o jgC^@s@dZddlmZmZmZddlmZddlZddl m Z ddl m Z ddl mZddlmZdd lmZmZdd l mZmZdd lmZmZdd lmZdd lmZmZm Z ddl!m"Z"ddl#m$Z$ddl%m&Z&ddl'm(Z(ddl)m*Z*m+Z+m,Z,ddl-m.Z.m/Z/ddl0m1Z1e dkrdgZ2Gddde3Z4dddZ5e56ddZ7e5dZ8eZ9ddZ:dd d!Z;dd"d#Zdd(d)Z?dd*d+Z@dd,d-ZAd d/d0ZBdd1d2ZCdd3d4ZDdd5d6ZEdd7d8ZFdd9d:ZGdd;d<ZHd d>d?ZIdd@dAZJddBdCZKddDdEZLdFdGZMdHdIZNdJdKZOddLdMZPdNdOZQdPdQZRddRdSZSdTdUZTdVdVdWdddXdYdZZUd[d\ZVd]d^ZWdd_d`ZXddadbZYddcddZZdedfZ[dgdhZ\didjZ]dkdlZ^idmdndodpdqdrdsdtdudvdwdxdydzd{d|d}d~dddddddddddddddd=iddddddddddddddddddddddddddddddddddiddddddudddddddddddddddd“ddēddƓddȓddʓdd̓ddΓddiZ_dd҄e_`DZad ddՄZbd ddׄZce1ddgdڍddZddd݄Zedd߄Zfd ddZgddZhdddZiddZjd ddZkddZlddZmddZndddZodddZpdddZqdddZrddZsddZtddZuddZvddZwdddZxddZydS( a This file contains some classical ciphers and routines implementing a linear-feedback shift register (LFSR) and the Diffie-Hellman key exchange. .. warning:: This module is intended for educational purposes only. Do not use the functions in this module for real cryptographic applications. If you wish to encrypt real data, we recommend using something like the `cryptography `_ module. ) whitespaceascii_uppercase printable)reduceN)cycle) GROUND_TYPESSymbol)Rational) _randrange_randint)gcdinvert)totientreduced_totient)Matrix)isprimeprimitive_root factorint) nextprime)crt)FF)Poly)as_int filldedent translate)uniqmultiset)doctest_depends_onflint lfsr_sequencec@s*eZdZdZddZddZd ddZd S) NonInvertibleCipherWarningz1A warning raised if the cipher is not invertible.cCs ||_dSN fullMessage)selfmsgr'K/var/www/html/zoom/venv/lib/python3.10/site-packages/sympy/crypto/crypto.py__init__- z#NonInvertibleCipherWarning.__init__cCs d|jS)Nz r#)r%r'r'r(__str__0r*z"NonInvertibleCipherWarning.__str__cCstj||ddS)N stacklevel)warningswarn)r%r.r'r'r(r03szNonInvertibleCipherWarning.warnN)r,)__name__ __module__ __qualname____doc__r)r+r0r'r'r'r(r!+s r!cCs:|stSt|t}|r|g}dd|D}|r|dS|S)aReturn the letters of ``s`` in uppercase. In case more than one string is passed, each of them will be processed and a list of upper case strings will be returned. Examples ======== >>> from sympy.crypto.crypto import AZ >>> AZ('Hello, world!') 'HELLOWORLD' >>> AZ('Hello, world!'.split()) ['HELLO', 'WORLD'] See Also ======== check_and_join cSs"g|] }t|tddqS)Tfilter)check_and_joinuppersplit uppercase.0ir'r'r( PszAZ..r)r: isinstancestr)strvr'r'r(AZ7s rDJ 0123456789cstt}t|tkr"dtfddD}td|t|t|}|r7tddt|dtt|}|td|d|S)aReturn a string of the distinct characters of ``symbols`` with those of ``key`` appearing first. A ValueError is raised if a) there are duplicate characters in ``symbols`` or b) there are characters in ``key`` that are not in ``symbols``. Examples ======== >>> from sympy.crypto.crypto import padded_key >>> padded_key('PUPPY', 'OPQRSTUVWXY') 'PUYOQRSTVWX' >>> padded_key('RSA', 'ARTIST') Traceback (most recent call last): ... ValueError: duplicate characters in symbols: T rFcsh|] }|dkr|qS))countr;symbolsr'r( oszpadded_key..z#duplicate characters in symbols: %sz%characters in key but not symbols: %sN)listrlenjoinsorted ValueErrorsetr)keyrKsymsextrakey0r'rJr( padded_key[s  rWcCs^dd|}|dur-t|}dtt|t|}|r-|s'td|t|d|}|S)a; Joins characters of ``phrase`` and if ``symbols`` is given, raises an error if any character in ``phrase`` is not in ``symbols``. Parameters ========== phrase String or list of strings to be returned as a string. symbols Iterable of characters allowed in ``phrase``. If ``symbols`` is ``None``, no checking is performed. Examples ======== >>> from sympy.crypto.crypto import check_and_join >>> check_and_join('a phrase') 'a phrase' >>> check_and_join('a phrase'.upper().split()) 'APHRASE' >>> check_and_join('a phrase!'.upper().split(), 'ARE', filter=True) 'ARAE' >>> check_and_join('a phrase!'.upper().split(), 'ARE') Traceback (most recent call last): ... ValueError: characters in phrase but not symbols: "!HPS" rFNz*characters in phrase but not symbols: "%s")rOr7rPrRrQr)phraserKr6rCmissingr'r'r(r7|s  r7cCsV|s|st}t|}t|}n|}nd|}t||dd}t||dd}|||fS)NrFTr5)rDrOr7)r&rSalpdefaultr'r'r(_preps   r\cCs"||}tt||tt|S)a Returns the elements of the list ``range(n)`` shifted to the left by ``k`` (so the list starts with ``k`` (mod ``n``)). Examples ======== >>> from sympy.crypto.crypto import cycle_list >>> cycle_list(3, 10) [3, 4, 5, 6, 7, 8, 9, 0, 1, 2] )rMrange)knr'r'r( cycle_lists r`cCsJt|d|\}}}t||t|}||d|d|}t|||S)a Performs shift cipher encryption on plaintext msg, and returns the ciphertext. Parameters ========== key : int The secret key. msg : str Plaintext of upper-case letters. Returns ======= str Ciphertext of upper-case letters. Examples ======== >>> from sympy.crypto.crypto import encipher_shift, decipher_shift >>> msg = "GONAVYBEATARMY" >>> ct = encipher_shift(msg, 1); ct 'HPOBWZCFBUBSNZ' To decipher the shifted text, change the sign of the key: >>> encipher_shift(ct, -1) 'GONAVYBEATARMY' There is also a convenience function that does this with the original key: >>> decipher_shift(ct, 1) 'GONAVYBEATARMY' Notes ===== ALGORITHM: STEPS: 0. Number the letters of the alphabet from 0, ..., N 1. Compute from the string ``msg`` a list ``L1`` of corresponding integers. 2. Compute from the list ``L1`` a new list ``L2``, given by adding ``(k mod 26)`` to each element in ``L1``. 3. Compute from the list ``L2`` a string ``ct`` of corresponding letters. The shift cipher is also called the Caesar cipher, after Julius Caesar, who, according to Suetonius, used it with a shift of three to protect messages of military significance. Caesar's nephew Augustus reportedly used a similar cipher, but with a right shift of 1. References ========== .. [1] https://en.wikipedia.org/wiki/Caesar_cipher .. [2] https://mathworld.wolfram.com/CaesarsMethod.html See Also ======== decipher_shift rFN)r\rNr)r&rSrK_Ashiftr'r'r(encipher_shiftsG rdcCst|| |S)a Return the text by shifting the characters of ``msg`` to the left by the amount given by ``key``. Examples ======== >>> from sympy.crypto.crypto import encipher_shift, decipher_shift >>> msg = "GONAVYBEATARMY" >>> ct = encipher_shift(msg, 1); ct 'HPOBWZCFBUBSNZ' To decipher the shifted text, change the sign of the key: >>> encipher_shift(ct, -1) 'GONAVYBEATARMY' Or use this function with the original key: >>> decipher_shift(ct, 1) 'GONAVYBEATARMY' rdr&rSrKr'r'r(decipher_shiftsrgcC t|d|S)a  Performs the ROT13 encryption on a given plaintext ``msg``. Explanation =========== ROT13 is a substitution cipher which substitutes each letter in the plaintext message for the letter furthest away from it in the English alphabet. Equivalently, it is just a Caeser (shift) cipher with a shift key of 13 (midway point of the alphabet). References ========== .. [1] https://en.wikipedia.org/wiki/ROT13 See Also ======== decipher_rot13 encipher_shift rer&rKr'r'r(encipher_rot132s rkcCrh)a Performs the ROT13 decryption on a given plaintext ``msg``. Explanation ============ ``decipher_rot13`` is equivalent to ``encipher_rot13`` as both ``decipher_shift`` with a key of 13 and ``encipher_shift`` key with a key of 13 will return the same results. Nonetheless, ``decipher_rot13`` has nonetheless been explicitly defined here for consistency. Examples ======== >>> from sympy.crypto.crypto import encipher_rot13, decipher_rot13 >>> msg = 'GONAVYBEATARMY' >>> ciphertext = encipher_rot13(msg);ciphertext 'TBANILORNGNEZL' >>> decipher_rot13(ciphertext) 'GONAVYBEATARMY' >>> encipher_rot13(msg) == decipher_rot13(msg) True >>> msg == decipher_rot13(ciphertext) True ri)rgrjr'r'r(decipher_rot13Ns rlFcst|d|\}}t|\tdksJ|r+t} |}||dfddtD}t||S)a  Performs the affine cipher encryption on plaintext ``msg``, and returns the ciphertext. Explanation =========== Encryption is based on the map `x \rightarrow ax+b` (mod `N`) where ``N`` is the number of characters in the alphabet. Decryption is based on the map `x \rightarrow cx+d` (mod `N`), where `c = a^{-1}` (mod `N`) and `d = -a^{-1}b` (mod `N`). In particular, for the map to be invertible, we need `\mathrm{gcd}(a, N) = 1` and an error will be raised if this is not true. Parameters ========== msg : str Characters that appear in ``symbols``. a, b : int, int A pair integers, with ``gcd(a, N) = 1`` (the secret key). symbols String of characters (default = uppercase letters). When no symbols are given, ``msg`` is converted to upper case letters and all other characters are ignored. Returns ======= ct String of characters (the ciphertext message) Notes ===== ALGORITHM: STEPS: 0. Number the letters of the alphabet from 0, ..., N 1. Compute from the string ``msg`` a list ``L1`` of corresponding integers. 2. Compute from the list ``L1`` a new list ``L2``, given by replacing ``x`` by ``a*x + b (mod N)``, for each element ``x`` in ``L1``. 3. Compute from the list ``L2`` a string ``ct`` of corresponding letters. This is a straightforward generalization of the shift cipher with the added complexity of requiring 2 characters to be deciphered in order to recover the key. References ========== .. [1] https://en.wikipedia.org/wiki/Affine_cipher See Also ======== decipher_affine rFrHcs g|] }|qSr'r'r;rbNabr'r(r> z#encipher_affine..)r\rNr rrOr]r)r&rSrK_inverseracdBr'rmr(encipher_affineosC   " rvcCst|||ddS)at Return the deciphered text that was made from the mapping, `x \rightarrow ax+b` (mod `N`), where ``N`` is the number of characters in the alphabet. Deciphering is done by reciphering with a new key: `x \rightarrow cx+d` (mod `N`), where `c = a^{-1}` (mod `N`) and `d = -a^{-1}b` (mod `N`). Examples ======== >>> from sympy.crypto.crypto import encipher_affine, decipher_affine >>> msg = "GO NAVY BEAT ARMY" >>> key = (3, 1) >>> encipher_affine(msg, key) 'TROBMVENBGBALV' >>> decipher_affine(_, key) 'GONAVYBEATARMY' See Also ======== encipher_affine T)rrrvrfr'r'r(decipher_affinesrxcCrh)a Enciphers a given ``msg`` into its Atbash ciphertext and returns it. Explanation =========== Atbash is a substitution cipher originally used to encrypt the Hebrew alphabet. Atbash works on the principle of mapping each alphabet to its reverse / counterpart (i.e. a would map to z, b to y etc.) Atbash is functionally equivalent to the affine cipher with ``a = 25`` and ``b = 25`` See Also ======== decipher_atbash rzrwrjr'r'r(encipher_atbashs r{cCrh)a Deciphers a given ``msg`` using Atbash cipher and returns it. Explanation =========== ``decipher_atbash`` is functionally equivalent to ``encipher_atbash``. However, it has still been added as a separate function to maintain consistency. Examples ======== >>> from sympy.crypto.crypto import encipher_atbash, decipher_atbash >>> msg = 'GONAVYBEATARMY' >>> encipher_atbash(msg) 'TLMZEBYVZGZINB' >>> decipher_atbash(msg) 'TLMZEBYVZGZINB' >>> encipher_atbash(msg) == decipher_atbash(msg) True >>> msg == encipher_atbash(encipher_atbash(msg)) True References ========== .. [1] https://en.wikipedia.org/wiki/Atbash See Also ======== encipher_atbash ry)rxrjr'r'r(decipher_atbashs $r|cCs t|||S)a Returns the ciphertext obtained by replacing each character that appears in ``old`` with the corresponding character in ``new``. If ``old`` is a mapping, then new is ignored and the replacements defined by ``old`` are used. Explanation =========== This is a more general than the affine cipher in that the key can only be recovered by determining the mapping for each symbol. Though in practice, once a few symbols are recognized the mappings for other characters can be quickly guessed. Examples ======== >>> from sympy.crypto.crypto import encipher_substitution, AZ >>> old = 'OEYAG' >>> new = '034^6' >>> msg = AZ("go navy! beat army!") >>> ct = encipher_substitution(msg, old, new); ct '60N^V4B3^T^RM4' To decrypt a substitution, reverse the last two arguments: >>> encipher_substitution(ct, new, old) 'GONAVYBEATARMY' In the special case where ``old`` and ``new`` are a permutation of order 2 (representing a transposition of characters) their order is immaterial: >>> old = 'NAVY' >>> new = 'ANYV' >>> encipher = lambda x: encipher_substitution(x, old, new) >>> encipher('NAVY') 'ANYV' >>> encipher(_) 'NAVY' The substitution cipher, in general, is a method whereby "units" (not necessarily single characters) of plaintext are replaced with ciphertext according to a regular system. >>> ords = dict(zip('abc', ['\\%i' % ord(i) for i in 'abc'])) >>> print(encipher_substitution('abc', ords)) \97\98\99 References ========== .. [1] https://en.wikipedia.org/wiki/Substitution_cipher )r)r&oldnewr'r'r(encipher_substitutions 8rc st|||\}}}ddt|Dfdd|D}t}t|}g}t|D]\}}|||||||q)d|}|S)a Performs the Vigenere cipher encryption on plaintext ``msg``, and returns the ciphertext. Examples ======== >>> from sympy.crypto.crypto import encipher_vigenere, AZ >>> key = "encrypt" >>> msg = "meet me on monday" >>> encipher_vigenere(msg, key) 'QRGKKTHRZQEBPR' Section 1 of the Kryptos sculpture at the CIA headquarters uses this cipher and also changes the order of the alphabet [2]_. Here is the first line of that section of the sculpture: >>> from sympy.crypto.crypto import decipher_vigenere, padded_key >>> alp = padded_key('KRYPTOS', AZ()) >>> key = 'PALIMPSEST' >>> msg = 'EMUFPHZLRFAXYUSDJKZLDKRNSHGNFIVJ' >>> decipher_vigenere(msg, key, alp) 'BETWEENSUBTLESHADINGANDTHEABSENC' Explanation =========== The Vigenere cipher is named after Blaise de Vigenere, a sixteenth century diplomat and cryptographer, by a historical accident. Vigenere actually invented a different and more complicated cipher. The so-called *Vigenere cipher* was actually invented by Giovan Batista Belaso in 1553. This cipher was used in the 1800's, for example, during the American Civil War. The Confederacy used a brass cipher disk to implement the Vigenere cipher (now on display in the NSA Museum in Fort Meade) [1]_. The Vigenere cipher is a generalization of the shift cipher. Whereas the shift cipher shifts each letter by the same amount (that amount being the key of the shift cipher) the Vigenere cipher shifts a letter by an amount determined by the key (which is a word or phrase known only to the sender and receiver). For example, if the key was a single letter, such as "C", then the so-called Vigenere cipher is actually a shift cipher with a shift of `2` (since "C" is the 2nd letter of the alphabet, if you start counting at `0`). If the key was a word with two letters, such as "CA", then the so-called Vigenere cipher will shift letters in even positions by `2` and letters in odd positions are left alone (shifted by `0`, since "A" is the 0th letter, if you start counting at `0`). ALGORITHM: INPUT: ``msg``: string of characters that appear in ``symbols`` (the plaintext) ``key``: a string of characters that appear in ``symbols`` (the secret key) ``symbols``: a string of letters defining the alphabet OUTPUT: ``ct``: string of characters (the ciphertext message) STEPS: 0. Number the letters of the alphabet from 0, ..., N 1. Compute from the string ``key`` a list ``L1`` of corresponding integers. Let ``n1 = len(L1)``. 2. Compute from the string ``msg`` a list ``L2`` of corresponding integers. Let ``n2 = len(L2)``. 3. Break ``L2`` up sequentially into sublists of size ``n1``; the last sublist may be smaller than ``n1`` 4. For each of these sublists ``L`` of ``L2``, compute a new list ``C`` given by ``C[i] = L[i] + L1[i] (mod N)`` to the ``i``-th element in the sublist, for each ``i``. 5. Assemble these lists ``C`` by concatenation into a new list of length ``n2``. 6. Compute from the new list a string ``ct`` of corresponding letters. Once it is known that the key is, say, `n` characters long, frequency analysis can be applied to every `n`-th letter of the ciphertext to determine the plaintext. This method is called *Kasiski examination* (although it was first discovered by Babbage). If they key is as long as the message and is comprised of randomly selected characters -- a one-time pad -- the message is theoretically unbreakable. The cipher Vigenere actually discovered is an "auto-key" cipher described as follows. ALGORITHM: INPUT: ``key``: a string of letters (the secret key) ``msg``: string of letters (the plaintext message) OUTPUT: ``ct``: string of upper-case letters (the ciphertext message) STEPS: 0. Number the letters of the alphabet from 0, ..., N 1. Compute from the string ``msg`` a list ``L2`` of corresponding integers. Let ``n2 = len(L2)``. 2. Let ``n1`` be the length of the key. Append to the string ``key`` the first ``n2 - n1`` characters of the plaintext message. Compute from this string (also of length ``n2``) a list ``L1`` of integers corresponding to the letter numbers in the first step. 3. Compute a new list ``C`` given by ``C[i] = L1[i] + L2[i] (mod N)``. 4. Compute from the new list a string ``ct`` of letters corresponding to the new integers. To decipher the auto-key ciphertext, the key is used to decipher the first ``n1`` characters and then those characters become the key to decipher the next ``n1`` characters, etc...: >>> m = AZ('go navy, beat army! yes you can'); m 'GONAVYBEATARMYYESYOUCAN' >>> key = AZ('gold bug'); n1 = len(key); n2 = len(m) >>> auto_key = key + m[:n2 - n1]; auto_key 'GOLDBUGGONAVYBEATARMYYE' >>> ct = encipher_vigenere(m, auto_key); ct 'MCYDWSHKOGAMKZCELYFGAYR' >>> n1 = len(key) >>> pt = [] >>> while ct: ... part, ct = ct[:n1], ct[n1:] ... pt.append(decipher_vigenere(part, key)) ... key = pt[-1] ... >>> ''.join(pt) == m True References ========== .. [1] https://en.wikipedia.org/wiki/Vigenere_cipher .. [2] https://web.archive.org/web/20071116100808/https://filebox.vt.edu/users/batman/kryptos.html (short URL: https://goo.gl/ijr22d) cSi|]\}}||qSr'r'r<r=rsr'r'r( z%encipher_vigenere..cg|]}|qSr'r'r<rsmapr'r(r>z%encipher_vigenere..rF)r\ enumeraterNappendrO) r&rSrKrbrnr^rCr=mr'rr(encipher_vigenereYs$ rcs~t|||\}}ddtDtfdd|Dtfdd|D}dfddt|D}|S)z Decode using the Vigenere cipher. Examples ======== >>> from sympy.crypto.crypto import decipher_vigenere >>> key = "encrypt" >>> ct = "QRGK kt HRZQE BPR" >>> decipher_vigenere(ct, key) 'MEETMEONMONDAY' cSrr'r'rr'r'r(rrz%decipher_vigenere..crr'r'rrr'r(r>rz%decipher_vigenere..crr'r'rrr'r(r>rrFcs*g|]\}}| |qSr'r'r)rbKrnr_r'r(r>s*)r\rrNrO)r&rSrKCrCr')rbrrnrr_r(decipher_vigeneres"rQcsjsJt|dks Jt|||\}}ddtDfdd|Dtjt}t|\}}|rK|g||d7}dfddt|D}|S)a9 Return the Hill cipher encryption of ``msg``. Explanation =========== The Hill cipher [1]_, invented by Lester S. Hill in the 1920's [2]_, was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once. The following discussion assumes an elementary knowledge of matrices. First, each letter is first encoded as a number starting with 0. Suppose your message `msg` consists of `n` capital letters, with no spaces. This may be regarded an `n`-tuple M of elements of `Z_{26}` (if the letters are those of the English alphabet). A key in the Hill cipher is a `k x k` matrix `K`, all of whose entries are in `Z_{26}`, such that the matrix `K` is invertible (i.e., the linear transformation `K: Z_{N}^k \rightarrow Z_{N}^k` is one-to-one). Parameters ========== msg Plaintext message of `n` upper-case letters. key A `k \times k` invertible matrix `K`, all of whose entries are in `Z_{26}` (or whatever number of symbols are being used). pad Character (default "Q") to use to make length of text be a multiple of ``k``. Returns ======= ct Ciphertext of upper-case letters. Notes ===== ALGORITHM: STEPS: 0. Number the letters of the alphabet from 0, ..., N 1. Compute from the string ``msg`` a list ``L`` of corresponding integers. Let ``n = len(L)``. 2. Break the list ``L`` up into ``t = ceiling(n/k)`` sublists ``L_1``, ..., ``L_t`` of size ``k`` (with the last list "padded" to ensure its size is ``k``). 3. Compute new list ``C_1``, ..., ``C_t`` given by ``C[i] = K*L_i`` (arithmetic is done mod N), for each ``i``. 4. Concatenate these into a list ``C = C_1 + ... + C_t``. 5. Compute from ``C`` a string ``ct`` of corresponding letters. This has length ``k*t``. References ========== .. [1] https://en.wikipedia.org/wiki/Hill_cipher .. [2] Lester S. Hill, Cryptography in an Algebraic Alphabet, The American Mathematical Monthly Vol.36, June-July 1929, pp.306-312. See Also ======== decipher_hill rHcSrr'r'rr'r'r(rkrz!encipher_hill..crr'r'rrr'r(r>lrz!encipher_hill..rFc sRg|]%}ttdfddt||dDD]}|qqS)rHcrr'r'r;)Pr'r(r>us z,encipher_hill...rMrr])r<jrs)rbrnrr^rSr'r(r>ts  ) is_squarerNr\rcolsdivmodrOr])r&rSrKpadr_rrrCr')rbrnrr^rSrr( encipher_hills M$rcs|jsJt|d|\}}ddtDfdd|Dt|jt}t|\}}|rAdg||d7}|dfddt|D}|S) ab Deciphering is the same as enciphering but using the inverse of the key matrix. Examples ======== >>> from sympy.crypto.crypto import encipher_hill, decipher_hill >>> from sympy import Matrix >>> key = Matrix([[1, 2], [3, 5]]) >>> encipher_hill("meet me on monday", key) 'UEQDUEODOCTCWQ' >>> decipher_hill(_, key) 'MEETMEONMONDAY' When the length of the plaintext (stripped of invalid characters) is not a multiple of the key dimension, extra characters will appear at the end of the enciphered and deciphered text. In order to decipher the text, those characters must be included in the text to be deciphered. In the following, the key has a dimension of 4 but the text is 2 short of being a multiple of 4 so two characters will be added. >>> key = Matrix([[1, 1, 1, 2], [0, 1, 1, 0], ... [2, 2, 3, 4], [1, 1, 0, 1]]) >>> msg = "ST" >>> encipher_hill(msg, key) 'HJEB' >>> decipher_hill(_, key) 'STQQ' >>> encipher_hill(msg, key, pad="Z") 'ISPK' >>> decipher_hill(_, key) 'STZZ' If the last two characters of the ciphertext were ignored in either case, the wrong plaintext would be recovered: >>> decipher_hill("HD", key) 'ORMV' >>> decipher_hill("IS", key) 'UIKY' See Also ======== encipher_hill rFcSrr'r'rr'r'r(rrz!decipher_hill..crr'r'rrr'r(r>rz!decipher_hill..rrHc sRg|]%}ttdfddt||dDD]}|qqS)rHcrr'r'r;)rr'r(r>rz,decipher_hill...r)r<rp)rbrrnr^key_invr'r(r>s &) rr\rrNrrinv_modrOr])r&rSrKrar_rrrCr')rbrrnr^rrr( decipher_hillzs 3 $rc st|||t\}}}dt|p|t|d}|t|kr'tdt|t|tdkr@tfdd|DfddtDt fd d|D\}}||}d d Ddfd d t |d d d|dd dD}|S)a' Performs the Bifid cipher encryption on plaintext ``msg``, and returns the ciphertext. This is the version of the Bifid cipher that uses an `n \times n` Polybius square. Parameters ========== msg Plaintext string. key Short string for key. Duplicate characters are ignored and then it is padded with the characters in ``symbols`` that were not in the short key. symbols `n \times n` characters defining the alphabet. (default is string.printable) Returns ======= ciphertext Ciphertext using Bifid5 cipher without spaces. See Also ======== decipher_bifid, encipher_bifid5, encipher_bifid6 References ========== .. [1] https://en.wikipedia.org/wiki/Bifid_cipher rF?/Length of alphabet (%s) is not a square number.cg|]}|vr|qSr'r'r<xlong_keyr'r(r>z"encipher_bifid..ci|] \}}|t|qSr'rr<r=chrnr'r(rsz"encipher_bifid..crr'r'rrow_colr'r(r>rcSrr'r'r<rr=r'r'r(rrc3|]}|VqdSr"r'r;rr'r( z!encipher_bifid..NrH r\bifid10rOrrNintrQrMrzipitems) r&rSrKrbr_rrsrcrCr'rnrrrr(encipher_bifids *   2rcst|d|t\}}}dt|p|t|d}|t|kr'tdt|t|tdkr@tfdd|DfddtDfd d|D}t|}t |d |||d f}d d Ddfd d |D}|S)aL Performs the Bifid cipher decryption on ciphertext ``msg``, and returns the plaintext. This is the version of the Bifid cipher that uses the `n \times n` Polybius square. Parameters ========== msg Ciphertext string. key Short string for key. Duplicate characters are ignored and then it is padded with the characters in symbols that were not in the short key. symbols `n \times n` characters defining the alphabet. (default=string.printable, a `10 \times 10` matrix) Returns ======= deciphered Deciphered text. Examples ======== >>> from sympy.crypto.crypto import ( ... encipher_bifid, decipher_bifid, AZ) Do an encryption using the bifid5 alphabet: >>> alp = AZ().replace('J', '') >>> ct = AZ("meet me on monday!") >>> key = AZ("gold bug") >>> encipher_bifid(ct, key, alp) 'IEILHHFSTSFQYE' When entering the text or ciphertext, spaces are ignored so it can be formatted as desired. Re-entering the ciphertext from the preceding, putting 4 characters per line and padding with an extra J, does not cause problems for the deciphering: >>> decipher_bifid(''' ... IEILH ... HFSTS ... FQYEJ''', key, alp) 'MEETMEONMONDAY' When no alphabet is given, all 100 printable characters will be used: >>> key = '' >>> encipher_bifid('hello world!', key) 'bmtwmg-bIo*w' >>> decipher_bifid(_, key) 'hello world!' If the key is changed, a different encryption is obtained: >>> key = 'gold bug' >>> encipher_bifid('hello world!', 'gold_bug') 'hg2sfuei7t}w' And if the key used to decrypt the message is not exact, the original text will not be perfectly obtained: >>> decipher_bifid(_, 'gold pug') 'heldo~wor6d!' rFrrrcrr'r'rrr'r(r>Wrz"decipher_bifid..crr'rrrr'r(rZsz"decipher_bifid..csg|] }|D]}|qqSr'r')r<rsr=rr'r(r>\sNcSrr'r'rr'r'r(r_rc3rr"r'r;rr'r(r`rz!decipher_bifid..r)r&rSrKrarbr_rrCr'rr(decipher_bifids&N    rcsbdtd|tdtkrtdttfdd}t|}|S)a/Return characters of ``key`` arranged in a square. Examples ======== >>> from sympy.crypto.crypto import ( ... bifid_square, AZ, padded_key, bifid5) >>> bifid_square(AZ().replace('J', '')) Matrix([ [A, B, C, D, E], [F, G, H, I, K], [L, M, N, O, P], [Q, R, S, T, U], [V, W, X, Y, Z]]) >>> bifid_square(padded_key(AZ('gold bug!'), bifid5)) Matrix([ [G, O, L, D, B], [U, A, C, E, F], [H, I, K, M, N], [P, Q, R, S, T], [V, W, X, Y, Z]]) See Also ======== padded_key rFrrcst||Sr"rr=rrbr_r'r(rzbifid_square..)rOrrNrrQr)rSfrCr'rr( bifid_squareds    rcC2t||dt\}}}t|t}t|d|S)aO Performs the Bifid cipher encryption on plaintext ``msg``, and returns the ciphertext. Explanation =========== This is the version of the Bifid cipher that uses the `5 \times 5` Polybius square. The letter "J" is ignored so it must be replaced with something else (traditionally an "I") before encryption. ALGORITHM: (5x5 case) STEPS: 0. Create the `5 \times 5` Polybius square ``S`` associated to ``key`` as follows: a) moving from left-to-right, top-to-bottom, place the letters of the key into a `5 \times 5` matrix, b) if the key has less than 25 letters, add the letters of the alphabet not in the key until the `5 \times 5` square is filled. 1. Create a list ``P`` of pairs of numbers which are the coordinates in the Polybius square of the letters in ``msg``. 2. Let ``L1`` be the list of all first coordinates of ``P`` (length of ``L1 = n``), let ``L2`` be the list of all second coordinates of ``P`` (so the length of ``L2`` is also ``n``). 3. Let ``L`` be the concatenation of ``L1`` and ``L2`` (length ``L = 2*n``), except that consecutive numbers are paired ``(L[2*i], L[2*i + 1])``. You can regard ``L`` as a list of pairs of length ``n``. 4. Let ``C`` be the list of all letters which are of the form ``S[i, j]``, for all ``(i, j)`` in ``L``. As a string, this is the ciphertext of ``msg``. Parameters ========== msg : str Plaintext string. Converted to upper case and filtered of anything but all letters except J. key Short string for key; non-alphabetic letters, J and duplicated characters are ignored and then, if the length is less than 25 characters, it is padded with other letters of the alphabet (in alphabetical order). Returns ======= ct Ciphertext (all caps, no spaces). Examples ======== >>> from sympy.crypto.crypto import ( ... encipher_bifid5, decipher_bifid5) "J" will be omitted unless it is replaced with something else: >>> round_trip = lambda m, k: \ ... decipher_bifid5(encipher_bifid5(m, k), k) >>> key = 'a' >>> msg = "JOSIE" >>> round_trip(msg, key) 'OSIE' >>> round_trip(msg.replace("J", "I"), key) 'IOSIE' >>> j = "QIQ" >>> round_trip(msg.replace("J", j), key).replace(j, "J") 'JOSIE' Notes ===== The Bifid cipher was invented around 1901 by Felix Delastelle. It is a *fractional substitution* cipher, where letters are replaced by pairs of symbols from a smaller alphabet. The cipher uses a `5 \times 5` square filled with some ordering of the alphabet, except that "J" is replaced with "I" (this is a so-called Polybius square; there is a `6 \times 6` analog if you add back in "J" and also append onto the usual 26 letter alphabet, the digits 0, 1, ..., 9). According to Helen Gaines' book *Cryptanalysis*, this type of cipher was used in the field by the German Army during World War I. See Also ======== decipher_bifid5, encipher_bifid NrF)r\r8bifid5rWrr&rSrar'r'r(encipher_bifid5sf  rcCr)a Return the Bifid cipher decryption of ``msg``. Explanation =========== This is the version of the Bifid cipher that uses the `5 \times 5` Polybius square; the letter "J" is ignored unless a ``key`` of length 25 is used. Parameters ========== msg Ciphertext string. key Short string for key; duplicated characters are ignored and if the length is less then 25 characters, it will be padded with other letters from the alphabet omitting "J". Non-alphabetic characters are ignored. Returns ======= plaintext Plaintext from Bifid5 cipher (all caps, no spaces). Examples ======== >>> from sympy.crypto.crypto import encipher_bifid5, decipher_bifid5 >>> key = "gold bug" >>> encipher_bifid5('meet me on friday', key) 'IEILEHFSTSFXEE' >>> encipher_bifid5('meet me on monday', key) 'IEILHHFSTSFQYE' >>> decipher_bifid5(_, key) 'MEETMEONMONDAY' NrF)r\r8rrWrrr'r'r(decipher_bifid5s*  rcC:|st}t|Std|dt\}}}t|t}t|S)aN 5x5 Polybius square. Produce the Polybius square for the `5 \times 5` Bifid cipher. Examples ======== >>> from sympy.crypto.crypto import bifid5_square >>> bifid5_square("gold bug") Matrix([ [G, O, L, D, B], [U, A, C, E, F], [H, I, K, M, N], [P, Q, R, S, T], [V, W, X, Y, Z]]) rFN)rr\r8rWrrSrar'r'r( bifid5_square's  rcCr)at Performs the Bifid cipher encryption on plaintext ``msg``, and returns the ciphertext. This is the version of the Bifid cipher that uses the `6 \times 6` Polybius square. Parameters ========== msg Plaintext string (digits okay). key Short string for key (digits okay). If ``key`` is less than 36 characters long, the square will be filled with letters A through Z and digits 0 through 9. Returns ======= ciphertext Ciphertext from Bifid cipher (all caps, no spaces). See Also ======== decipher_bifid6, encipher_bifid NrF)r\r8bifid6rWrrr'r'r(encipher_bifid6Bs  rcCr)a Performs the Bifid cipher decryption on ciphertext ``msg``, and returns the plaintext. This is the version of the Bifid cipher that uses the `6 \times 6` Polybius square. Parameters ========== msg Ciphertext string (digits okay); converted to upper case key Short string for key (digits okay). If ``key`` is less than 36 characters long, the square will be filled with letters A through Z and digits 0 through 9. All letters are converted to uppercase. Returns ======= plaintext Plaintext from Bifid cipher (all caps, no spaces). Examples ======== >>> from sympy.crypto.crypto import encipher_bifid6, decipher_bifid6 >>> key = "gold bug" >>> encipher_bifid6('meet me on monday at 8am', key) 'KFKLJJHF5MMMKTFRGPL' >>> decipher_bifid6(_, key) 'MEETMEONMONDAYAT8AM' NrF)r\r8rrWrrr'r'r(decipher_bifid6gs&  rcCr)a 6x6 Polybius square. Produces the Polybius square for the `6 \times 6` Bifid cipher. Assumes alphabet of symbols is "A", ..., "Z", "0", ..., "9". Examples ======== >>> from sympy.crypto.crypto import bifid6_square >>> key = "gold bug" >>> bifid6_square(key) Matrix([ [G, O, L, D, B, U], [A, C, E, F, H, I], [J, K, M, N, P, Q], [R, S, T, V, W, X], [Y, Z, 0, 1, 2, 3], [4, 5, 6, 7, 8, 9]]) rFN)rr\r8rWrrr'r'r( bifid6_squares  rcs2fdd|D}t||}|std|dS)aDecipher RSA using chinese remainder theorem from the information of the relatively-prime factors of the modulus. Parameters ========== i : integer Ciphertext d : integer The exponent component. factors : list of relatively-prime integers The integers given must be coprime and the product must equal the modulus component of the original RSA key. Examples ======== How to decrypt RSA with CRT: >>> from sympy.crypto.crypto import rsa_public_key, rsa_private_key >>> primes = [61, 53] >>> e = 17 >>> args = primes + [e] >>> puk = rsa_public_key(*args) >>> prk = rsa_private_key(*args) >>> from sympy.crypto.crypto import encipher_rsa, _decipher_rsa_crt >>> msg = 65 >>> crt_primes = primes >>> encrypted = encipher_rsa(msg, puk) >>> decrypted = _decipher_rsa_crt(encrypted, prk[1], primes) >>> decrypted 65 csg|]}t|qSr'powr<rrtr=r'r(r>rz%_decipher_rsa_crt..z CRT failedr)rrQ)r=rtfactors modulusesresultr'rr(_decipher_rsa_crts % rTEuler)publicprivaterindex multipowercGsxt|dkrdS|dvrtd||dkrt}nt}|dur,t|}|dkr,td|dd |d }}td d |DsRg} |D] } | t| d d qD| }t dd|} t |} tdd | Drot || } n|s~t d|| | jddt || } t|| dkr|r|st|t r|| }||| 7}| |fS|r|st|| }t|t r||| 7}| |fSdS)aWA private subroutine to generate RSA key Parameters ========== public, private : bool, optional Flag to generate either a public key, a private key. totient : 'Euler' or 'Carmichael' Different notation used for totient. multipower : bool, optional Flag to bypass warning for multipower RSA. rF)r Carmichaelz@The argument totient={} should either be 'Euler', 'Carmichalel'.rNrz^Setting the 'index' keyword argument requires totientnotation to be specified as 'Carmichael'.css|]}t|VqdSr")rrr'r'r(rrz_rsa_key..T)multiplecS||Sr"r'rr'r'r(r z_rsa_key..css|]}|dkVqdS)rHNr')r<vr'r'r(rra:Non-distinctive primes found in the factors {}. The cipher may not be decryptable for some numbers in the complete residue system Z[{}], but the cipher can still be valid if you restrict the domain to be the reduced residue system Z*[{}]. You can pass the flag multipower=True if you want to suppress this warning.r-rH)rNrQformat_euler _carmichaelrallextendrrrvaluesrr!r0r r?r)rrrrrargs_totientprimese new_primesr=r_tallyphirtr'r'r(_rsa_keys\         rcOt|ddd|S)aReturn the RSA *public key* pair, `(n, e)` Parameters ========== args : naturals If specified as `p, q, e` where `p` and `q` are distinct primes and `e` is a desired public exponent of the RSA, `n = p q` and `e` will be verified against the totient `\phi(n)` (Euler totient) or `\lambda(n)` (Carmichael totient) to be `\gcd(e, \phi(n)) = 1` or `\gcd(e, \lambda(n)) = 1`. If specified as `p_1, p_2, \dots, p_n, e` where `p_1, p_2, \dots, p_n` are specified as primes, and `e` is specified as a desired public exponent of the RSA, it will be able to form a multi-prime RSA, which is a more generalized form of the popular 2-prime RSA. It can also be possible to form a single-prime RSA by specifying the argument as `p, e`, which can be considered a trivial case of a multiprime RSA. Furthermore, it can be possible to form a multi-power RSA by specifying two or more pairs of the primes to be same. However, unlike the two-distinct prime RSA or multi-prime RSA, not every numbers in the complete residue system (`\mathbb{Z}_n`) will be decryptable since the mapping `\mathbb{Z}_{n} \rightarrow \mathbb{Z}_{n}` will not be bijective. (Only except for the trivial case when `e = 1` or more generally, .. math:: e \in \left \{ 1 + k \lambda(n) \mid k \in \mathbb{Z} \land k \geq 0 \right \} when RSA reduces to the identity.) However, the RSA can still be decryptable for the numbers in the reduced residue system (`\mathbb{Z}_n^{\times}`), since the mapping `\mathbb{Z}_{n}^{\times} \rightarrow \mathbb{Z}_{n}^{\times}` can still be bijective. If you pass a non-prime integer to the arguments `p_1, p_2, \dots, p_n`, the particular number will be prime-factored and it will become either a multi-prime RSA or a multi-power RSA in its canonical form, depending on whether the product equals its radical or not. `p_1 p_2 \dots p_n = \text{rad}(p_1 p_2 \dots p_n)` totient : bool, optional If ``'Euler'``, it uses Euler's totient `\phi(n)` which is :meth:`sympy.functions.combinatorial.numbers.totient` in SymPy. If ``'Carmichael'``, it uses Carmichael's totient `\lambda(n)` which is :meth:`sympy.functions.combinatorial.numbers.reduced_totient` in SymPy. Unlike private key generation, this is a trivial keyword for public key generation because `\gcd(e, \phi(n)) = 1 \iff \gcd(e, \lambda(n)) = 1`. index : nonnegative integer, optional Returns an arbitrary solution of a RSA public key at the index specified at `0, 1, 2, \dots`. This parameter needs to be specified along with ``totient='Carmichael'``. Similarly to the non-uniquenss of a RSA private key as described in the ``index`` parameter documentation in :meth:`rsa_private_key`, RSA public key is also not unique and there is an infinite number of RSA public exponents which can behave in the same manner. From any given RSA public exponent `e`, there are can be an another RSA public exponent `e + k \lambda(n)` where `k` is an integer, `\lambda` is a Carmichael's totient function. However, considering only the positive cases, there can be a principal solution of a RSA public exponent `e_0` in `0 < e_0 < \lambda(n)`, and all the other solutions can be canonicalzed in a form of `e_0 + k \lambda(n)`. ``index`` specifies the `k` notation to yield any possible value an RSA public key can have. An example of computing any arbitrary RSA public key: >>> from sympy.crypto.crypto import rsa_public_key >>> rsa_public_key(61, 53, 17, totient='Carmichael', index=0) (3233, 17) >>> rsa_public_key(61, 53, 17, totient='Carmichael', index=1) (3233, 797) >>> rsa_public_key(61, 53, 17, totient='Carmichael', index=2) (3233, 1577) multipower : bool, optional Any pair of non-distinct primes found in the RSA specification will restrict the domain of the cryptosystem, as noted in the explanation of the parameter ``args``. SymPy RSA key generator may give a warning before dispatching it as a multi-power RSA, however, you can disable the warning if you pass ``True`` to this keyword. Returns ======= (n, e) : int, int `n` is a product of any arbitrary number of primes given as the argument. `e` is relatively prime (coprime) to the Euler totient `\phi(n)`. False Returned if less than two arguments are given, or `e` is not relatively prime to the modulus. Examples ======== >>> from sympy.crypto.crypto import rsa_public_key A public key of a two-prime RSA: >>> p, q, e = 3, 5, 7 >>> rsa_public_key(p, q, e) (15, 7) >>> rsa_public_key(p, q, 30) False A public key of a multiprime RSA: >>> primes = [2, 3, 5, 7, 11, 13] >>> e = 7 >>> args = primes + [e] >>> rsa_public_key(*args) (30030, 7) Notes ===== Although the RSA can be generalized over any modulus `n`, using two large primes had became the most popular specification because a product of two large primes is usually the hardest to factor relatively to the digits of `n` can have. However, it may need further understanding of the time complexities of each prime-factoring algorithms to verify the claim. See Also ======== rsa_private_key encipher_rsa decipher_rsa References ========== .. [1] https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29 .. [2] https://cacr.uwaterloo.ca/techreports/2006/cacr2006-16.pdf .. [3] https://link.springer.com/content/pdf/10.1007/BFb0055738.pdf .. [4] https://www.itiis.org/digital-library/manuscript/1381 TFrrrrkwargsr'r'r(rsa_public_key2s*rcOr)a/Return the RSA *private key* pair, `(n, d)` Parameters ========== args : naturals The keyword is identical to the ``args`` in :meth:`rsa_public_key`. totient : bool, optional If ``'Euler'``, it uses Euler's totient convention `\phi(n)` which is :meth:`sympy.functions.combinatorial.numbers.totient` in SymPy. If ``'Carmichael'``, it uses Carmichael's totient convention `\lambda(n)` which is :meth:`sympy.functions.combinatorial.numbers.reduced_totient` in SymPy. There can be some output differences for private key generation as examples below. Example using Euler's totient: >>> from sympy.crypto.crypto import rsa_private_key >>> rsa_private_key(61, 53, 17, totient='Euler') (3233, 2753) Example using Carmichael's totient: >>> from sympy.crypto.crypto import rsa_private_key >>> rsa_private_key(61, 53, 17, totient='Carmichael') (3233, 413) index : nonnegative integer, optional Returns an arbitrary solution of a RSA private key at the index specified at `0, 1, 2, \dots`. This parameter needs to be specified along with ``totient='Carmichael'``. RSA private exponent is a non-unique solution of `e d \mod \lambda(n) = 1` and it is possible in any form of `d + k \lambda(n)`, where `d` is an another already-computed private exponent, and `\lambda` is a Carmichael's totient function, and `k` is any integer. However, considering only the positive cases, there can be a principal solution of a RSA private exponent `d_0` in `0 < d_0 < \lambda(n)`, and all the other solutions can be canonicalzed in a form of `d_0 + k \lambda(n)`. ``index`` specifies the `k` notation to yield any possible value an RSA private key can have. An example of computing any arbitrary RSA private key: >>> from sympy.crypto.crypto import rsa_private_key >>> rsa_private_key(61, 53, 17, totient='Carmichael', index=0) (3233, 413) >>> rsa_private_key(61, 53, 17, totient='Carmichael', index=1) (3233, 1193) >>> rsa_private_key(61, 53, 17, totient='Carmichael', index=2) (3233, 1973) multipower : bool, optional The keyword is identical to the ``multipower`` in :meth:`rsa_public_key`. Returns ======= (n, d) : int, int `n` is a product of any arbitrary number of primes given as the argument. `d` is the inverse of `e` (mod `\phi(n)`) where `e` is the exponent given, and `\phi` is a Euler totient. False Returned if less than two arguments are given, or `e` is not relatively prime to the totient of the modulus. Examples ======== >>> from sympy.crypto.crypto import rsa_private_key A private key of a two-prime RSA: >>> p, q, e = 3, 5, 7 >>> rsa_private_key(p, q, e) (15, 7) >>> rsa_private_key(p, q, 30) False A private key of a multiprime RSA: >>> primes = [2, 3, 5, 7, 11, 13] >>> e = 7 >>> args = primes + [e] >>> rsa_private_key(*args) (30030, 823) See Also ======== rsa_public_key encipher_rsa decipher_rsa References ========== .. [1] https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29 .. [2] https://cacr.uwaterloo.ca/techreports/2006/cacr2006-16.pdf .. [3] https://link.springer.com/content/pdf/10.1007/BFb0055738.pdf .. [4] https://www.itiis.org/digital-library/manuscript/1381 FTrrrr'r'r(rsa_private_keyswrcCsX|\}}|s t|||Sdd}tdd|}||kr%||r%t|||St||ddS)NcSsPd}tt|D]}t|dt|D]}t||||dkr$d}nqq|S)NTrHF)r]rNr )lis_coprime_setr=rr'r'r(_is_coprime_set]sz/_encipher_decipher_rsa.._is_coprime_setcSrr"r'rr'r'r(rfrz(_encipher_decipher_rsa..r)rrr_encipher_decipher_rsa)r=rSrr_rtrprodr'r'r(rXs   rcCt|||dS)aEncrypt the plaintext with RSA. Parameters ========== i : integer The plaintext to be encrypted for. key : (n, e) where n, e are integers `n` is the modulus of the key and `e` is the exponent of the key. The encryption is computed by `i^e \bmod n`. The key can either be a public key or a private key, however, the message encrypted by a public key can only be decrypted by a private key, and vice versa, as RSA is an asymmetric cryptography system. factors : list of coprime integers This is identical to the keyword ``factors`` in :meth:`decipher_rsa`. Notes ===== Some specifications may make the RSA not cryptographically meaningful. For example, `0`, `1` will remain always same after taking any number of exponentiation, thus, should be avoided. Furthermore, if `i^e < n`, `i` may easily be figured out by taking `e` th root. And also, specifying the exponent as `1` or in more generalized form as `1 + k \lambda(n)` where `k` is an nonnegative integer, `\lambda` is a carmichael totient, the RSA becomes an identity mapping. Examples ======== >>> from sympy.crypto.crypto import encipher_rsa >>> from sympy.crypto.crypto import rsa_public_key, rsa_private_key Public Key Encryption: >>> p, q, e = 3, 5, 7 >>> puk = rsa_public_key(p, q, e) >>> msg = 12 >>> encipher_rsa(msg, puk) 3 Private Key Encryption: >>> p, q, e = 3, 5, 7 >>> prk = rsa_private_key(p, q, e) >>> msg = 12 >>> encipher_rsa(msg, prk) 3 Encryption using chinese remainder theorem: >>> encipher_rsa(msg, prk, factors=[p, q]) 3 rrr=rSrr'r'r( encipher_rsalsBrcCr)ap Decrypt the ciphertext with RSA. Parameters ========== i : integer The ciphertext to be decrypted for. key : (n, d) where n, d are integers `n` is the modulus of the key and `d` is the exponent of the key. The decryption is computed by `i^d \bmod n`. The key can either be a public key or a private key, however, the message encrypted by a public key can only be decrypted by a private key, and vice versa, as RSA is an asymmetric cryptography system. factors : list of coprime integers As the modulus `n` created from RSA key generation is composed of arbitrary prime factors `n = {p_1}^{k_1}{p_2}^{k_2}\dots{p_n}^{k_n}` where `p_1, p_2, \dots, p_n` are distinct primes and `k_1, k_2, \dots, k_n` are positive integers, chinese remainder theorem can be used to compute `i^d \bmod n` from the fragmented modulo operations like .. math:: i^d \bmod {p_1}^{k_1}, i^d \bmod {p_2}^{k_2}, \dots, i^d \bmod {p_n}^{k_n} or like .. math:: i^d \bmod {p_1}^{k_1}{p_2}^{k_2}, i^d \bmod {p_3}^{k_3}, \dots , i^d \bmod {p_n}^{k_n} as long as every moduli does not share any common divisor each other. The raw primes used in generating the RSA key pair can be a good option. Note that the speed advantage of using this is only viable for very large cases (Like 2048-bit RSA keys) since the overhead of using pure Python implementation of :meth:`sympy.ntheory.modular.crt` may overcompensate the theoretical speed advantage. Notes ===== See the ``Notes`` section in the documentation of :meth:`encipher_rsa` Examples ======== >>> from sympy.crypto.crypto import decipher_rsa, encipher_rsa >>> from sympy.crypto.crypto import rsa_public_key, rsa_private_key Public Key Encryption and Decryption: >>> p, q, e = 3, 5, 7 >>> prk = rsa_private_key(p, q, e) >>> puk = rsa_public_key(p, q, e) >>> msg = 12 >>> new_msg = encipher_rsa(msg, prk) >>> new_msg 3 >>> decipher_rsa(new_msg, puk) 12 Private Key Encryption and Decryption: >>> p, q, e = 3, 5, 7 >>> prk = rsa_private_key(p, q, e) >>> puk = rsa_public_key(p, q, e) >>> msg = 12 >>> new_msg = encipher_rsa(msg, puk) >>> new_msg 3 >>> decipher_rsa(new_msg, prk) 12 Decryption using chinese remainder theorem: >>> decipher_rsa(new_msg, prk, factors=[p, q]) 12 See Also ======== encipher_rsa rrrr'r'r( decipher_rsas`rcCs<||d}|||}|||}||d|}||fS)aa Kid RSA is a version of RSA useful to teach grade school children since it does not involve exponentiation. Explanation =========== Alice wants to talk to Bob. Bob generates keys as follows. Key generation: * Select positive integers `a, b, A, B` at random. * Compute `M = a b - 1`, `e = A M + a`, `d = B M + b`, `n = (e d - 1)//M`. * The *public key* is `(n, e)`. Bob sends these to Alice. * The *private key* is `(n, d)`, which Bob keeps secret. Encryption: If `p` is the plaintext message then the ciphertext is `c = p e \pmod n`. Decryption: If `c` is the ciphertext message then the plaintext is `p = c d \pmod n`. Examples ======== >>> from sympy.crypto.crypto import kid_rsa_public_key >>> a, b, A, B = 3, 4, 5, 6 >>> kid_rsa_public_key(a, b, A, B) (369, 58) rHr'rorprbruMrrtr_r'r'r(kid_rsa_public_keys  rcCs<||d}|||}|||}||d|}||fS)a< Compute `M = a b - 1`, `e = A M + a`, `d = B M + b`, `n = (e d - 1) / M`. The *private key* is `d`, which Bob keeps secret. Examples ======== >>> from sympy.crypto.crypto import kid_rsa_private_key >>> a, b, A, B = 3, 4, 5, 6 >>> kid_rsa_private_key(a, b, A, B) (369, 70) rHr'rr'r'r(kid_rsa_private_key>s   rcC|\}}|||S)aI Here ``msg`` is the plaintext and ``key`` is the public key. Examples ======== >>> from sympy.crypto.crypto import ( ... encipher_kid_rsa, kid_rsa_public_key) >>> msg = 200 >>> a, b, A, B = 3, 4, 5, 6 >>> key = kid_rsa_public_key(a, b, A, B) >>> encipher_kid_rsa(msg, key) 161 r')r&rSr_rr'r'r(encipher_kid_rsaTs rcCr)a Here ``msg`` is the plaintext and ``key`` is the private key. Examples ======== >>> from sympy.crypto.crypto import ( ... kid_rsa_public_key, kid_rsa_private_key, ... decipher_kid_rsa, encipher_kid_rsa) >>> a, b, A, B = 3, 4, 5, 6 >>> d = kid_rsa_private_key(a, b, A, B) >>> msg = 200 >>> pub = kid_rsa_public_key(a, b, A, B) >>> pri = kid_rsa_private_key(a, b, A, B) >>> ct = encipher_kid_rsa(msg, pub) >>> decipher_kid_rsa(ct, pri) 200 r')r&rSr_rtr'r'r(decipher_kid_rsahs rz.-rbz-...ruz-.-.rz-..D.Ez..-.Fz--.Gz....Hz..Iz.---z-.-rz.-..Lz--rz-.rnz---Oz.--.rz--.-z.-.Rz...S-Tz..-Uz...-Vz.--Wz-..-Xz-.--Yz--..Zz-----0z.----1z..---2z...--3z....-4z.....5z-....6z--...7z---..8z----.9z.-.-.-z--..--,z---...:z-.-.-.;z..--..?z-....-z..--.-raz-.--.(z-.--.-)z.----.'z-...-=z.-.-.+z-..-./z.--.-.@z...-..-$z-.-.--!cCrr'r')r<r^rr'r'r(rrr|c Cs|pt}||vs Jd|}||d<|o|dtv}|rdnd|}td|}t|}t|dd||}g}|}|D]} g} | D] } || } | | qM|| } || qG|||rm|SdS)a Encodes a plaintext into popular Morse Code with letters separated by ``sep`` and words by a double ``sep``. Examples ======== >>> from sympy.crypto.crypto import encode_morse >>> msg = 'ATTACK RIGHT FLANK' >>> encode_morse(msg) '.-|-|-|.-|-.-.|-.-||.-.|..|--.|....|-||..-.|.-..|.-|-.|-.-' References ========== .. [1] https://en.wikipedia.org/wiki/Morse_code r rrFN) char_morserrOr9rRkeysrr) r&sepmappingword_sepsuffixcharsok morsestringwordsword morsewordletter morseletterr'r'r( encode_morses&     rBc slptd|}g}|||}|D]}||}fdd|D}d|}||qd|} | S)a Decodes a Morse Code with letters separated by ``sep`` (default is '|') and words by `word_sep` (default is '||) into plaintext. Examples ======== >>> from sympy.crypto.crypto import decode_morse >>> mc = '--|---|...-|.||.|.-|...|-' >>> decode_morse(mc) 'MOVE EAST' References ========== .. [1] https://en.wikipedia.org/wiki/Morse_code rcrr'r'rr7r'r(r>rz decode_morse..rFr3) morse_charstripr9rOr) r&r6r7r8characterstringr=r>lettersr:rCr'rCr( decode_morses    rHpythongmpy) ground_typesc stts tdt|tstdd}t|}|}t|}g}t|D]*}|dd||d|d|}tfddt|D} ||| q(|S)a+ This function creates an LFSR sequence. Parameters ========== key : list A list of finite field elements, `[c_0, c_1, \ldots, c_k].` fill : list The list of the initial terms of the LFSR sequence, `[x_0, x_1, \ldots, x_k].` n Number of terms of the sequence that the function returns. Returns ======= L The LFSR sequence defined by `x_{n+1} = c_k x_n + \ldots + c_0 x_{n-k}`, for `n \leq k`. Notes ===== S. Golomb [G]_ gives a list of three statistical properties a sequence of numbers `a = \{a_n\}_{n=1}^\infty`, `a_n \in \{0,1\}`, should display to be considered "random". Define the autocorrelation of `a` to be .. math:: C(k) = C(k,a) = \lim_{N\rightarrow \infty} {1\over N}\sum_{n=1}^N (-1)^{a_n + a_{n+k}}. In the case where `a` is periodic with period `P` then this reduces to .. math:: C(k) = {1\over P}\sum_{n=1}^P (-1)^{a_n + a_{n+k}}. Assume `a` is periodic with period `P`. - balance: .. math:: \left|\sum_{n=1}^P(-1)^{a_n}\right| \leq 1. - low autocorrelation: .. math:: C(k) = \left\{ \begin{array}{cc} 1,& k = 0,\\ \epsilon, & k \ne 0. \end{array} \right. (For sequences satisfying these first two properties, it is known that `\epsilon = -1/P` must hold.) - proportional runs property: In each period, half the runs have length `1`, one-fourth have length `2`, etc. Moreover, there are as many runs of `1`'s as there are of `0`'s. Examples ======== >>> from sympy.crypto.crypto import lfsr_sequence >>> from sympy.polys.domains import FF >>> F = FF(2) >>> fill = [F(1), F(1), F(0), F(1)] >>> key = [F(1), F(0), F(0), F(1)] >>> lfsr_sequence(key, fill, 10) [1 mod 2, 1 mod 2, 0 mod 2, 1 mod 2, 0 mod 2, 1 mod 2, 1 mod 2, 0 mod 2, 0 mod 2, 1 mod 2] References ========== .. [G] Solomon Golomb, Shift register sequences, Aegean Park Press, Laguna Hills, Ca, 1967 zkey must be a listzfill must be a listrNrHc3s$|] }t||VqdSr"rr;rSs0r'r(rW s"z lfsr_sequence..) r?rM TypeErrormodulusrrNr]rsum) rSfillr_rr rAr^rr=rr'rMr(r s V     cslt|ts td|t|}t|d|}||dfddt|D}t|}t||S)a This function computes the LFSR autocorrelation function. Parameters ========== L A periodic sequence of elements of `GF(2)`. L must have length larger than P. P The period of L. k : int An integer `k` (`0 < k < P`). Returns ======= autocorrelation The k-th value of the autocorrelation of the LFSR L. Examples ======== >>> from sympy.crypto.crypto import ( ... lfsr_sequence, lfsr_autocorrelation) >>> from sympy.polys.domains import FF >>> F = FF(2) >>> fill = [F(1), F(1), F(0), F(1)] >>> key = [F(1), F(0), F(0), F(1)] >>> s = lfsr_sequence(key, fill, 20) >>> lfsr_autocorrelation(s, 15, 7) -1/15 >>> lfsr_autocorrelation(s, 15, 0) 1 zL (=%s) must be a listNcs,g|]}dt|t|qS)rrLr;L1r^r'r(r> s,z(lfsr_autocorrelation..)r?rMrOrr]rQr )rrr^L0L2totr'rSr(lfsr_autocorrelation\ s '   rXc sdtddddd}d}dd}d}dtkr|dkrit}t|d|d}dgfddtd|dDtt fddtd|D}|dkrwtd}|dkr|d7}d7|dkrd|kr||d|| |d7}d7n&}||d|| d|}d}|}|}d7tks(t}dgfd dtd|dDt fd dt|dDS) a This function computes the LFSR connection polynomial. Parameters ========== s A sequence of elements of even length, with entries in a finite field. Returns ======= C(x) The connection polynomial of a minimal LFSR yielding s. This implements the algorithm in section 3 of J. L. Massey's article [M]_. Examples ======== >>> from sympy.crypto.crypto import ( ... lfsr_sequence, lfsr_connection_polynomial) >>> from sympy.polys.domains import FF >>> F = FF(2) >>> fill = [F(1), F(1), F(0), F(1)] >>> key = [F(1), F(0), F(0), F(1)] >>> s = lfsr_sequence(key, fill, 20) >>> lfsr_connection_polynomial(s) x**4 + x + 1 >>> fill = [F(1), F(0), F(0), F(1)] >>> key = [F(1), F(1), F(0), F(1)] >>> s = lfsr_sequence(key, fill, 20) >>> lfsr_connection_polynomial(s) x**3 + 1 >>> fill = [F(1), F(0), F(1)] >>> key = [F(1), F(1), F(0)] >>> s = lfsr_sequence(key, fill, 20) >>> lfsr_connection_polynomial(s) x**3 + x**2 + 1 >>> fill = [F(1), F(0), F(1)] >>> key = [F(1), F(0), F(1)] >>> s = lfsr_sequence(key, fill, 20) >>> lfsr_connection_polynomial(s) x**3 + x + 1 References ========== .. [M] James L. Massey, "Shift-Register Synthesis and BCH Decoding." IEEE Trans. on Information Theory, vol. 15(1), pp. 122-127, Jan 1969. rrrHcg|] }|qSr'coeffr;rrr'r(r> sz.lfsr_connection_polynomial..c3s(|]}|t|VqdSr"rLr;)rncoeffsCrAr'r(r s z-lfsr_connection_polynomial..rcrYr'rZr;r\r'r(r> c3s0|]}|dur||VqdSr"r'r;)r]rrr'r(r s  ) rPr rNrdegreeminsubsr]rrQexpand) rArurrprdCrrtrr')rrnr]rrArr(lfsr_connection_polynomial sR 9       ( (  ,"rd cCs(t|}td|}|t||d|fS)a  Return three number tuple as private key. Explanation =========== Elgamal encryption is based on the mathematical problem called the Discrete Logarithm Problem (DLP). For example, `a^{b} \equiv c \pmod p` In general, if ``a`` and ``b`` are known, ``ct`` is easily calculated. If ``b`` is unknown, it is hard to use ``a`` and ``ct`` to get ``b``. Parameters ========== digit : int Minimum number of binary digits for key. Returns ======= tuple : (p, r, d) p = prime number. r = primitive root. d = random number. Notes ===== For testing purposes, the ``seed`` parameter may be set to control the output of this routine. See sympy.core.random._randrange. Examples ======== >>> from sympy.crypto.crypto import elgamal_private_key >>> from sympy.ntheory import is_primitive_root, isprime >>> a, b, _ = elgamal_private_key() >>> isprime(a) True >>> is_primitive_root(b, a) True r)r rr)digitseed randrangerr'r'r(elgamal_private_key s2 ricC|\}}}||t|||fS)a Return three number tuple as public key. Parameters ========== key : (p, r, e) Tuple generated by ``elgamal_private_key``. Returns ======= tuple : (p, r, e) `e = r**d \bmod p` `d` is a random number in private key. Examples ======== >>> from sympy.crypto.crypto import elgamal_public_key >>> elgamal_public_key((1031, 14, 636)) (1031, 14, 212) r)rSrrrr'r'r(elgamal_public_key) s rkcCs\|\}}}|dks ||krtd||ft|}|d|}t||||t||||fS)a/ Encrypt message with public key. Explanation =========== ``i`` is a plaintext message expressed as an integer. ``key`` is public key (p, r, e). In order to encrypt a message, a random number ``a`` in ``range(2, p)`` is generated and the encryped message is returned as `c_{1}` and `c_{2}` where: `c_{1} \equiv r^{a} \pmod p` `c_{2} \equiv m e^{a} \pmod p` Parameters ========== msg int of encoded message. key Public key. Returns ======= tuple : (c1, c2) Encipher into two number. Notes ===== For testing purposes, the ``seed`` parameter may be set to control the output of this routine. See sympy.core.random._randrange. Examples ======== >>> from sympy.crypto.crypto import encipher_elgamal, elgamal_private_key, elgamal_public_key >>> pri = elgamal_private_key(5, seed=[3]); pri (37, 2, 3) >>> pub = elgamal_public_key(pri); pub (37, 2, 8) >>> msg = 36 >>> encipher_elgamal(msg, pub, seed=[3]) (8, 6) rz#Message (%s) should be in range(%s)r)rQr r)r=rSrgrrrrhror'r'r(encipher_elgamalG s 3   rlcCs,|\}}}|\}}t|| |}|||S)a? Decrypt message with private key. `msg = (c_{1}, c_{2})` `key = (p, r, d)` According to extended Eucliden theorem, `u c_{1}^{d} + p n = 1` `u \equiv 1/{{c_{1}}^d} \pmod p` `u c_{2} \equiv \frac{1}{c_{1}^d} c_{2} \equiv \frac{1}{r^{ad}} c_{2} \pmod p` `\frac{1}{r^{ad}} m e^a \equiv \frac{1}{r^{ad}} m {r^{d a}} \equiv m \pmod p` Examples ======== >>> from sympy.crypto.crypto import decipher_elgamal >>> from sympy.crypto.crypto import encipher_elgamal >>> from sympy.crypto.crypto import elgamal_private_key >>> from sympy.crypto.crypto import elgamal_public_key >>> pri = elgamal_private_key(5, seed=[3]) >>> pub = elgamal_public_key(pri); pub (37, 2, 8) >>> msg = 17 >>> decipher_elgamal(encipher_elgamal(msg, pub), pri) == msg True r)r&rSrrartc1c2ur'r'r(decipher_elgamal s ! rpcCs0td|}t|}t|}|d|}|||fS)aF Return three integer tuple as private key. Explanation =========== Diffie-Hellman key exchange is based on the mathematical problem called the Discrete Logarithm Problem (see ElGamal). Diffie-Hellman key exchange is divided into the following steps: * Alice and Bob agree on a base that consist of a prime ``p`` and a primitive root of ``p`` called ``g`` * Alice choses a number ``a`` and Bob choses a number ``b`` where ``a`` and ``b`` are random numbers in range `[2, p)`. These are their private keys. * Alice then publicly sends Bob `g^{a} \pmod p` while Bob sends Alice `g^{b} \pmod p` * They both raise the received value to their secretly chosen number (``a`` or ``b``) and now have both as their shared key `g^{ab} \pmod p` Parameters ========== digit Minimum number of binary digits required in key. Returns ======= tuple : (p, g, a) p = prime number. g = primitive root of p. a = random number from 2 through p - 1. Notes ===== For testing purposes, the ``seed`` parameter may be set to control the output of this routine. See sympy.core.random._randrange. Examples ======== >>> from sympy.crypto.crypto import dh_private_key >>> from sympy.ntheory import isprime, is_primitive_root >>> p, g, _ = dh_private_key() >>> isprime(p) True >>> is_primitive_root(g, p) True >>> p, g, _ = dh_private_key(5) >>> isprime(p) True >>> is_primitive_root(g, p) True r)rrr )rfrgrgrhror'r'r(dh_private_key s >  rrcCrj)aS Return three number tuple as public key. This is the tuple that Alice sends to Bob. Parameters ========== key : (p, g, a) A tuple generated by ``dh_private_key``. Returns ======= tuple : int, int, int A tuple of `(p, g, g^a \mod p)` with `p`, `g` and `a` given as parameters.s Examples ======== >>> from sympy.crypto.crypto import dh_private_key, dh_public_key >>> p, g, a = dh_private_key(); >>> _p, _g, x = dh_public_key((p, g, a)) >>> p == _p and g == _g True >>> x == pow(g, a, p) True r)rSrrqror'r'r( dh_public_key s rscCs6|\}}}d|ks ||krttd|t|||S)a Return an integer that is the shared key. This is what Bob and Alice can both calculate using the public keys they received from each other and their private keys. Parameters ========== key : (p, g, x) Tuple `(p, g, x)` generated by ``dh_public_key``. b Random number in the range of `2` to `p - 1` (Chosen by second key exchange member (Bob)). Returns ======= int A shared key. Examples ======== >>> from sympy.crypto.crypto import ( ... dh_private_key, dh_public_key, dh_shared_key) >>> prk = dh_private_key(); >>> p, g, x = dh_public_key(prk); >>> sk = dh_shared_key((p, g, x), 1000) >>> sk == pow(x, 1000, p) True rHzO Value of b should be greater 1 and less than prime %s.)rQrr)rSrprrarr'r'r( dh_shared_key s # rtcCs0t||dd|}|dkrdS|dkrdSdS)a Returns the legendre symbol of a and p assuming that p is a prime. i.e. 1 if a is a quadratic residue mod p -1 if a is not a quadratic residue mod p 0 if a is divisible by p Parameters ========== a : int The number to test. p : prime The prime to test ``a`` against. Returns ======= int Legendre symbol (a / p). rHrrrr)rorsigr'r'r( _legendreC s rvccs*t|} ||}t||dkr|Vq)NTrH)r r )r_rgrhyr'r'r(_random_coprime_streame srxcCs\||kr td|t|rt|std||f|dks"|dkr*td||f||fS)a] Check if ``p`` and ``q`` can be used as private keys for the Goldwasser-Micali encryption. The method works roughly as follows. Explanation =========== #. Pick two large primes $p$ and $q$. #. Call their product $N$. #. Given a message as an integer $i$, write $i$ in its bit representation $b_0, \dots, b_n$. #. For each $k$, if $b_k = 0$: let $a_k$ be a random square (quadratic residue) modulo $p q$ such that ``jacobi_symbol(a, p*q) = 1`` if $b_k = 1$: let $a_k$ be a random non-square (non-quadratic residue) modulo $p q$ such that ``jacobi_symbol(a, p*q) = 1`` returns $\left[a_1, a_2, \dots\right]$ $b_k$ can be recovered by checking whether or not $a_k$ is a residue. And from the $b_k$'s, the message can be reconstructed. The idea is that, while ``jacobi_symbol(a, p*q)`` can be easily computed (and when it is equal to $-1$ will tell you that $a$ is not a square mod $p q$), quadratic residuosity modulo a composite number is hard to compute without knowing its factorization. Moreover, approximately half the numbers coprime to $p q$ have :func:`~.jacobi_symbol` equal to $1$ . And among those, approximately half are residues and approximately half are not. This maximizes the entropy of the code. Parameters ========== p, q, a Initialization variables. Returns ======= tuple : (p, q) The input value ``p`` and ``q``. Raises ====== ValueError If ``p`` and ``q`` are not distinct odd primes. z.expected distinct primes, got two copies of %iz/first two arguments must be prime, got %i of %irz2first two arguments must not be even, got %i of %i)rQr)rqror'r'r(gm_private_keym s;rzcCst||\}}||}|dur0t|} ||}t||t||kr(dkr/nn ||fSqt||dks>t||dkr@dS||fS)a Compute public keys for ``p`` and ``q``. Note that in Goldwasser-Micali Encryption, public keys are randomly selected. Parameters ========== p, q, a : int, int, int Initialization variables. Returns ======= tuple : (a, N) ``a`` is the input ``a`` if it is not ``None`` otherwise some random integer coprime to ``p`` and ``q``. ``N`` is the product of ``p`` and ``q``. NTrF)rzr rv)rryrorgrnrhr'r'r( gm_public_key s$r{csz|dkr td||\g}|dkr#||d|d}|dkst|t|}fddfdd|DS)a/ Encrypt integer 'i' using public_key 'key' Note that gm uses random encryption. Parameters ========== i : int The message to encrypt. key : (a, N) The public key. Returns ======= list : list of int The randomized encrypted message. r6message must be a non-negative integer: got %d insteadrcstdt|S)Nr)nextr)rp)rnrogenr'r(r r^zencipher_gm..csg|]}|qSr'r')r<rp)encoder'r(r> rzencipher_gm..)rQrrxreversed)r=rSrgbitsrevr')rnrorr~r( encipher_gm s  rcsJ|\ddfdd|D}d}|D] }|dK}|| 7}q|S)a  Decrypt message 'message' using public_key 'key'. Parameters ========== message : list of int The randomized encrypted message. key : (p, q) The private key. Returns ======= int The encrypted message. cSst||dkS)Nr)rv)rrr'r'r(r szdecipher_gm..cs g|] }||qSr'r')r<rrryresr'r(r> rqzdecipher_gm..rrHr')messagerSrrrpr'rr( decipher_gm s rcs<tt|}t||ddddt|fdddS)a Performs Railfence Encryption on plaintext and returns ciphertext Examples ======== >>> from sympy.crypto.crypto import encipher_railfence >>> message = "hello world" >>> encipher_railfence(message,3) 'horel ollwd' Parameters ========== message : string, the message to encrypt. rails : int, the number of rails. Returns ======= The Encrypted string message. References ========== .. [1] https://en.wikipedia.org/wiki/Rail_fence_cipher rrrFctSr"r}r=rr'r(r? rz$encipher_railfence..rS)rMr]rrOrP)rrailsrr'rr(encipher_railfence! s rcsrtt|}t||dddttt|fddd}dgt|}t||D]\}}|||<q+d|S)ay Decrypt the message using the given rails Examples ======== >>> from sympy.crypto.crypto import decipher_railfence >>> decipher_railfence("horel ollwd",3) 'hello world' Parameters ========== message : string, the message to encrypt. rails : int, the number of rails. Returns ======= The Decrypted string message. rrrcrr"rrrr'r(r\ rz$decipher_railfence..rrF)rMr]rrPrNrrO) ciphertextrridxrr=rsr'rr(decipher_railfenceB s   rcCslt|rt|std||f||krtd||dddks*|dddkr2td||f||fS)a= Check if p and q can be used as private keys for the Blum-Goldwasser cryptosystem. Explanation =========== The three necessary checks for p and q to pass so that they can be used as private keys: 1. p and q must both be prime 2. p and q must be distinct 3. p and q must be congruent to 3 mod 4 Parameters ========== p, q The keys to be checked. Returns ======= p, q Input values. Raises ====== ValueError If p and q do not pass the above conditions. z.the two arguments must be prime, got %i and %iz:the two arguments must be distinct, got two copies of %i. r,rrz=the two arguments must be congruent to 3 mod 4, got %i and %i)rrQ)rryr'r'r(bg_private_keye s# rcCst||\}}||}|S)aH Calculates public keys from private keys. Explanation =========== The function first checks the validity of private keys passed as arguments and then returns their product. Parameters ========== p, q The private keys. Returns ======= N The public key. )r)rryrnr'r'r( bg_public_key src Cs|dkr td|g}|dkr||d|d}|dks|t|}t|d|d}|d|}tt|td|t|}g}t|D]} ||d|d|}qJddt||D} | |fS)a Encrypts the message using public key and seed. Explanation =========== ALGORITHM: 1. Encodes i as a string of L bits, m. 2. Select a random element r, where 1 < r < key, and computes x = r^2 mod key. 3. Use BBS pseudo-random number generator to generate L random bits, b, using the initial seed as x. 4. Encrypted message, c_i = m_i XOR b_i, 1 <= i <= L. 5. x_L = x^(2^L) mod key. 6. Return (c, x_L) Parameters ========== i Message, a non-negative integer key The public key Returns ======= Tuple (encrypted_message, x_L) Raises ====== ValueError If i is negative. rr|rrHcSsg|]\}}||AqSr'r')r<rrpr'r'r(r> rzencipher_bg..) rQrreverserNr rrr]r) r=rSrgenc_msgrrrx_L rand_bitsra encrypt_msgr'r'r( encipher_bg s,(  rcCs|\}}|\}}||}t|}|dd|}|dd|} tt|t|t|} tt|t| t|} |t||| |t||| |} g} t|D]}| | d| d|} qRd}t|| D]\}}|d}|||A7}qi|S)a Decrypts the message using private keys. Explanation =========== ALGORITHM: 1. Let, c be the encrypted message, y the second number received, and p and q be the private keys. 2. Compute, r_p = y^((p+1)/4 ^ L) mod p and r_q = y^((q+1)/4 ^ L) mod q. 3. Compute x_0 = (q(q^-1 mod p)r_p + p(p^-1 mod q)r_q) mod N. 4. From, recompute the bits using the BBS generator, as in the encryption algorithm. 5. Compute original message by XORing c and b. Parameters ========== message Tuple of encrypted message and a non-negative integer. key Tuple of private keys. Returns ======= orig_msg The original message rHrrr)rNrrrr]rr)rrSrryrrw public_keyrp_tq_tr_pr_qr orig_bitsraorig_msgrrpr'r'r( decipher_bg s$"( rr")NN)NF)Nr)r2N)reN)zr4stringrrr:r functoolsrr/ itertoolsrsympy.external.gmpyr sympy.corer sympy.core.numbersr sympy.core.randomr r r r%sympy.functions.combinatorial.numbersrrrrsympy.matricesr sympy.ntheoryrrrsympy.ntheory.generatersympy.ntheory.modularrsympy.polys.domainsrsympy.polys.polytoolsrsympy.utilities.miscrrrsympy.utilities.iterablesrrsympy.utilities.decoratorr__doctest_skip__RuntimeWarningr!rDreplacerrrrWr7r\r`rdrgrkrlrvrxr{r|rrrrrrrrrrrrrrrrrrrrrrrrrrDrr4rBrHr rXrdrirkrlrprrrsrtrvrxrzr{rrrrrrrrr'r'r'r(s              ! ,  M   ! O   ) ? (  _ H >d)k /% + -S- z  Ef'            / % g2 d7 < )E#/ "  G &&!! # .A