Coverage for local/lib/python2.7/site-packages/sage/crypto/util.py : 97%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
""" Utility Functions for Cryptography
Miscellaneous utility functions for cryptographic purposes.
AUTHORS:
- Minh Van Nguyen (2009-12): initial version with the following functions: ``ascii_integer``, ``ascii_to_bin``, ``bin_to_ascii``, ``has_blum_prime``, ``is_blum_prime``, ``least_significant_bits``, ``random_blum_prime``. """
#***************************************************************************** # Copyright (c) 2009, 2010 Minh Van Nguyen <nguyenminh2@gmail.com> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
r""" Return the ASCII integer corresponding to the binary string ``B``.
INPUT:
- ``B`` -- a non-empty binary string or a non-empty list of bits. The number of bits in ``B`` must be 8.
OUTPUT:
- The ASCII integer corresponding to the 8-bit block ``B``.
EXAMPLES:
The ASCII integers of some binary strings::
sage: from sage.crypto.util import ascii_integer sage: bin = BinaryStrings() sage: B = bin.encoding("A"); B 01000001 sage: ascii_integer(B) 65 sage: B = bin.encoding("C"); list(B) [0, 1, 0, 0, 0, 0, 1, 1] sage: ascii_integer(list(B)) 67 sage: ascii_integer("01000100") 68 sage: ascii_integer([0, 1, 0, 0, 0, 1, 0, 1]) 69
TESTS:
The input ``B`` must be a non-empty string or a non-empty list of bits::
sage: from sage.crypto.util import ascii_integer sage: ascii_integer("") Traceback (most recent call last): ... ValueError: B must consist of 8 bits. sage: ascii_integer([]) Traceback (most recent call last): ... ValueError: B must consist of 8 bits.
The input ``B`` must be an 8-bit string or a list of 8 bits::
sage: from sage.crypto.util import ascii_integer sage: ascii_integer("101") Traceback (most recent call last): ... ValueError: B must consist of 8 bits. sage: ascii_integer([1, 0, 1, 1]) Traceback (most recent call last): ... ValueError: B must consist of 8 bits. """ L[3]*16, L[2]*32, L[1]*64, L[0]*128])
r""" Return the binary representation of the ASCII string ``A``.
INPUT:
- ``A`` -- a string or list of ASCII characters.
OUTPUT:
- The binary representation of ``A``.
ALGORITHM:
Let `A = a_0 a_1 \cdots a_{n-1}` be an ASCII string, where each `a_i` is an ASCII character. Let `c_i` be the ASCII integer corresponding to `a_i` and let `b_i` be the binary representation of `c_i`. The binary representation `B` of `A` is `B = b_0 b_1 \cdots b_{n-1}`.
EXAMPLES:
The binary representation of some ASCII strings::
sage: from sage.crypto.util import ascii_to_bin sage: ascii_to_bin("A") 01000001 sage: ascii_to_bin("Abc123") 010000010110001001100011001100010011001000110011
The empty string is different from the string with one space character. For the empty string and the empty list, this function returns the same result::
sage: from sage.crypto.util import ascii_to_bin sage: ascii_to_bin("") <BLANKLINE> sage: ascii_to_bin(" ") 00100000 sage: ascii_to_bin([]) <BLANKLINE>
This function also accepts a list of ASCII characters. You can also pass in a list of strings::
sage: from sage.crypto.util import ascii_to_bin sage: ascii_to_bin(["A", "b", "c", "1", "2", "3"]) 010000010110001001100011001100010011001000110011 sage: ascii_to_bin(["A", "bc", "1", "23"]) 010000010110001001100011001100010011001000110011
TESTS:
For a list of ASCII characters or strings, do not mix characters or strings with integers::
sage: from sage.crypto.util import ascii_to_bin sage: ascii_to_bin(["A", "b", "c", 1, 2, 3]) Traceback (most recent call last): ... TypeError: sequence item 3: expected string, sage.rings.integer.Integer found sage: ascii_to_bin(["Abc", 1, 2, 3]) Traceback (most recent call last): ... TypeError: sequence item 1: expected string, sage.rings.integer.Integer found """
r""" Return the ASCII representation of the binary string ``B``.
INPUT:
- ``B`` -- a non-empty binary string or a non-empty list of bits. The number of bits in ``B`` must be a multiple of 8.
OUTPUT:
- The ASCII string corresponding to ``B``.
ALGORITHM:
Consider a block of bits `B = b_0 b_1 \cdots b_{n-1}` where each sub-block `b_i` is a binary string of length 8. Then the total number of bits is a multiple of 8 and is given by `8n`. Let `c_i` be the integer representation of `b_i`. We can consider `c_i` as the integer representation of an ASCII character. Then the ASCII representation `A` of `B` is `A = a_0 a_1 \cdots a_{n-1}`.
EXAMPLES:
Convert some ASCII strings to their binary representations and recover the ASCII strings from the binary representations::
sage: from sage.crypto.util import ascii_to_bin sage: from sage.crypto.util import bin_to_ascii sage: A = "Abc" sage: B = ascii_to_bin(A); B 010000010110001001100011 sage: bin_to_ascii(B) 'Abc' sage: bin_to_ascii(B) == A True
::
sage: A = "123 \" #" sage: B = ascii_to_bin(A); B 00110001001100100011001100100000001000100010000000100011 sage: bin_to_ascii(B) '123 " #' sage: bin_to_ascii(B) == A True
This function also accepts strings and lists of bits::
sage: from sage.crypto.util import bin_to_ascii sage: bin_to_ascii("010000010110001001100011") 'Abc' sage: bin_to_ascii([0, 1, 0, 0, 0, 0, 0, 1]) 'A'
TESTS:
The number of bits in ``B`` must be non-empty and a multiple of 8::
sage: from sage.crypto.util import bin_to_ascii sage: bin_to_ascii("") Traceback (most recent call last): ... ValueError: B must be a non-empty binary string. sage: bin_to_ascii([]) Traceback (most recent call last): ... ValueError: B must be a non-empty binary string. sage: bin_to_ascii(" ") Traceback (most recent call last): ... ValueError: The number of bits in B must be a multiple of 8. sage: bin_to_ascii("101") Traceback (most recent call last): ... ValueError: The number of bits in B must be a multiple of 8. sage: bin_to_ascii([1, 0, 1]) Traceback (most recent call last): ... ValueError: The number of bits in B must be a multiple of 8. """ # sanity checks # perform conversion to ASCII string # the number of 8-bit blocks # Convert from 8-bit string to ASCII integer. Then convert the # ASCII integer to the corresponding ASCII character.
r""" Return the Carmichael function of a positive integer ``n``.
The Carmichael function of `n`, denoted `\lambda(n)`, is the smallest positive integer `k` such that `a^k \equiv 1 \pmod{n}` for all `a \in \ZZ/n\ZZ` satisfying `\gcd(a, n) = 1`. Thus, `\lambda(n) = k` is the exponent of the multiplicative group `(\ZZ/n\ZZ)^{\ast}`.
INPUT:
- ``n`` -- a positive integer.
OUTPUT:
- The Carmichael function of ``n``.
ALGORITHM:
If `n = 2, 4` then `\lambda(n) = \varphi(n)`. Let `p \geq 3` be an odd prime and let `k` be a positive integer. Then `\lambda(p^k) = p^{k - 1}(p - 1) = \varphi(p^k)`. If `k \geq 3`, then `\lambda(2^k) = 2^{k - 2}`. Now consider the case where `n > 3` is composite and let `n = p_1^{k_1} p_2^{k_2} \cdots p_t^{k_t}` be the prime factorization of `n`. Then
.. MATH::
\lambda(n) = \lambda(p_1^{k_1} p_2^{k_2} \cdots p_t^{k_t}) = \text{lcm}(\lambda(p_1^{k_1}), \lambda(p_2^{k_2}), \dots, \lambda(p_t^{k_t}))
EXAMPLES:
The Carmichael function of all positive integers up to and including 10::
sage: from sage.crypto.util import carmichael_lambda sage: list(map(carmichael_lambda, [1..10])) [1, 1, 2, 2, 4, 2, 6, 2, 6, 4]
The Carmichael function of the first ten primes::
sage: list(map(carmichael_lambda, primes_first_n(10))) [1, 2, 4, 6, 10, 12, 16, 18, 22, 28]
Cases where the Carmichael function is equivalent to the Euler phi function::
sage: carmichael_lambda(2) == euler_phi(2) True sage: carmichael_lambda(4) == euler_phi(4) True sage: p = random_prime(1000, lbound=3, proof=True) sage: k = randint(1, 1000) sage: carmichael_lambda(p^k) == euler_phi(p^k) True
A case where `\lambda(n) \neq \varphi(n)`::
sage: k = randint(1, 1000) sage: carmichael_lambda(2^k) == 2^(k - 2) True sage: carmichael_lambda(2^k) == 2^(k - 2) == euler_phi(2^k) False
Verifying the current implementation of the Carmichael function using another implementation. The other implementation that we use for verification is an exhaustive search for the exponent of the multiplicative group `(\ZZ/n\ZZ)^{\ast}`. ::
sage: from sage.crypto.util import carmichael_lambda sage: n = randint(1, 500) sage: c = carmichael_lambda(n) sage: def coprime(n): ....: return [i for i in range(n) if gcd(i, n) == 1] sage: def znpower(n, k): ....: L = coprime(n) ....: return list(map(power_mod, L, [k]*len(L), [n]*len(L))) sage: def my_carmichael(n): ....: for k in range(1, n): ....: L = znpower(n, k) ....: ones = [1] * len(L) ....: T = [L[i] == ones[i] for i in range(len(L))] ....: if all(T): ....: return k sage: c == my_carmichael(n) True
Carmichael's theorem states that `a^{\lambda(n)} \equiv 1 \pmod{n}` for all elements `a` of the multiplicative group `(\ZZ/n\ZZ)^{\ast}`. Here, we verify Carmichael's theorem. ::
sage: from sage.crypto.util import carmichael_lambda sage: n = randint(1, 1000) sage: c = carmichael_lambda(n) sage: ZnZ = IntegerModRing(n) sage: M = ZnZ.list_of_elements_of_multiplicative_group() sage: ones = [1] * len(M) sage: P = [power_mod(a, c, n) for a in M] sage: P == ones True
TESTS:
The input ``n`` must be a positive integer::
sage: from sage.crypto.util import carmichael_lambda sage: carmichael_lambda(0) Traceback (most recent call last): ... ValueError: Input n must be a positive integer. sage: carmichael_lambda(randint(-10, 0)) Traceback (most recent call last): ... ValueError: Input n must be a positive integer.
Bug reported in :trac:`8283`::
sage: from sage.crypto.util import carmichael_lambda sage: type(carmichael_lambda(16)) <type 'sage.rings.integer.Integer'>
REFERENCES:
- :wikipedia:`Carmichael_function` """ # sanity check
# first get rid of the prime factor 2 else: # for k >= 3, lambda(2**k) = 2**(k - 2)
# then other prime factors
# finish the job
""" Determine whether or not there is a Blum prime within the specified closed interval.
INPUT:
- ``lbound`` -- positive integer; the lower bound on how small a Blum prime can be. The lower bound must be distinct from the upper bound.
- ``ubound`` -- positive integer; the upper bound on how large a Blum prime can be. The lower bound must be distinct from the upper bound.
OUTPUT:
- ``True`` if there is a Blum prime ``p`` such that ``lbound <= p <= ubound``. ``False`` otherwise.
ALGORITHM:
Let `L` and `U` be distinct positive integers. Let `P` be the set of all odd primes `p` such that `L \leq p \leq U`. Our main focus is on Blum primes, i.e. odd primes that are congruent to 3 modulo 4, so we assume that the lower bound `L > 2`. The closed interval `[L, U]` has a Blum prime if and only if the set `P` has a Blum prime.
EXAMPLES:
Testing for the presence of Blum primes within some closed intervals. The interval `[4, 100]` has a Blum prime, the smallest such prime being 7. The interval `[24, 28]` has no primes, hence no Blum primes. ::
sage: from sage.crypto.util import has_blum_prime sage: from sage.crypto.util import is_blum_prime sage: has_blum_prime(4, 100) True sage: for n in range(4, 100): ....: if is_blum_prime(n): ....: print(n) ....: break 7 sage: has_blum_prime(24, 28) False
TESTS:
Both the lower and upper bounds must be greater than 2::
sage: from sage.crypto.util import has_blum_prime sage: has_blum_prime(2, 3) Traceback (most recent call last): ... ValueError: Both the lower and upper bounds must be > 2. sage: has_blum_prime(3, 2) Traceback (most recent call last): ... ValueError: Both the lower and upper bounds must be > 2. sage: has_blum_prime(2, 2) Traceback (most recent call last): ... ValueError: Both the lower and upper bounds must be > 2.
The lower and upper bounds must be distinct from each other::
sage: has_blum_prime(3, 3) Traceback (most recent call last): ... ValueError: The lower and upper bounds must be distinct.
The lower bound must be less than the upper bound::
sage: has_blum_prime(4, 3) Traceback (most recent call last): ... ValueError: The lower bound must be less than the upper bound. """ # sanity checks # now test for presence of a Blum prime
r""" Determine whether or not ``n`` is a Blum prime.
INPUT:
- ``n`` a positive prime.
OUTPUT:
- ``True`` if ``n`` is a Blum prime; ``False`` otherwise.
Let `n` be a positive prime. Then `n` is a Blum prime if `n` is congruent to 3 modulo 4, i.e. `n \equiv 3 \pmod{4}`.
EXAMPLES:
Testing some integers to see if they are Blum primes::
sage: from sage.crypto.util import is_blum_prime sage: from sage.crypto.util import random_blum_prime sage: is_blum_prime(101) False sage: is_blum_prime(7) True sage: p = random_blum_prime(10**3, 10**5) sage: is_blum_prime(p) True """ return False else: else:
r""" Return the ``k`` least significant bits of ``n``.
INPUT:
- ``n`` -- an integer.
- ``k`` -- a positive integer.
OUTPUT:
- The ``k`` least significant bits of the integer ``n``. If ``k=1``, then return the parity bit of the integer ``n``. Let `b` be the binary representation of ``n``, where `m` is the length of the binary string `b`. If `k \geq m`, then return the binary representation of ``n``.
EXAMPLES:
Obtain the parity bits of some integers::
sage: from sage.crypto.util import least_significant_bits sage: least_significant_bits(0, 1) [0] sage: least_significant_bits(2, 1) [0] sage: least_significant_bits(3, 1) [1] sage: least_significant_bits(-2, 1) [0] sage: least_significant_bits(-3, 1) [1]
Obtain the 4 least significant bits of some integers::
sage: least_significant_bits(101, 4) [0, 1, 0, 1] sage: least_significant_bits(-101, 4) [0, 1, 0, 1] sage: least_significant_bits(124, 4) [1, 1, 0, 0] sage: least_significant_bits(-124, 4) [1, 1, 0, 0]
The binary representation of 123::
sage: n = 123; b = n.binary(); b '1111011' sage: least_significant_bits(n, len(b)) [1, 1, 1, 1, 0, 1, 1] """
r""" A random Blum prime within the specified bounds.
Let `p` be a positive prime. Then `p` is a Blum prime if `p` is congruent to 3 modulo 4, i.e. `p \equiv 3 \pmod{4}`.
INPUT:
- ``lbound`` -- positive integer; the lower bound on how small a random Blum prime `p` can be. So we have ``0 < lbound <= p <= ubound``. The lower bound must be distinct from the upper bound.
- ``ubound`` -- positive integer; the upper bound on how large a random Blum prime `p` can be. So we have ``0 < lbound <= p <= ubound``. The lower bound must be distinct from the upper bound.
- ``ntries`` -- (default: ``100``) the number of attempts to generate a random Blum prime. If ``ntries`` is a positive integer, then perform that many attempts at generating a random Blum prime. This might or might not result in a Blum prime.
OUTPUT:
- A random Blum prime within the specified lower and upper bounds.
.. NOTE::
Beware that there might not be any primes between the lower and upper bounds. So make sure that these two bounds are "sufficiently" far apart from each other for there to be primes congruent to 3 modulo 4. In particular, there should be at least two distinct Blum primes within the specified bounds.
EXAMPLES:
Choose a random prime and check that it is a Blum prime::
sage: from sage.crypto.util import random_blum_prime sage: p = random_blum_prime(10**4, 10**5) sage: is_prime(p) True sage: mod(p, 4) == 3 True
TESTS:
Make sure that there is at least one Blum prime between the lower and upper bounds. In the following example, we have ``lbound=24`` and ``ubound=30`` with 29 being the only prime within those bounds. But 29 is not a Blum prime. ::
sage: from sage.crypto.util import random_blum_prime sage: random_blum_prime(24, 30, ntries=10) Traceback (most recent call last): ... ValueError: No Blum primes within the specified closed interval. sage: random_blum_prime(24, 28) Traceback (most recent call last): ... ValueError: No Blum primes within the specified closed interval. """ # sanity check # Now we know that there is a Blum prime within the closed interval # [lbound, ubound]. Pick one such prime at random. raise ValueError("Maximum number of attempts exceeded.") |