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authorReto <reto.fritsche@ost.ch>2021-04-24 14:11:14 +0200
committerReto <reto.fritsche@ost.ch>2021-04-24 14:11:14 +0200
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downloadSeminarMatrizen-0d8053d944fd7a01ba634fd26fb946ee2f7d4f4e.tar.gz
SeminarMatrizen-0d8053d944fd7a01ba634fd26fb946ee2f7d4f4e.zip
added simple code example of mceliece cryptosystem
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+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Sun Apr 18 08:58:30 2021
+
+@author: terminator
+"""
+import numpy as np
+from numpy.polynomial import Polynomial as Poly
+
+
+def gen_perm_M(n):
+ '''
+ generating random binary permutation matrix: https://en.wikipedia.org/wiki/Permutation_matrix
+
+ Parameters
+ ----------
+ n : int
+ size of output matrix.
+
+ Returns
+ -------
+ perm_M : np.ndarray
+ binary permutation matrix.
+
+ '''
+ perm_M=np.zeros((n, n), dtype=int)
+ perm_V=np.random.default_rng().permutation(n)
+ perm_M[perm_V, np.r_[0:n]]=1
+ return perm_M
+
+
+
+def create_linear_code_matrix(n, k, g):
+ '''
+ create generator matrix for linear encoding
+ use this matrix to create code_vector: matrix @ data_vector=code_vector
+
+ Parameters
+ ----------
+ n : int
+ len of code_vector (matrix @ data_vector).
+ k : int
+ len of data_vector.
+ g : numpy.Polynominal
+ polynomina wich defines the the constellations of .
+
+ Returns
+ -------
+ np.ndarray
+ Generator matrix
+
+ '''
+ if n != k+len(g)-1:
+ raise Exception("n, k not compatible with degree of g")
+ rows=[]
+ for i in range(k): #create potences of p
+ row=np.r_[np.zeros(i), g.coef, np.zeros(n-k-i)]
+ rows.append(row)
+ return np.array(rows, dtype=int).T
+
+
+def gf2_inv(M, get_full=False):
+ '''
+ create inverse of matrix M in gf2
+
+ Parameters
+ ----------
+ M : numpy.ndarray
+ input Matrix.
+ get_full : Bool, optional
+ if Ture, return inverse of G with I on the left (if gaussian inversion successful) If True, vaidity is not proven. The default is False.
+
+ Returns
+ -------
+ np.ndarray or None
+ returns inverse if M was not singular in gf2, else None.
+
+ '''
+ size=len(M)
+ G=np.hstack((M, np.eye(size)))
+ G=np.array(G, dtype=int)
+ for n in range(size): #forward reduction
+ if G[n, n] == 0: #swap line if necessary
+ for i in range(n+1, size):
+ if G[i, n]:
+ G[i, :], G[n, :] = G[n, :].copy(), G[i, :].copy() #swap
+
+ for i in range(n+1, size): #downward reduction
+ if G[i, n]:
+ G[i, :] = G[i, :] ^ G[n, :]
+ #reached buttom_right with pivo
+ for n in range(size-1, -1, -1): #backwards
+ for i in range(n):
+ if G[i, n]:
+ G[i, :]= G[i, :] ^ G[n, :]
+
+
+
+ if get_full:
+ return G
+ else:
+ valid=np.sum(np.abs(G[:, :size]-np.eye(size))) == 0 #reduction successfull when eye left on the left of G
+ if not valid:
+ return None
+ else:
+ return G[:, size:]
+
+def create_rand_bin_M(n, with_inverse=False):
+ '''
+ create random binary matrix
+
+ Parameters
+ ----------
+ n : int
+ size.
+ with_inverse : bool, optional
+ if False, return only random binary matrix.
+ if True, return also inverse of random martix. for this purpose, random matrix will not be singular.
+ The default is False.
+
+ Returns
+ -------
+ M : TYPE
+ if with_inverse is True: return tuple(random_matrix, inverse_of_random_matrix)
+ else: return random_matrix
+
+ '''
+ inv=None
+ while type(inv) == type(None): #do it until inversion of m is successful wich means det(M)%2 != 0
+ M=np.random.randint(0,2, (n,n))
+ inv=gf2_inv(M)
+ if with_inverse:
+ return(M, inv)
+ else:
+ return M
+
+def create_syndrome_table(n, g):
+ '''
+ create syndrome table for correcting errors in linear code
+
+ Parameters
+ ----------
+ n : int
+ len of linear-code code_vector.
+ g : numpy.Polinominal
+ generator polynominal used by linear-code-encoder.
+
+ Returns
+ -------
+ list of error_vectors, one per syndrome.
+ get the corresponding error_vector by using the value represented by your syndrome as index of this list.
+
+ '''
+ zeros=np.zeros(n, dtype=int)
+ syndrome_table=[0 for i in range(n+1)]
+ syndrome_table[0]=zeros #when syndrome = 0, no bit-error to correct
+ for i in range(n):
+ faulty_vector=zeros.copy()
+ faulty_vector[i]=1
+ q, r=divmod(Poly(faulty_vector), g)
+ r=np.r_[r.coef%2, np.zeros(len(g)-len(r))]
+ index=np.sum([int(a*2**i) for i, a in enumerate(r)])
+ syndrome_table[index]=faulty_vector
+ return np.array(syndrome_table)
+
+
+def decode_linear_code(c, g, syndrome_table):
+ '''
+ function to decode codeword encoded with linear_code
+
+ Parameters
+ ----------
+ c : list or np.ndarray
+ code_vector.
+ g : numpy.Polynominal
+ generator polynominal, used to create code_vector.
+ syndrome_table : list of error_vectors
+ if codeword contains an error, syndrome_table is used to correct wrong bit.
+
+ Returns
+ -------
+ numpy.ndarray
+ data_vector.
+
+ '''
+ q, r=divmod(Poly(c), g)
+ q=np.r_[q.coef%2, np.zeros(len(c)-len(q)-len(g)+1)]
+ r=np.r_[r.coef%2, np.zeros(len(g)-len(r))]
+ syndrome_index=np.sum([int(a*2**i) for i, a in enumerate(r)])
+ while syndrome_index > 0:
+ c=c ^ syndrome_table[syndrome_index]
+ q, r=divmod(Poly(c), g)
+ q=np.r_[q.coef%2, np.zeros(len(c)-len(q)-len(g)+1)]
+ r=np.r_[r.coef%2, np.zeros(len(g)-len(r))]
+ syndrome_index=np.sum([int(a*2**i) for i, a in enumerate(r)])
+ return np.array(q, dtype=int)
+
+def encode_linear_code(d, G):
+ '''
+ uses generator matrix G to encode d
+
+ Parameters
+ ----------
+ d : numpy.ndarray
+ data_vector.
+ G : numpy.ndarray
+ generator matrix.
+
+ Returns
+ -------
+ c : numpy.ndarray
+ G @ d.
+
+ '''
+ c=(G @ d)%2
+ return c
+
+def encrypt(d, pub_key, t):
+ '''
+ encrypt data with public key, adding t bit-errors
+
+ Parameters
+ ----------
+ d : numpy.ndarray or list of numpy.ndarray
+ data_vector or list of data vectors to encrypt.
+ pub_key : numpy.ndarray
+ public key matrix used to encrypt data.
+ t : int
+ number of random errors to add to code_vector.
+
+ Returns
+ -------
+ numpy.ndarray or list of numpy.ndarray (depending on d)
+ encrypted data.
+
+ '''
+ if type(d) in (list,):
+ encrypted_list=[encrypt(data, pub_key, t) for data in d]
+ return encrypted_list
+ else:
+ c = np.array((pub_key @ d)%2, dtype=int)
+
+ indexes_of_errors=np.random.default_rng().permutation(pub_key.shape[0])[:t] #add t random errors to codeword
+ e=np.zeros(pub_key.shape[0], dtype=int)
+ e[indexes_of_errors]=1
+ c=c ^ e
+ return np.array(c, dtype=int)
+
+def decrypt(c, P_inv, linear_code_decoder, S_inv):
+ '''
+ decrypt encrypted message
+
+ Parameters
+ ----------
+ c : numpy.ndarray or list of numpy.ndarray
+ code_vector or list of code_vectors to decrypt.
+ P_inv : numpy.ndarray
+ inverted permutation matrix.
+ linear_code_decoder : function(x)
+ function to use to decode linear code.
+ S_inv : numpy.ndarray
+ inverted random binary matrix.
+
+ Returns
+ -------
+ numpy.ndarray or list of numpy.ndarray (depending on d)
+ decrypted data.
+
+ '''
+ if type(c) in (list,):
+ decrypted_list=[decrypt(codew, P_inv, linear_code_decoder, S_inv) for codew in c]
+ return decrypted_list
+ else:
+ c=np.array(P_inv @ c, dtype=int)%2
+ d=linear_code_decoder(c)
+ d=(S_inv @ d)%2
+ return np.array(d, dtype=int)
+
+def str_to_blocks4(string):
+ blocks=[]
+ for char in string:
+ bits=[int(value) for value in np.binary_repr(ord(char), 8)[::-1]] #lsb @ index 0
+ blocks.append(np.array(bits[:4], dtype=int)) #lower nibble first
+ blocks.append(np.array(bits[4:], dtype=int))
+ return blocks
+
+def blocks4_to_str(blocks):
+ string=''
+ for i in range(0, len(blocks), 2):
+ char=np.sum([b*2**i for i, b in enumerate(blocks[i])]) + \
+ np.sum([b*2**(i+4) for i, b in enumerate(blocks[i+1])])
+ string+=chr(char)
+ return string
+
+#shared attributes:
+n=7
+k=4
+t=1
+
+#private key(s):
+g=Poly([1,1,0,1]) #generator polynom for 7/4 linear code (from table, 1.0 + 1.0·x¹ + 0.0·x² + 1.0·x³)
+P_M=gen_perm_M(n) #create permutation matrix
+G_M=create_linear_code_matrix(n, k, g) #linear code generator matrix
+S_M, S_inv=create_rand_bin_M(k, True) #random binary matrix and its inverse
+P_M_inv=P_M.T #inverse permutation matrix
+
+syndrome_table=create_syndrome_table(n, g) #part of linear-code decoder
+linear_code_decoder=lambda c:decode_linear_code(c, g, syndrome_table)
+
+#public key:
+pub_key=(P_M @ G_M @ S_M)%2
+
+
+msg_tx='Hello World?'
+
+blocks_tx=str_to_blocks4(msg_tx)
+encrypted=encrypt(blocks_tx, pub_key, t)
+
+blocks_rx=decrypt(encrypted, P_M_inv, linear_code_decoder, S_inv)
+msg_rx=blocks4_to_str(blocks_rx)
+
+print(f'msg_rx: {msg_rx}')
+
+ \ No newline at end of file