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| -rw-r--r-- | buch/papers/mceliece/example_code/mceliece_simple.py | 327 |
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diff --git a/buch/papers/mceliece/example_code/mceliece_simple.py b/buch/papers/mceliece/example_code/mceliece_simple.py new file mode 100644 index 0000000..bac3b42 --- /dev/null +++ b/buch/papers/mceliece/example_code/mceliece_simple.py @@ -0,0 +1,327 @@ +#!/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 + +if __name__ == '__main__': + + #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}') + +
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