问题描述
最近我一直在从事QR码项目，并且一直在进行多项式操作以进行纠错
说明和示例如下所示：
尽管这些指示可以在纸上完成，但我很难编程。
为了解决这个问题，我将Sympy库用于多项式操作，但是遇到了一些问题，
目前，我的代码使我进入了需要使用a变量并将其转换为galios字段的步骤，但是，我很难抓取多项式的某些部分并抓取LC和/或进行指数加法运算。我将在下面链接我的代码，将不胜感激。
import sympy as s
a,x = s.symbols('a,x')
def polynomial_generator(value,steps,step):
step += 1
step_polynomial = (x + a ** step)
# Multiply Terms out & Combined Terms
value = value * step_polynomial
# GF Addition -Unsure how to grab the Exponent/LC
# XOR
# Result
if step == steps - 1:
return value
else:
polynomial_generator(value,step)
ec_generation = 2
first_value = (x + a**0).as_poly()
polynomial = polynomial_generator(first_value,ec_generation,0)
# https://www.thonky.com/qr-code-tutorial/error-correction-coding
我的代码将返回以下多项式类型：
poly(x**2 + (a + 1)*x + a,x,domain='ZZ[a]')
我也对阻止Sympy仅仅出于问题的性质以及使用alpha表示法处理对数和XOR转换而阻止将a^0转移到1的方式
解决方法
我不知道如何保留a^0，因为poly似乎总是简化了表达式。我什至尝试使用自Symbol继承但没有用的自定义类。对于这种高度数值的计算，SymPy似乎也是一个不错的选择。当您可以简单地创建一个将2个代表多项式的列表并相乘的函数时，就不需要计算机代数系统。
这就是为我工作的东西，但是我敢肯定，只要自己进行硬编码，它就会更快，更灵活。
from sympy import *
alpha_to_int = {0: 1,1: 2,2: 4,3: 8,4: 16,5: 32,6: 64,7: 128,8: 29,9: 58,10: 116,11: 232,12: 205,13: 135,14: 19,15: 38,16: 76,17: 152,18: 45,19: 90,20: 180,21: 117,22: 234,23: 201,24: 143,25: 3,26: 6,27: 12,28: 24,29: 48,30: 96,31: 192,32: 157,33: 39,34: 78,35: 156,36: 37,37: 74,38: 148,39: 53,40: 106,41: 212,42: 181,43: 119,44: 238,45: 193,46: 159,47: 35,48: 70,49: 140,50: 5,51: 10,52: 20,53: 40,54: 80,55: 160,56: 93,57: 186,58: 105,59: 210,60: 185,61: 111,62: 222,63: 161,64: 95,65: 190,66: 97,67: 194,68: 153,69: 47,70: 94,71: 188,72: 101,73: 202,74: 137,75: 15,76: 30,77: 60,78: 120,79: 240,80: 253,81: 231,82: 211,83: 187,84: 107,85: 214,86: 177,87: 127,88: 254,89: 225,90: 223,91: 163,92: 91,93: 182,94: 113,95: 226,96: 217,97: 175,98: 67,99: 134,100: 17,101: 34,102: 68,103: 136,104: 13,105: 26,106: 52,107: 104,108: 208,109: 189,110: 103,111: 206,112: 129,113: 31,114: 62,115: 124,116: 248,117: 237,118: 199,119: 147,120: 59,121: 118,122: 236,123: 197,124: 151,125: 51,126: 102,127: 204,128: 133,129: 23,130: 46,131: 92,132: 184,133: 109,134: 218,135: 169,136: 79,137: 158,138: 33,139: 66,140: 132,141: 21,142: 42,143: 84,144: 168,145: 77,146: 154,147: 41,148: 82,149: 164,150: 85,151: 170,152: 73,153: 146,154: 57,155: 114,156: 228,157: 213,158: 183,159: 115,160: 230,161: 209,162: 191,163: 99,164: 198,165: 145,166: 63,167: 126,168: 252,169: 229,170: 215,171: 179,172: 123,173: 246,174: 241,175: 255,176: 227,177: 219,178: 171,179: 75,180: 150,181: 49,182: 98,183: 196,184: 149,185: 55,186: 110,187: 220,188: 165,189: 87,190: 174,191: 65,192: 130,193: 25,194: 50,195: 100,196: 200,197: 141,198: 7,199: 14,200: 28,201: 56,202: 112,203: 224,204: 221,205: 167,206: 83,207: 166,208: 81,209: 162,210: 89,211: 178,212: 121,213: 242,214: 249,215: 239,216: 195,217: 155,218: 43,219: 86,220: 172,221: 69,222: 138,223: 9,224: 18,225: 36,226: 72,227: 144,228: 61,229: 122,230: 244,231: 245,232: 247,233: 243,234: 251,235: 235,236: 203,237: 139,238: 11,239: 22,240: 44,241: 88,242: 176,243: 125,244: 250,245: 233,246: 207,247: 131,248: 27,249: 54,250: 108,251: 216,252: 173,253: 71,254: 142,255: 1}
int_to_alpha = {1: 0,2: 1,3: 25,4: 2,5: 50,6: 26,7: 198,8: 3,9: 223,10: 51,11: 238,12: 27,13: 104,14: 199,15: 75,16: 4,17: 100,18: 224,19: 14,20: 52,21: 141,22: 239,23: 129,24: 28,25: 193,26: 105,27: 248,28: 200,29: 8,30: 76,31: 113,32: 5,33: 138,34: 101,35: 47,36: 225,37: 36,38: 15,39: 33,40: 53,41: 147,42: 142,43: 218,44: 240,45: 18,46: 130,47: 69,48: 29,49: 181,50: 194,51: 125,52: 106,53: 39,54: 249,55: 185,56: 201,57: 154,58: 9,59: 120,60: 77,61: 228,62: 114,63: 166,64: 6,65: 191,66: 139,67: 98,68: 102,69: 221,70: 48,71: 253,72: 226,73: 152,74: 37,75: 179,76: 16,77: 145,78: 34,79: 136,80: 54,81: 208,82: 148,83: 206,84: 143,85: 150,86: 219,87: 189,88: 241,89: 210,90: 19,91: 92,92: 131,93: 56,94: 70,95: 64,96: 30,97: 66,98: 182,99: 163,100: 195,101: 72,102: 126,103: 110,104: 107,105: 58,106: 40,107: 84,108: 250,109: 133,110: 186,111: 61,112: 202,113: 94,114: 155,115: 159,116: 10,117: 21,118: 121,119: 43,120: 78,121: 212,122: 229,123: 172,124: 115,125: 243,126: 167,127: 87,128: 7,129: 112,130: 192,131: 247,132: 140,133: 128,134: 99,135: 13,136: 103,137: 74,138: 222,139: 237,140: 49,141: 197,142: 254,143: 24,144: 227,145: 165,146: 153,147: 119,148: 38,149: 184,150: 180,151: 124,152: 17,153: 68,154: 146,155: 217,156: 35,157: 32,158: 137,159: 46,160: 55,161: 63,162: 209,163: 91,164: 149,165: 188,166: 207,167: 205,168: 144,169: 135,170: 151,171: 178,172: 220,173: 252,174: 190,175: 97,176: 242,177: 86,178: 211,179: 171,180: 20,181: 42,182: 93,183: 158,184: 132,185: 60,186: 57,187: 83,188: 71,189: 109,190: 65,191: 162,192: 31,193: 45,194: 67,195: 216,196: 183,197: 123,198: 164,199: 118,200: 196,201: 23,202: 73,203: 236,204: 127,205: 12,206: 111,207: 246,208: 108,209: 161,210: 59,211: 82,212: 41,213: 157,214: 85,215: 170,216: 251,217: 96,218: 134,219: 177,220: 187,221: 204,222: 62,223: 90,224: 203,225: 89,226: 95,227: 176,228: 156,229: 169,230: 160,231: 81,232: 11,233: 245,234: 22,236: 122,237: 117,238: 44,239: 215,240: 79,241: 174,242: 213,243: 233,244: 230,245: 231,246: 173,247: 232,248: 116,249: 214,250: 244,251: 234,252: 168,253: 80,254: 88,255: 175}
# out of curiosity,why doesn't 0 map to 255 in int_to_alpha?
# Even then it wouldn't be a bijection. Very weird that both 0 and 255 map to 1 in int_to_alpha
# Xor in sympy is for logic not for integers
def xor_numeric(a,b):
return S(int(a) ^ int(b))
xor_symbolic = Function('xor')
a,x = symbols('a x')
def polynomial_generator(value,steps,step):
step += 1
step_polynomial = (x + a ** step)
# Multiply Terms out & Combined Terms
value: Poly = value * step_polynomial
# GF Addition -Unsure how to grab the Exponent/LC
coeffs = value.coeffs()
# change + to XOR before substitution otherwise sympy will simplify the addition
coeffs = [coeff.replace(Add,xor_symbolic) for coeff in coeffs]
# convert a**k to an integer
coeffs = [coeff.xreplace({a ** k: v for k,v in alpha_to_int.items()})
if isinstance(coeff,Add) else coeff.replace(a,alpha_to_int[1])
for coeff in coeffs]
# evaluate the expression with the numeric xor
coeffs = [coeff.replace(xor_symbolic,xor_numeric) for coeff in coeffs]
# convert the integers back to a**k
coeffs = [coeff.xreplace({k: Pow(a,k,evaluate=False) for k,v in int_to_alpha.items()})
for coeff in coeffs]
# Result
value = Poly(coeffs,x)
if step == steps - 1:
return value
else:
polynomial_generator(value,step)
ec_generation = 2
first_value = (x + a ** 0).as_poly(x) # poly in x
polynomial = polynomial_generator(first_value,ec_generation,0)
print(polynomial)
哪个给Poly(x**2 + a**25*x + a,x,domain='ZZ[a]')。
您可以使用一次性进行以上所有操作
values = [(x + a ** step) for step in range(ec_generation)]
polynomial = prod(values).as_poly(x,auto=False)
，然后在其下方插入coeffs = ...行。再次，从头开始编码多项式乘法可能会导致更有效的代码。