i i ( t divmodR ( R! t mt cR' t uct vct udt vdt q( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt inverse_mod s
2 c C s" x | r | | | } } q W| S( s1 Greatest common divisor using Euclid's algorithm.( ( R! R( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt gcd2 s c G sK t | d k r t t | St | d d rC t t | d S| d S( sP Greatest common divisor.
Usage: gcd( [ 2, 4, 6 ] )
or: gcd( 2, 4, 6 )
i i t __iter__( R R R5 t hasattr( R! ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt gcd s
c C s | | t | | S( s& Least common multiple of two integers.( R8 ( R! R( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt lcm2 s c G sK t | d k r t t | St | d d rC t t | d S| d S( sN Least common multiple.
Usage: lcm( [ 3, 4, 5 ] )
or: lcm( 3, 4, 5 )
i i R6 ( R R R9 R7 ( R! ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt lcm s
c C s t | t s t | d k r% g Sg } d } x t D] } | | k rN Pn t | | \ } } | d k r8 d } xE | | k r | } t | | \ } } | d k r Pn | d } qx W| j | | f q8 q8 W| t d k rt | r | j | d f qt d } x | d } t | | \ } } | | k rEPn | d k rd } | } xE | | k rt | | \ } } | d k rPn | } | d } q`W| j | | f qqW| d k r| j | d f qn | S( s2 Decompose n into a list of (prime,exponent) pairs.i i i i( t
isinstanceR R t smallprimesR, t appendt is_prime( R" t resultR' R3 t rt count( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt
factorization sP
c C s t | t s t | d k r% d Sd } t | } x[ | D]S } | d } | d k r | | d | d | d d } q> | | d d } q> W| S( s' Return the Euler totient function of n.i i i ( R; R R RB ( R" R? R* R) R$ ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt phi' s
%c C s t t | S( s Return Carmichael function of n.
Carmichael(n) is the smallest integer x such that
m**x = 1 mod n for all m relatively prime to n.
( t carmichael_of_factorizedRB ( R" ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt
carmichael9 s c C sc t | d k r d St | d } x6 t d t | D] } t | t | | } q< W| S( sh Return the Carmichael function of a number that is
represented as a list of (prime,exponent) pairs.
i i ( R t carmichael_of_ppowerR R: ( t f_listR? R ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyRD C s c C sH | \ } } | d k r0 | d k r0 d | d S| d | | d Sd S( s= Carmichael function of the given power of the given prime.
i i N( ( t ppR R! ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyRF P s c C sf | d k r d St | | d k s+ t | } d } x( | d k ra | | | } | d } q: W| S( s; Return the order of x in the multiplicative group mod m.
i i ( R8 R ( t xR- t zR? ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt order_modZ s c C sb x[ t | | } | d k r" Pn | } x/ t | | \ } } | d k rP Pn | } q+ Wq W| S( s8 Return the largest factor of a relatively prime to b.
i i ( R8 R, ( R! R( R' R3 R@ ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt largest_factor_relatively_primem s c C s t | t | | S( sy Return the order of x in the multiplicative group mod m',
where m' is the largest factor of m relatively prime to x.
( RK RL ( RI R- ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt kinda_order_mod} s c
C s d a | t d k r- | t k r& t St Sn t | d! d k rF t Sd } d t t j | d } xN d" d# d$ d% d& d' d( d) d* d+ d, d- f D]" \ } } | | k r Pn | } q Wd } | d } x( | d d k r | d } | d } q Wx t | D] } t | } t | | | } | d k r| | d k rd }
xZ |
| d k r| | d k rt | d | } | d k r| d a t S|
d }
qHW| | d k r| d a t SqqWt S(. s* Return True if x is prime, False otherwise.
We use the Miller-Rabin test, as given in Menezes et al. p. 138.
This test is not exact: there are composite values n for which
it returns True.
In testing the odd numbers from 10000001 to 19999999,
about 66 composites got past the first test,
5 got past the second test, and none got past the third.
Since factors of 2, 3, 5, 7, and 11 were detected during
preliminary screening, the number of numbers tested by
Miller-Rabin was (19999999 - 10000001)*(2/3)*(4/5)*(6/7)
= 4.57 million.
i ii i i i i i i( id i i i i i i i i, i i^ i i i i i& i i iR i i i i i ( id i ( i i ( i i ( i i ( i, i ( i^ i ( i i ( i i ( i& i ( i i ( iR i ( i i (
t miller_rabin_test_countR< t Truet FalseR8 t intt matht logR R ( R" t tt n_bitsR t ttR R@ R R! t yR ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyR> sX
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c C s? | d k r d S| d d B} x t | s: | d } q! W| S( s9 Return the smallest prime larger than the starting value.i i ( R> ( t starting_valueR? ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt
next_prime s i i i i i i
i i i i i i% i) i+ i/ i5 i; i= iC iG iI iO iS iY ia ie ig ik im iq i i i i i i i i i i i i i i i i i i i i i i i i i i i
i i i i i% i3 i7 i9 i= iK iQ i[ i] ia ig io iu i{ i i i i i i i i i i i i i i i i i i i i i i i i i i# i- i3 i9 i; iA iK iQ iW iY i_ ie ii ik iw i i i i i i i i i i i i i i i i i i i i i i i i i) i+ i5 i7 i; i= iG iU iY i[ i_ im iq is iw i i i i i i i i i i i i i i i i i i i i i i i i% i' i- i? iC iE iI iO iU i] ic ii i i i i i i i i i i i i i c C s/ t d t d5 d7 d9 d: k s( t t d< d> d@ g dA k sI t t d d k sa t t d t d dB dC dE k s t t d dF dG g dI k s t t d d k s t t d dJ } xB t t | d D]* } t | | | | d k s t q Wd } xt D]} t d | g } x t d d | d D]k } | | | } | j | t | | } | | | | k rV| d } t d! | | | | f qVqVWxk t d | D]Z } | | k ry t | | } Wn t
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q/X| d } t d" | | f qqWq$Wxt d d# d D]} t d$ | t | r>g } xi t d | D]X } t | | | d k r| d } t d% | | | f n | j | | | qWx t d | D]K } | | k rt | | d& k r7| d } t d' | | f q7qqWqJt
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x t d | D]u } d } x- |
D]% } | t | | d | d } qmW| t | | k rZ| d } t d( | | | f qZqZWqJWt d) d& d l }
d } x t d* D] } |
j d+ d, } x t d* D] } |
j d | d } t | | d k r%| d } t | | } | d k s| | k s| | | d k r| d } t d- | | | f qq%q%Wq W| d. k st t | d/ d0 t f d1 Y} t | d2 | d k r+| d3 | n d S(K Ns Testing gcd...i i i i i
s Testing lcm...s Testing next_prime...i@ iA i)A i/A iEA iSA iYA i]A iA iA iA iA iA iA iA iB iB iB i+B i/B i i s1 Testing square_root_mod_prime for modulus p = %d.i s0 Failed to find %d as sqrt( %d ) mod %d. Said %d.s/ Failed to report no root for sqrt( %d ) mod %d.i s" Testing jacobi for modulus m = %d.s jacobi( %d * %d, %d ) != 1is jacobi( %d, %d ) != -1s %d != jacobi( %d, %d )s Testing inverse_mod . . .id i i' s$ %d = inverse_mod( %d, %d ) is wrong.i s tests of inverse_mod completed.t
FailedTestc B s e Z RS( ( R R ( ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyRZ _ s s errors detected.s %d errors detectedi ii i i i i i i ii i i i i i i i i ii i i i ii ( i@ iA i)A i/A iEA iSA iYA i]A iA iA iA iA iA iA iA iB iB iB i+B i/B ( R R8 R R: R R RY R< R= R+ R R> R RB t randomt randintR4 t Exception( t bigprimesR t error_tallyR t squarest roott sqt
calculatedt nonsquareR- R) R! R. R[ t n_testsR t invRZ ( ( sX C:\Program Files\MySQL\MySQL Workbench 6.3 CE/python/site-packages\ecdsa\numbertheory.pyt __main__ s
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