reed_solomon.py

#!/usr/bin/env python
# -*- coding: utf-8 -*-

###################################################
### Universal Reed-Solomon Codec
### initially released at Wikiversity
###################################################

################### INIT and stuff ###################

import sys as sys

DARTH_PLAGUEIS_SCRIPT = "Did you ever hear the tragedy of Darth Plagueis The Wise? I thought not. It's not a story the Jedi would tell you. It's a Sith legend. Darth Plagueis was a Dark Lord of the Sith, so powerful and so wise he could use the Force to influence the midichlorians to create life... He had such a knowledge of the dark side that he could even keep the ones he cared about from dying. The dark side of the Force is a pathway to many abilities some consider to be unnatural. He became so powerful... the only thing he was afraid of was losing his power, which eventually, of course, he did. Unfortunately, he taught his apprentice everything he knew, then his apprentice killed him in his sleep. Ironic. He could save others from death, but not himself."

try: # compatibility with Python 3+
    xrange
except NameError:
    xrange = range

class ReedSolomonError(Exception):
    pass
    #def __init__(self, arg):
    #    print(arg)
    #    sys.exit(1)

gf_exp = [0] * 512 # For efficiency, gf_exp[] has size 2*GF_SIZE, so that a simple multiplication of two numbers can be resolved without calling % 255. For more infos on how to generate this extended exponentiation table, see paper: "Fast software implementation of finite field operations", Cheng Huang and Lihao Xu, Washington University in St. Louis, Tech. Rep (2003).
gf_log = [0] * 256
field_charac = int(2**8 - 1)


################### GALOIS FIELD ELEMENTS MATHS ###################

def rwh_primes1(n):
    # http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
    ''' Returns  a list of primes < n '''
    sieve = [True] * (n/2)
    for i in xrange(3,int(n**0.5)+1,2):
        if sieve[i/2]:
            sieve[i*i/2::i] = [False] * ((n-i*i-1)/(2*i)+1)
    return [2] + [2*i+1 for i in xrange(1,n/2) if sieve[i]]

def find_prime_polys(generator=2, c_exp=8, fast_primes=False, single=False):
    '''Compute the list of prime polynomials for the given generator and galois field characteristic exponent.'''
    # fast_primes will output less results but will be significantly faster.
    # single will output the first prime polynomial found, so if all you want is to just find one prime polynomial to generate the LUT for Reed-Solomon to work, then just use that.

    # A prime polynomial (necessarily irreducible) is necessary to reduce the multiplications in the Galois Field, so as to avoid overflows.
    # Why do we need a "prime polynomial"? Can't we just reduce modulo 255 (for GF(2^8) for example)? Because we need the values to be unique.
    # For example: if the generator (alpha) = 2 and c_exp = 8 (GF(2^8) == GF(256)), then the generated Galois Field (0, 1, a, a^1, a^2, ..., a^(p-1)) will be galois field it becomes 0, 1, 2, 4, 8, 16, etc. However, upon reaching 128, the next value will be doubled (ie, next power of 2), which will give 256. Then we must reduce, because we have overflowed above the maximum value of 255. But, if we modulo 255, this will generate 256 == 1. Then 2, 4, 8, 16, etc. giving us a repeating pattern of numbers. This is very bad, as it's then not anymore a bijection (ie, a non-zero value doesn't have a unique index). That's why we can't just modulo 255, but we need another number above 255, which is called the prime polynomial.
    # Why so much hassle? Because we are using precomputed look-up tables for multiplication: instead of multiplying a*b, we precompute alpha^a, alpha^b and alpha^(a+b), so that we can just use our lookup table at alpha^(a+b) and get our result. But just like in our original field we had 0,1,2,...,p-1 distinct unique values, in our "LUT" field using alpha we must have unique distinct values (we don't care that they are different from the original field as long as they are unique and distinct). That's why we need to avoid duplicated values, and to avoid duplicated values we need to use a prime irreducible polynomial.

    # Here is implemented a bruteforce approach to find all these prime polynomials, by generating every possible prime polynomials (ie, every integers between field_charac+1 and field_charac*2), and then we build the whole Galois Field, and we reject the candidate prime polynomial if it duplicates even one value or if it generates a value above field_charac (ie, cause an overflow).
    # Note that this algorithm is slow if the field is too big (above 12), because it's an exhaustive search algorithm. There are probabilistic approaches, and almost surely prime approaches, but there is no determistic polynomial time algorithm to find irreducible monic polynomials. More info can be found at: http://people.mpi-inf.mpg.de/~csaha/lectures/lec9.pdf
    # Another faster algorithm may be found at Adleman, Leonard M., and Hendrik W. Lenstra. "Finding irreducible polynomials over finite fields." Proceedings of the eighteenth annual ACM symposium on Theory of computing. ACM, 1986.

    # Prepare the finite field characteristic (2^p - 1), this also represent the maximum possible value in this field
    root_charac = 2 # we're in GF(2)
    field_charac = int(root_charac**c_exp - 1)
    field_charac_next = int(root_charac**(c_exp+1) - 1)

    prim_candidates = []
    if fast_primes:
        prim_candidates = rwh_primes1(field_charac_next) # generate maybe prime polynomials and check later if they really are irreducible
        prim_candidates = [x for x in prim_candidates if x > field_charac] # filter out too small primes
    else:
        prim_candidates = xrange(field_charac+2, field_charac_next, root_charac) # try each possible prime polynomial, but skip even numbers (because divisible by 2 so necessarily not irreducible)

    # Start of the main loop
    correct_primes = []
    for prim in prim_candidates: # try potential candidates primitive irreducible polys
        seen = [0] * (field_charac+1) # memory variable to indicate if a value was already generated in the field (value at index x is set to 1) or not (set to 0 by default)
        conflict = False # flag to know if there was at least one conflict

        # Second loop, build the whole Galois Field
        x = 1
        for i in xrange(field_charac):
            # Compute the next value in the field (ie, the next power of alpha/generator)
            x = gf_mult_noLUT(x, generator, prim, field_charac+1)

            # Rejection criterion: if the value overflowed (above field_charac) or is a duplicate of a previously generated power of alpha, then we reject this polynomial (not prime)
            if x > field_charac or seen[x] == 1:
                conflict = True
                break
            # Else we flag this value as seen (to maybe detect future duplicates), and we continue onto the next power of alpha
            else:
                seen[x] = 1

        # End of the second loop: if there's no conflict (no overflow nor duplicated value), this is a prime polynomial!
        if not conflict: 
            correct_primes.append(prim)
            if single: return prim

    # Return the list of all prime polynomials
    return correct_primes # you can use the following to print the hexadecimal representation of each prime polynomial: print [hex(i) for i in correct_primes]

def init_tables(prim=0x11d, generator=2, c_exp=8):
    '''Precompute the logarithm and anti-log tables for faster computation later, using the provided primitive polynomial.
    These tables are used for multiplication/division since addition/substraction are simple XOR operations inside GF of characteristic 2.
    The basic idea is quite simple: since b**(log_b(x), log_b(y)) == x * y given any number b (the base or generator of the logarithm), then we can use any number b to precompute logarithm and anti-log (exponentiation) tables to use for multiplying two numbers x and y.
    That's why when we use a different base/generator number, the log and anti-log tables are drastically different, but the resulting computations are the same given any such tables.
    For more infos, see https://en.wikipedia.org/wiki/Finite_field_arithmetic#Implementation_tricks
    '''
    # generator is the generator number (the "increment" that will be used to walk through the field by multiplication, this must be a prime number). This is basically the base of the logarithm/anti-log tables. Also often noted "alpha" in academic books.
    # prim is the primitive/prime (binary) polynomial and must be irreducible (ie, it can't represented as the product of two smaller polynomials). It's a polynomial in the binary sense: each bit is a coefficient, but in fact it's an integer between field_charac+1 and field_charac*2, and not a list of gf values. The prime polynomial will be used to reduce the overflows back into the range of the Galois Field without duplicating values (all values should be unique). See the function find_prime_polys() and: http://research.swtch.com/field and http://www.pclviewer.com/rs2/galois.html
    # note that the choice of generator or prime polynomial doesn't matter very much: any two finite fields of size p^n have identical structure, even if they give the individual elements different names (ie, the coefficients of the codeword will be different, but the final result will be the same: you can always correct as many errors/erasures with any choice for those parameters). That's why it makes sense to refer to all the finite fields, and all decoders based on Reed-Solomon, of size p^n as one concept: GF(p^n). It can however impact sensibly the speed (because some parameters will generate sparser tables).
    # c_exp is the exponent for the field's characteristic GF(2^c_exp)

    global gf_exp, gf_log, field_charac
    field_charac = int(2**c_exp - 1)
    gf_exp = [0] * (field_charac * 2) # anti-log (exponential) table. The first two elements will always be [GF256int(1), generator]
    gf_log = [0] * (field_charac+1) # log table, log[0] is impossible and thus unused

    # For each possible value in the galois field 2^8, we will pre-compute the logarithm and anti-logarithm (exponential) of this value
    # To do that, we generate the Galois Field F(2^p) by building a list starting with the element 0 followed by the (p-1) successive powers of the generator a : 1, a, a^1, a^2, ..., a^(p-1).
    x = 1
    for i in xrange(field_charac): # we could skip index 255 which is equal to index 0 because of modulo: g^255==g^0 but either way, this does not change the later outputs (ie, the ecc symbols will be the same either way)
        gf_exp[i] = x # compute anti-log for this value and store it in a table
        gf_log[x] = i # compute log at the same time
        x = gf_mult_noLUT(x, generator, prim, field_charac+1)

        # If you use only generator==2 or a power of 2, you can use the following which is faster than gf_mult_noLUT():
        #x <<= 1 # multiply by 2 (change 1 by another number y to multiply by a power of 2^y)
        #if x & 0x100: # similar to x >= 256, but a lot faster (because 0x100 == 256)
            #x ^= prim # substract the primary polynomial to the current value (instead of 255, so that we get a unique set made of coprime numbers), this is the core of the tables generation

    # Optimization: double the size of the anti-log table so that we don't need to mod 255 to stay inside the bounds (because we will mainly use this table for the multiplication of two GF numbers, no more).
    for i in xrange(field_charac, field_charac * 2):
        gf_exp[i] = gf_exp[i - field_charac]

    return [gf_log, gf_exp]

def gf_add(x, y):
    return x ^ y

def gf_sub(x, y):
    return x ^ y # in binary galois field, substraction is just the same as addition (since we mod 2)

def gf_neg(x):
    return x

def gf_mul(x, y):
    if x == 0 or y == 0:
        return 0
    return gf_exp[(gf_log[x] + gf_log[y]) % field_charac]

def gf_div(x, y):
    if y == 0:
        raise ZeroDivisionError()
    if x == 0:
        return 0
    return gf_exp[(gf_log[x] + field_charac - gf_log[y]) % field_charac]

def gf_pow(x, power):
    return gf_exp[(gf_log[x] * power) % field_charac]

def gf_inverse(x):
    return gf_exp[field_charac - gf_log[x]] # gf_inverse(x) == gf_div(1, x)

def gf_mult_noLUT(x, y, prim=0, field_charac_full=256, carryless=True):
    '''Galois Field integer multiplication using Russian Peasant Multiplication algorithm (faster than the standard multiplication + modular reduction).
    If prim is 0 and carryless=False, then the function produces the result for a standard integers multiplication (no carry-less arithmetics nor modular reduction).'''
    r = 0
    while y: # while y is above 0
        if y & 1: r = r ^ x if carryless else r + x # y is odd, then add the corresponding x to r (the sum of all x's corresponding to odd y's will give the final product). Note that since we're in GF(2), the addition is in fact an XOR (very important because in GF(2) the multiplication and additions are carry-less, thus it changes the result!).
        y = y >> 1 # equivalent to y // 2
        x = x << 1 # equivalent to x*2
        if prim > 0 and x & field_charac_full: x = x ^ prim # GF modulo: if x >= 256 then apply modular reduction using the primitive polynomial (we just substract, but since the primitive number can be above 256 then we directly XOR).

    return r


################### GALOIS FIELD POLYNOMIALS MATHS ###################

def gf_poly_scale(p, x):
    return [gf_mul(p[i], x) for i in xrange(len(p))]

def gf_poly_add(p,q):
    r = [0] * max(len(p),len(q))
    for i in xrange(len(p)):
        r[i+len(r)-len(p)] = p[i]
    for i in xrange(len(q)):
        r[i+len(r)-len(q)] ^= q[i]
    return r

def gf_poly_mul(p, q):
    '''Multiply two polynomials, inside Galois Field (but the procedure is generic). Optimized function by precomputation of log.'''
    # Pre-allocate the result array
    r = [0] * (len(p) + len(q) - 1)
    # Precompute the logarithm of p
    lp = [gf_log[p[i]] for i in xrange(len(p))]
    # Compute the polynomial multiplication (just like the outer product of two vectors, we multiply each coefficients of p with all coefficients of q)
    for j in xrange(len(q)):
        qj = q[j] # optimization: load the coefficient once
        if qj != 0: # log(0) is undefined, we need to check that
            lq = gf_log[qj] # Optimization: precache the logarithm of the current coefficient of q
            for i in xrange(len(p)):
                if p[i] != 0: # log(0) is undefined, need to check that...
                    r[i + j] ^= gf_exp[lp[i] + lq] # equivalent to: r[i + j] = gf_add(r[i+j], gf_mul(p[i], q[j]))
    return r

def gf_poly_mul_simple(p, q): # simple equivalent way of multiplying two polynomials without precomputation, but thus it's slower
    '''Multiply two polynomials, inside Galois Field'''
    # Pre-allocate the result array
    r = [0] * (len(p) + len(q) - 1)
    # Compute the polynomial multiplication (just like the outer product of two vectors, we multiply each coefficients of p with all coefficients of q)
    for j in xrange(len(q)):
        for i in xrange(len(p)):
            r[i + j] ^= gf_mul(p[i], q[j]) # equivalent to: r[i + j] = gf_add(r[i+j], gf_mul(p[i], q[j])) -- you can see it's your usual polynomial multiplication
    return r

def gf_poly_neg(poly):
    '''Returns the polynomial with all coefficients negated. In GF(2^p), negation does not change the coefficient, so we return the polynomial as-is.'''
    return poly

def gf_poly_div(dividend, divisor):
    '''Fast polynomial division by using Extended Synthetic Division and optimized for GF(2^p) computations
    (doesn't work with standard polynomials outside of this galois field, see the Wikipedia article for generic algorithm).'''
    # CAUTION: this function expects polynomials to follow the opposite convention at decoding:
    # the terms must go from the biggest to lowest degree (while most other functions here expect
    # a list from lowest to biggest degree). eg: 1 + 2x + 5x^2 = [5, 2, 1], NOT [1, 2, 5]

    msg_out = list(dividend) # Copy the dividend list and pad with 0 where the ecc bytes will be computed
    #normalizer = divisor[0] # precomputing for performance
    for i in xrange(len(dividend) - (len(divisor)-1)):
        #msg_out[i] /= normalizer # for general polynomial division (when polynomials are non-monic), the usual way of using
                                  # synthetic division is to divide the divisor g(x) with its leading coefficient, but not needed here.
        coef = msg_out[i] # precaching
        if coef != 0: # log(0) is undefined, so we need to avoid that case explicitly (and it's also a good optimization).
            for j in xrange(1, len(divisor)): # in synthetic division, we always skip the first coefficient of the divisior,
                                              # because it's only used to normalize the dividend coefficient
                if divisor[j] != 0: # log(0) is undefined
                    msg_out[i + j] ^= gf_mul(divisor[j], coef) # equivalent to the more mathematically correct
                                                               # (but xoring directly is faster): msg_out[i + j] += -divisor[j] * coef

    # The resulting msg_out contains both the quotient and the remainder, the remainder being the size of the divisor
    # (the remainder has necessarily the same degree as the divisor -- not length but degree == length-1 -- since it's
    # what we couldn't divide from the dividend), so we compute the index where this separation is, and return the quotient and remainder.
    separator = -(len(divisor)-1)
    return msg_out[:separator], msg_out[separator:] # return quotient, remainder.

def gf_poly_eval(poly, x):
    '''Evaluates a polynomial in GF(2^p) given the value for x. This is based on Horner's scheme for maximum efficiency.'''
    y = poly[0]
    for i in xrange(1, len(poly)):
        y = gf_mul(y, x) ^ poly[i]
    return y


################### REED-SOLOMON ENCODING ###################

def rs_generator_poly(nsym, fcr=0, generator=2):
    '''Generate an irreducible generator polynomial (necessary to encode a message into Reed-Solomon)'''
    g = [1]
    for i in xrange(nsym):
        g = gf_poly_mul(g, [1, gf_pow(generator, i+fcr)])
    return g

def rs_generator_poly_all(max_nsym, fcr=0, generator=2):
    '''Generate all irreducible generator polynomials up to max_nsym (usually you can use n, the length of the message+ecc). Very useful to reduce processing time if you want to encode using variable schemes and nsym rates.'''
    g_all = {}
    g_all[0] = g_all[1] = [1]
    for nsym in xrange(max_nsym):
        g_all[nsym] = rs_generator_poly(nsym, fcr, generator)
    return g_all

def rs_simple_encode_msg(msg_in, nsym, fcr=0, generator=2, gen=None):
    '''Simple Reed-Solomon encoding (mainly an example for you to understand how it works, because it's slower than the inlined function below)'''
    global field_charac
    if (len(msg_in) + nsym) > field_charac: raise ValueError("Message is too long (%i when max is %i)" % (len(msg_in)+nsym, field_charac))
    if gen is None: gen = rs_generator_poly(nsym, fcr, generator)

    # Pad the message, then divide it by the irreducible generator polynomial
    _, remainder = gf_poly_div(msg_in + [0] * (len(gen)-1), gen)
    # The remainder is our RS code! Just append it to our original message to get our full codeword (this represents a polynomial of max 256 terms)
    msg_out = msg_in + remainder
    # Return the codeword
    return msg_out

def rs_encode_msg(msg_in, nsym, fcr=0, generator=2, gen=None):
    '''Reed-Solomon main encoding function, using polynomial division (Extended Synthetic Division, the fastest algorithm available to my knowledge), better explained at http://research.swtch.com/field'''
    global field_charac
    if (len(msg_in) + nsym) > field_charac: raise ValueError("Message is too long (%i when max is %i)" % (len(msg_in)+nsym, field_charac))
    if gen is None: gen = rs_generator_poly(nsym, fcr, generator)
    # Init msg_out with the values inside msg_in and pad with len(gen)-1 bytes (which is the number of ecc symbols).
    msg_out = [0] * (len(msg_in) + len(gen)-1)
    # Initializing the Synthetic Division with the dividend (= input message polynomial)
    msg_out[:len(msg_in)] = msg_in

    # Synthetic division main loop
    for i in xrange(len(msg_in)):
        # Note that it's msg_out here, not msg_in. Thus, we reuse the updated value at each iteration
        # (this is how Synthetic Division works: instead of storing in a temporary register the intermediate values,
        # we directly commit them to the output).
        coef = msg_out[i]

        # log(0) is undefined, so we need to manually check for this case.
        if coef != 0:
            # in synthetic division, we always skip the first coefficient of the divisior, because it's only used to normalize the dividend coefficient (which is here useless since the divisor, the generator polynomial, is always monic)
            for j in xrange(1, len(gen)):
                #if gen[j] != 0: # log(0) is undefined so we need to check that, but it slow things down in fact and it's useless in our case (reed-solomon encoding) since we know that all coefficients in the generator are not 0
                msg_out[i+j] ^= gf_mul(gen[j], coef) # equivalent to msg_out[i+j] += gf_mul(gen[j], coef)

    # At this point, the Extended Synthetic Divison is done, msg_out contains the quotient in msg_out[:len(msg_in)]
    # and the remainder in msg_out[len(msg_in):]. Here for RS encoding, we don't need the quotient but only the remainder
    # (which represents the RS code), so we can just overwrite the quotient with the input message, so that we get
    # our complete codeword composed of the message + code.
    msg_out[:len(msg_in)] = msg_in

    return msg_out


################### REED-SOLOMON DECODING ###################

def rs_calc_syndromes(msg, nsym, fcr=0, generator=2):
    '''Given the received codeword msg and the number of error correcting symbols (nsym), computes the syndromes polynomial.
    Mathematically, it's essentially equivalent to a Fourrier Transform (Chien search being the inverse).
    '''
    # Note the "[0] +" : we add a 0 coefficient for the lowest degree (the constant). This effectively shifts the syndrome, and will shift every computations depending on the syndromes (such as the errors locator polynomial, errors evaluator polynomial, etc. but not the errors positions).
    # This is not necessary, you can adapt subsequent computations to start from 0 instead of skipping the first iteration (ie, the often seen range(1, n-k+1)),
    synd = [0] * nsym
    for i in xrange(nsym):
        synd[i] = gf_poly_eval(msg, gf_pow(generator, i+fcr))
    return [0] + synd # pad with one 0 for mathematical precision (else we can end up with weird calculations sometimes)

def rs_correct_errata(msg_in, synd, err_pos, fcr=0, generator=2): # err_pos is a list of the positions of the errors/erasures/errata
    '''Forney algorithm, computes the values (error magnitude) to correct the input message.'''
    global field_charac
    # calculate errata locator polynomial to correct both errors and erasures (by combining the errors positions given by the error locator polynomial found by BM with the erasures positions given by caller)
    coef_pos = [len(msg_in) - 1 - p for p in err_pos] # need to convert the positions to coefficients degrees for the errata locator algo to work (eg: instead of [0, 1, 2] it will become [len(msg)-1, len(msg)-2, len(msg) -3])
    err_loc = rs_find_errata_locator(coef_pos, generator)
    # calculate errata evaluator polynomial (often called Omega or Gamma in academic papers)
    err_eval = rs_find_error_evaluator(synd[::-1], err_loc, len(err_loc)-1)[::-1]

    # Second part of Chien search to get the error location polynomial X from the error positions in err_pos (the roots of the error locator polynomial, ie, where it evaluates to 0)
    X = [] # will store the position of the errors
    for i in xrange(len(coef_pos)):
        l = field_charac - coef_pos[i]
        X.append( gf_pow(generator, -l) )

    # Forney algorithm: compute the magnitudes
    E = [0] * (len(msg_in)) # will store the values that need to be corrected (substracted) to the message containing errors. This is sometimes called the error magnitude polynomial.
    Xlength = len(X)
    for i, Xi in enumerate(X):

        Xi_inv = gf_inverse(Xi)

        # Compute the formal derivative of the error locator polynomial (see Blahut, Algebraic codes for data transmission, pp 196-197).
        # the formal derivative of the errata locator is used as the denominator of the Forney Algorithm, which simply says that the ith error value is given by error_evaluator(gf_inverse(Xi)) / error_locator_derivative(gf_inverse(Xi)). See Blahut, Algebraic codes for data transmission, pp 196-197.
        err_loc_prime_tmp = []
        for j in xrange(Xlength):
            if j != i:
                err_loc_prime_tmp.append( gf_sub(1, gf_mul(Xi_inv, X[j])) )
        # compute the product, which is the denominator of the Forney algorithm (errata locator derivative)
        err_loc_prime = 1
        for coef in err_loc_prime_tmp:
            err_loc_prime = gf_mul(err_loc_prime, coef)
        # equivalent to: err_loc_prime = functools.reduce(gf_mul, err_loc_prime_tmp, 1)

        # Compute y (evaluation of the errata evaluator polynomial)
        # This is a more faithful translation of the theoretical equation contrary to the old forney method. Here it is an exact reproduction:
        # Yl = omega(Xl.inverse()) / prod(1 - Xj*Xl.inverse()) for j in len(X)
        y = gf_poly_eval(err_eval[::-1], Xi_inv) # numerator of the Forney algorithm (errata evaluator evaluated)
        y = gf_mul(gf_pow(Xi, 1-fcr), y) # adjust to fcr parameter
        
        # Check: err_loc_prime (the divisor) should not be zero.
        if err_loc_prime == 0:
            raise ReedSolomonError("Could not find error magnitude")    # Could not find error magnitude

        # Compute the magnitude
        magnitude = gf_div(y, err_loc_prime) # magnitude value of the error, calculated by the Forney algorithm (an equation in fact): dividing the errata evaluator with the errata locator derivative gives us the errata magnitude (ie, value to repair) the ith symbol
        E[err_pos[i]] = magnitude # store the magnitude for this error into the magnitude polynomial

    # Apply the correction of values to get our message corrected! (note that the ecc bytes also gets corrected!)
    # (this isn't the Forney algorithm, we just apply the result of decoding here)
    msg_in = gf_poly_add(msg_in, E) # equivalent to Ci = Ri - Ei where Ci is the correct message, Ri the received (senseword) message, and Ei the errata magnitudes (minus is replaced by XOR since it's equivalent in GF(2^p)). So in fact here we substract from the received message the errors magnitude, which logically corrects the value to what it should be.
    return msg_in

def rs_find_error_locator(synd, nsym, erase_loc=None, erase_count=0):
    '''Find error/errata locator and evaluator polynomials with Berlekamp-Massey algorithm'''
    # The idea is that BM will iteratively estimate the error locator polynomial.
    # To do this, it will compute a Discrepancy term called Delta, which will tell us if the error locator polynomial needs an update or not
    # (hence why it's called discrepancy: it tells us when we are getting off board from the correct value).

    # Init the polynomials
    if erase_loc: # if the erasure locator polynomial is supplied, we init with its value, so that we include erasures in the final locator polynomial
        err_loc = list(erase_loc)
        old_loc = list(erase_loc)
    else:
        err_loc = [1] # This is the main variable we want to fill, also called Sigma in other notations or more formally the errors/errata locator polynomial.
        old_loc = [1] # BM is an iterative algorithm, and we need the errata locator polynomial of the previous iteration in order to update other necessary variables.
    #L = 0 # update flag variable, not needed here because we use an alternative equivalent way of checking if update is needed (but using the flag could potentially be faster depending on if using length(list) is taking linear time in your language, here in Python it's constant so it's as fast.

    # Fix the syndrome shifting: when computing the syndrome, some implementations may prepend a 0 coefficient for the lowest degree term (the constant). This is a case of syndrome shifting, thus the syndrome will be bigger than the number of ecc symbols (I don't know what purpose serves this shifting). If that's the case, then we need to account for the syndrome shifting when we use the syndrome such as inside BM, by skipping those prepended coefficients.
    # Another way to detect the shifting is to detect the 0 coefficients: by definition, a syndrome does not contain any 0 coefficient (except if there are no errors/erasures, in this case they are all 0). This however doesn't work with the modified Forney syndrome, which set to 0 the coefficients corresponding to erasures, leaving only the coefficients corresponding to errors.
    synd_shift = 0
    if len(synd) > nsym: synd_shift = len(synd) - nsym

    for i in xrange(nsym-erase_count): # generally: nsym-erase_count == len(synd), except when you input a partial erase_loc and using the full syndrome instead of the Forney syndrome, in which case nsym-erase_count is more correct (len(synd) will fail badly with IndexError).
        if erase_loc: # if an erasures locator polynomial was provided to init the errors locator polynomial, then we must skip the FIRST erase_count iterations (not the last iterations, this is very important!)
            K = erase_count+i+synd_shift
        else: # if erasures locator is not provided, then either there's no erasures to account or we use the Forney syndromes, so we don't need to use erase_count nor erase_loc (the erasures have been trimmed out of the Forney syndromes).
            K = i+synd_shift

        # Compute the discrepancy Delta
        # Here is the close-to-the-books operation to compute the discrepancy Delta: it's a simple polynomial multiplication of error locator with the syndromes, and then we get the Kth element.
        #delta = gf_poly_mul(err_loc[::-1], synd)[K] # theoretically it should be gf_poly_add(synd[::-1], [1])[::-1] instead of just synd, but it seems it's not absolutely necessary to correctly decode.
        # But this can be optimized: since we only need the Kth element, we don't need to compute the polynomial multiplication for any other element but the Kth. Thus to optimize, we compute the polymul only at the item we need, skipping the rest (avoiding a nested loop, thus we are linear time instead of quadratic).
        # This optimization is actually described in several figures of the book "Algebraic codes for data transmission", Blahut, Richard E., 2003, Cambridge university press.
        delta = synd[K]
        for j in xrange(1, len(err_loc)):
            delta ^= gf_mul(err_loc[-(j+1)], synd[K - j]) # delta is also called discrepancy. Here we do a partial polynomial multiplication (ie, we compute the polynomial multiplication only for the term of degree K). Should be equivalent to brownanrs.polynomial.mul_at().
        #print "delta", K, delta, list(gf_poly_mul(err_loc[::-1], synd)) # debugline

        # Shift polynomials to compute the next degree
        old_loc = old_loc + [0]

        # Iteratively estimate the errata locator and evaluator polynomials
        if delta != 0: # Update only if there's a discrepancy
            if len(old_loc) > len(err_loc): # Rule B (rule A is implicitly defined because rule A just says that we skip any modification for this iteration)
            #if 2*L <= K+erase_count: # equivalent to len(old_loc) > len(err_loc), as long as L is correctly computed
                # Computing errata locator polynomial Sigma
                new_loc = gf_poly_scale(old_loc, delta)
                old_loc = gf_poly_scale(err_loc, gf_inverse(delta)) # effectively we are doing err_loc * 1/delta = err_loc // delta
                err_loc = new_loc
                # Update the update flag
                #L = K - L # the update flag L is tricky: in Blahut's schema, it's mandatory to use `L = K - L - erase_count` (and indeed in a previous draft of this function, if you forgot to do `- erase_count` it would lead to correcting only 2*(errors+erasures) <= (n-k) instead of 2*errors+erasures <= (n-k)), but in this latest draft, this will lead to a wrong decoding in some cases where it should correctly decode! Thus you should try with and without `- erase_count` to update L on your own implementation and see which one works OK without producing wrong decoding failures.

            # Update with the discrepancy
            err_loc = gf_poly_add(err_loc, gf_poly_scale(old_loc, delta))

    # Check if the result is correct, that there's not too many errors to correct
    while len(err_loc) and err_loc[0] == 0: del err_loc[0] # drop leading 0s, else errs will not be of the correct size
    errs = len(err_loc) - 1
    if (errs-erase_count) * 2 + erase_count > nsym:
        raise ReedSolomonError("Too many errors to correct")    # too many errors to correct

    return err_loc

def rs_find_errata_locator(e_pos, generator=2):
    '''Compute the erasures/errors/errata locator polynomial from the erasures/errors/errata positions
       (the positions must be relative to the x coefficient, eg: "hello worldxxxxxxxxx" is tampered to "h_ll_ worldxxxxxxxxx"
       with xxxxxxxxx being the ecc of length n-k=9, here the string positions are [1, 4], but the coefficients are reversed
       since the ecc characters are placed as the first coefficients of the polynomial, thus the coefficients of the
       erased characters are n-1 - [1, 4] = [18, 15] = erasures_loc to be specified as an argument.'''

    e_loc = [1] # just to init because we will multiply, so it must be 1 so that the multiplication starts correctly without nulling any term
    # erasures_loc = product(1 - x*alpha**i) for i in erasures_pos and where alpha is the alpha chosen to evaluate polynomials.
    for i in e_pos:
        e_loc = gf_poly_mul( e_loc, gf_poly_add([1], [gf_pow(generator, i), 0]) )
    return e_loc

def rs_find_error_evaluator(synd, err_loc, nsym):
    '''Compute the error (or erasures if you supply sigma=erasures locator polynomial, or errata) evaluator polynomial Omega
       from the syndrome and the error/erasures/errata locator Sigma.'''

    # Omega(x) = [ Synd(x) * Error_loc(x) ] mod x^(n-k+1)
    _, remainder = gf_poly_div( gf_poly_mul(synd, err_loc), ([1] + [0]*(nsym+1)) ) # first multiply syndromes * errata_locator, then do a
                                                                                   # polynomial division to truncate the polynomial to the
                                                                                   # required length

    # Faster way that is equivalent
    #remainder = gf_poly_mul(synd, err_loc) # first multiply the syndromes with the errata locator polynomial
    #remainder = remainder[len(remainder)-(nsym+1):] # then slice the list to truncate it (which represents the polynomial), which
                                                     # is equivalent to dividing by a polynomial of the length we want

    return remainder

def rs_find_errors(err_loc, nmess, generator=2): # nmess is len(msg_in)
    '''Find the roots (ie, where evaluation = zero) of error polynomial by brute-force trial, this is a sort of Chien's search
    (but less efficient, Chien's search is a way to evaluate the polynomial such that each evaluation only takes constant time).'''
    errs = len(err_loc) - 1
    err_pos = []
    for i in xrange(nmess): # normally we should try all 2^8 possible values, but here we optimize to just check the interesting symbols
        if gf_poly_eval(err_loc, gf_pow(generator, i)) == 0: # It's a 0? Bingo, it's a root of the error locator polynomial,
                                                        # in other terms this is the location of an error
            err_pos.append(nmess - 1 - i)
    # Sanity check: the number of errors/errata positions found should be exactly the same as the length of the errata locator polynomial
    if len(err_pos) != errs:
        # couldn't find error locations
        raise ReedSolomonError("Too many (or few) errors found by Chien Search for the errata locator polynomial!")
    return err_pos

def rs_forney_syndromes(synd, pos, nmess, generator=2):
    # Compute Forney syndromes, which computes a modified syndromes to compute only errors (erasures are trimmed out). Do not confuse this with Forney algorithm, which allows to correct the message based on the location of errors.
    erase_pos_reversed = [nmess-1-p for p in pos] # prepare the coefficient degree positions (instead of the erasures positions)

    # Optimized method, all operations are inlined
    fsynd = list(synd[1:])      # make a copy and trim the first coefficient which is always 0 by definition
    for i in xrange(len(pos)):
        x = gf_pow(generator, erase_pos_reversed[i])
        for j in xrange(len(fsynd) - 1):
            fsynd[j] = gf_mul(fsynd[j], x) ^ fsynd[j + 1]
        #fsynd.pop() # useless? it doesn't change the results of computations to leave it there

    # Theoretical way of computing the modified Forney syndromes: fsynd = (erase_loc * synd) % x^(n-k)
    # See Shao, H. M., Truong, T. K., Deutsch, L. J., & Reed, I. S. (1986, April). A single chip VLSI Reed-Solomon decoder. In Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP'86. (Vol. 11, pp. 2151-2154). IEEE.ISO 690
    #erase_loc = rs_find_errata_locator(erase_pos_reversed, generator=generator) # computing the erasures locator polynomial
    #fsynd = gf_poly_mul(erase_loc[::-1], synd[1:]) # then multiply with the syndrome to get the untrimmed forney syndrome
    #fsynd = fsynd[len(pos):] # then trim the first erase_pos coefficients which are useless. Seems to be not necessary, but this reduces the computation time later in BM (thus it's an optimization).

    return fsynd

def rs_correct_msg(msg_in, nsym, fcr=0, generator=2, erase_pos=None, only_erasures=False):
    '''Reed-Solomon main decoding function'''
    global field_charac
    if len(msg_in) > field_charac:
        # Note that it is in fact possible to encode/decode messages that are longer than field_charac, but because this will be above the field, this will generate more error positions during Chien Search than it should, because this will generate duplicate values, which should normally be prevented thank's to the prime polynomial reduction (eg, because it can't discriminate between error at position 1 or 256, both being exactly equal under galois field 2^8). So it's really not advised to do it, but it's possible (but then you're not guaranted to be able to correct any error/erasure on symbols with a position above the length of field_charac -- if you really need a bigger message without chunking, then you should better enlarge c_exp so that you get a bigger field).
        raise ValueError("Message is too long (%i when max is %i)" % (len(msg_in), field_charac))

    msg_out = list(msg_in)     # copy of message
    # erasures: set them to null bytes for easier decoding (but this is not necessary, they will be corrected anyway, but debugging will be easier with null bytes because the error locator polynomial values will only depend on the errors locations, not their values)
    if erase_pos is None:
        erase_pos = []
    else:
        for e_pos in erase_pos:
            msg_out[e_pos] = 0
    # check if there are too many erasures to correct (beyond the Singleton bound)
    if len(erase_pos) > nsym: raise ReedSolomonError("Too many erasures to correct")
    # prepare the syndrome polynomial using only errors (ie: errors = characters that were either replaced by null byte
    # or changed to another character, but we don't know their positions)
    synd = rs_calc_syndromes(msg_out, nsym, fcr, generator)
    # check if there's any error/erasure in the input codeword. If not (all syndromes coefficients are 0), then just return the message as-is.
    if max(synd) == 0:
        return msg_out[:-nsym], msg_out[-nsym:]  # no errors

    # Find errors locations
    if only_erasures:
        err_pos = []
    else:
        # compute the Forney syndromes, which hide the erasures from the original syndrome (so that BM will just have to deal with errors, not erasures)
        fsynd = rs_forney_syndromes(synd, erase_pos, len(msg_out), generator)
        # compute the error locator polynomial using Berlekamp-Massey
        err_loc = rs_find_error_locator(fsynd, nsym, erase_count=len(erase_pos))
        # locate the message errors using Chien search (or bruteforce search)
        err_pos = rs_find_errors(err_loc[::-1], len(msg_out), generator)
        if err_pos is None:
            raise ReedSolomonError("Could not locate error")


    # Find errors values and apply them to correct the message
    # compute errata evaluator and errata magnitude polynomials, then correct errors and erasures
    msg_out = rs_correct_errata(msg_out, synd, (erase_pos + err_pos), fcr, generator) # note that we here use the original syndrome, not the forney syndrome
                                                         # (because we will correct both errors and erasures, so we need the full syndrome)
    # check if the final message is fully repaired
    synd = rs_calc_syndromes(msg_out, nsym, fcr, generator)
    if max(synd) > 0:
        raise ReedSolomonError("Could not correct message")     # message could not be repaired
    # return the successfully decoded message
    return msg_out[:-nsym], msg_out[-nsym:] # also return the corrected ecc block so that the user can check()

def rs_correct_msg_nofsynd(msg_in, nsym, fcr=0, generator=2, erase_pos=None, only_erasures=False):
    '''Reed-Solomon main decoding function, without using the modified Forney syndromes
    This demonstrates how the decoding process is done without using the Forney syndromes
    (this is the most common way nowadays, avoiding Forney syndromes require to use a modified Berlekamp-Massey
    that will take care of the erasures by itself, it's a simple matter of modifying some initialization variables and the loop ranges)'''
    global field_charac
    if len(msg_in) > field_charac:
        raise ValueError("Message is too long (%i when max is %i)" % (len(msg_in), field_charac))

    msg_out = list(msg_in)     # copy of message
    # erasures: set them to null bytes for easier decoding (but this is not necessary, they will be corrected anyway, but debugging will be easier with null bytes because the error locator polynomial values will only depend on the errors locations, not their values)
    if erase_pos is None:
        erase_pos = []
    else:
        for e_pos in erase_pos:
            msg_out[e_pos] = 0
    # check if there are too many erasures
    if len(erase_pos) > nsym: raise ReedSolomonError("Too many erasures to correct")
    # prepare the syndrome polynomial using only errors (ie: errors = characters that were either replaced by null byte or changed to another character, but we don't know their positions)
    synd = rs_calc_syndromes(msg_out, nsym, fcr, generator)
    # check if there's any error/erasure in the input codeword. If not (all syndromes coefficients are 0), then just return the codeword as-is.
    if max(synd) == 0:
        return msg_out[:-nsym], msg_out[-nsym:]  # no errors

    # prepare erasures locator and evaluator polynomials
    erase_loc = None
    #erase_eval = None
    erase_count = 0
    if erase_pos:
        erase_count = len(erase_pos)
        erase_pos_reversed = [len(msg_out)-1-eras for eras in erase_pos]
        erase_loc = rs_find_errata_locator(erase_pos_reversed, generator=generator)
        #erase_eval = rs_find_error_evaluator(synd[::-1], erase_loc, len(erase_loc)-1)

    # prepare errors/errata locator polynomial
    if only_erasures:
        err_loc = erase_loc[::-1]
        #err_eval = erase_eval[::-1]
    else:
        err_loc = rs_find_error_locator(synd, nsym, erase_loc=erase_loc, erase_count=erase_count)
        err_loc = err_loc[::-1]
        #err_eval = rs_find_error_evaluator(synd[::-1], err_loc[::-1], len(err_loc)-1)[::-1] # find error/errata evaluator polynomial (not really necessary since we already compute it at the same time as the error locator poly in BM)

    # locate the message errors
    err_pos = rs_find_errors(err_loc, len(msg_out), generator) # find the roots of the errata locator polynomial (ie: the positions of the errors/errata)
    if err_pos is None:
        raise ReedSolomonError("Could not locate error")

    # compute errata evaluator and errata magnitude polynomials, then correct errors and erasures
    msg_out = rs_correct_errata(msg_out, synd, err_pos, fcr=fcr, generator=generator)
    # check if the final message is fully repaired
    synd = rs_calc_syndromes(msg_out, nsym, fcr, generator)
    if max(synd) > 0:
        raise ReedSolomonError("Could not correct message")
    # return the successfully decoded message
    return msg_out[:-nsym], msg_out[-nsym:] # also return the corrected ecc block so that the user can check()

def rs_check(msg, nsym, fcr=0, generator=2):
    '''Returns true if the message + ecc has no error of false otherwise (may not always catch a wrong decoding or a wrong message, particularly if there are too many errors -- above the Singleton bound --, but it usually does)'''
    return ( max(rs_calc_syndromes(msg, nsym, fcr, generator)) == 0 )

def sample(message):
    # Configuration of the parameters and input message
    prim = 0x11d
    n = 20 # set the size you want, it must be > k, the remaining n-k symbols will be the ECC code (more is better)
    k = 11 # k = len(message)
    #message = "hello world" # input message

    # Initializing the log/antilog tables
    init_tables(prim)

    # Encoding the input message
    mesecc = rs_encode_msg([ord(x) for x in message], n-k)
    print("Original: %s" % mesecc)

    # Tampering 6 characters of the message (over 9 ecc symbols, so we are above the Singleton Bound)
    mesecc[0] = 0
    mesecc[1] = 2
    mesecc[2] = 2
    mesecc[3] = 2
    mesecc[4] = 2
    mesecc[5] = 2
    print("Corrupted: %s" % mesecc)

    # Decoding/repairing the corrupted message, by providing the locations of a few erasures, we get below the Singleton Bound
    # Remember that the Singleton Bound is: 2*e+v <= (n-k)
    corrected_message, corrected_ecc = rs_correct_msg(mesecc, n-k, erase_pos=[0, 1, 2])
    print("Repaired: %s" % (corrected_message+corrected_ecc))
    print(''.join([chr(x) for x in corrected_message]))

N_MAX = 255

def test():

    prim = 0x11d
    n = 20
    k = 2
    message = DARTH_PLAGUEIS_SCRIPT
    message_list = []

    init_tables(prim)

    for i in range(0, len(message), k):
        message_list.append(message[i : i + k])
    
    for msg in message_list:
      
        print("String: <%s>, Length = %d" % (msg, len(msg)))
        mesecc = rs_encode_msg([ord(x) for x in msg], n-k)
        print("\tEncoded String, Length = %d" % len(mesecc))
        for s in mesecc:
            print("<| (%c), 0x%02X |>" % (s, s))
    print("NUMBER OF ECC SYMBOLS = %d" % (n - k))

if __name__ == '__main__':

    sample("Te saluto. Augustus sum, imperator et pontifex maximus romae. Si tu es Romae amicus, es gratus.")
    #test()