File: //opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyc
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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__( ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyR s t SquareRootErrorc B s e Z RS( ( R R ( ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyR s t NegativeExponentErrorc B s e Z RS( ( R R ( ( ( sC /opt/alt/python27/lib/python2.7/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( ( sC /opt/alt/python27/lib/python2.7/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.i����i i i ( t AssertionErrort lent range( t polyt polymodt pt i( ( sC /opt/alt/python27/lib/python2.7/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( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyt polynomial_multiply_modF 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( ( sC /opt/alt/python27/lib/python2.7/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 i����i ( R t jacobi( t at nt a1t eR ( ( sC /opt/alt/python27/lib/python2.7/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 s�t � | d SqUWt d � � d S( s) Modular square root of a, mod p, p prime.i i i i����s %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( ( sC /opt/alt/python27/lib/python2.7/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( ( sC /opt/alt/python27/lib/python2.7/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( ( ( sC /opt/alt/python27/lib/python2.7/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! ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyt gcd� s
c C s | | t | | � S( s& Least common multiple of two integers.( R8 ( R! R( ( ( sC /opt/alt/python27/lib/python2.7/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! ( ( sC /opt/alt/python27/lib/python2.7/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 r�t | � r | j | d f � q�t d } x� | d } t | | � \ } } | | k rEPn | d k rd } | } xE | | k r�t | | � \ } } | d k r�Pn | } | d } q`W| j | | f � qqW| d k r�| j | d f � q�n | 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( ( sC /opt/alt/python27/lib/python2.7/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$ ( ( sC /opt/alt/python27/lib/python2.7/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.
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carmichael8 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 ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyRD B 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! ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyRF O 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? ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyt order_modY 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@ ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyt largest_factor_relatively_primel 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- ( ( sC /opt/alt/python27/lib/python2.7/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 }
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| d k r�| | d k r�t | d | � } | d k r�| d a t S|
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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.
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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 ( ( sC /opt/alt/python27/lib/python2.7/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? ( ( sC /opt/alt/python27/lib/python2.7/site-packages/ecdsa/numbertheory.pyt
next_prime� s i i i i i i
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j d | d � } t | | � d k r%| d } t | | � } | d k s�| | k s�| | | d k r�| d } t d- | | | f � q�q%q%Wq W| d. k s�t � 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 i�A i�A i�A i�A i�A i�A i�A 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 ) != 1i����s 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
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