我想使用Damm algorithm为32个字符的字母代码生成校验位。算法本身很容易应用于任何基础（2或6除外）。难点是必要的查找表，它必须是一个完全反对称的拟群，在主对角线下面有一个字符（通常为0）。
Wikipedia page为基数10提供了一个表格，Python implementation给出了table for base-16，但我还没有找到基数为32的示例。有没有人有一个适合base-32的表？
答案 0 :(得分：11)
受David Eisenstat的回答（以及他引用的original dissertation by Damm的启发），这里有一个简单的Python例程来计算/验证任何基数2的
# reduction bitmasks for GF(2^n), 2 <= n <= 32
masks = (3, 3, 3, 5, 3, 3, 27, 3, 9, 5, 9, 27, 33, 3, 43, 9,
9, 39, 9, 5, 3, 33, 27, 9, 27, 39, 3, 5, 3, 9, 141)
# calculate Damm checksum for base 2^n
def damm2n (digits, n):
modulus = (1 << n)
mask = modulus | masks[n - 2]
checksum = 0
for digit in digits:
checksum ^= digit
checksum <<= 1
if checksum >= modulus: checksum ^= mask
return checksum
该例程采用一个列表（或更一般地，一个可迭代的）数字，假定为0到2 n -1范围内的整数，以及每位数的位数 n （假设在2到32的范围内）。
Damm算法的这种实现使用的完全不对称的拟群由地图给出（ a ， b ）↦ 2 ⊗（ a ⊕ b ），其中⊕表示finite field GF中的加法（2 n ）（这只是按位XOR），⊗表示multiplication in GF(2n)， 2 表示通常中由位串0 ... 010 2 表示的元素n -bit表示GF（2 n ）。
这相当于地图（ a ， b ）↦（ 2 ⊗ a ）⊕<在他的论文的例子5.2中由Damm给出的em> b ，除了输入数字 b 置换（通过将它们与GF中的2相乘（2 n < / em> ））确保所有 a 的（ a ， a ）↦0。这相当于置换拟群操作表的列，使对角线全为零，并允许仅通过将校验和附加到原始输入并检查扩展输入的新校验和为零来验证校验和。
GF（2 n ）乘以 2 是使用通常的左移一个技巧实现的，如果 n 结果的第n位置位，使用对应于 n 的monic不可约多项式的位掩码对其进行异或。使用的特定位掩码取自Table of Low-Weight Binary Irreducible Polynomials by Gadiel Seroussi (1998)。如果您（由于某种原因）需要大于2 32 的基数的校验和，则他们的表达到高达2 10,000 。 Seroussi表列出了每个归约多项式的非零系数的指数，不包括常数项;上面代码中的位掩码是通过丢弃最高指数（总是 n ）获得的，将其他指数的2 k 加在一起< em> k 并添加1.（因此，例如，条目＆＃34; 8,4,3,1和＃34;对于 n = 8，得到掩码2 4 + 2 3 + 2 1 + 1 = 16 + 8 + 2 + 1 = 27。）
特别是，上面的代码为 n = 4产生了与Johannes Spielmann匹配base-16 Damm checksum implementation的结果。这一点并不能保证，因为有很多可能的方法可以为给定的基础实现Damm校验和，但在这种情况下，两个实现使用的拟群恰好匹配。
Ps。这里有一些Python代码以类似于维基百科文章使用的格式打印出查找表。 （感谢CJM的初始版本。）
alphabet = '0123456789ABCDEFGHJKLMNPQRTUVWXY' # avoids easy-to-confuse characters
bits = 5
# find out which first single-digit input gives which checksum
r = [-1] * 2**bits
for i in range(2**bits): r[damm2n([i], bits)] = i
# print header
print ' |', ' '.join(alphabet)
print '--+' + '--' * len(alphabet)
# print rest of table
for i in range(2**bits):
row = (alphabet[damm2n([r[i], j], bits)] for j in range(2**bits))
print alphabet[i], '|', ' '.join(row)
答案 1 :(得分：4)
总结下面的讨论：我们想要一个在主对角线上有零的表。尼克拉斯和我的印象是，这个属性不仅仅是算法的一个重要部分，而是仅仅是为了避免在给定x的y中求解方程x * y = 0，其中*是拟群运算。在主对角线上使用零，我们有x = y，但是没有，我们可以通过32个元素表中的一次查找来计算y。
Damm描述的结构是有问题的，因为当且仅当a = -1时它具有所需的属性，但是，在特征2中，我们有1 = -1。 约束求解器Z3无效。
以下是n = 32的此方法的实例化。
[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31],
[2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 18, 19, 16, 17, 22, 23, 20, 21, 26, 27, 24, 25, 30, 31, 28, 29],
[4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11, 20, 21, 22, 23, 16, 17, 18, 19, 28, 29, 30, 31, 24, 25, 26, 27],
[6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9, 22, 23, 20, 21, 18, 19, 16, 17, 30, 31, 28, 29, 26, 27, 24, 25],
[8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23],
[10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21],
[12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3, 28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 16, 17, 18, 19],
[14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1, 30, 31, 28, 29, 26, 27, 24, 25, 22, 23, 20, 21, 18, 19, 16, 17],
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15],
[18, 19, 16, 17, 22, 23, 20, 21, 26, 27, 24, 25, 30, 31, 28, 29, 2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13],
[20, 21, 22, 23, 16, 17, 18, 19, 28, 29, 30, 31, 24, 25, 26, 27, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 8, 9, 10, 11],
[22, 23, 20, 21, 18, 19, 16, 17, 30, 31, 28, 29, 26, 27, 24, 25, 6, 7, 4, 5, 2, 3, 0, 1, 14, 15, 12, 13, 10, 11, 8, 9],
[24, 25, 26, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7],
[26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5],
[28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 16, 17, 18, 19, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 0, 1, 2, 3],
[30, 31, 28, 29, 26, 27, 24, 25, 22, 23, 20, 21, 18, 19, 16, 17, 14, 15, 12, 13, 10, 11, 8, 9, 6, 7, 4, 5, 2, 3, 0, 1],
[5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10, 21, 20, 23, 22, 17, 16, 19, 18, 29, 28, 31, 30, 25, 24, 27, 26],
[7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24],
[1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30],
[3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12, 19, 18, 17, 16, 23, 22, 21, 20, 27, 26, 25, 24, 31, 30, 29, 28],
[13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2, 29, 28, 31, 30, 25, 24, 27, 26, 21, 20, 23, 22, 17, 16, 19, 18],
[15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16],
[9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6, 25, 24, 27, 26, 29, 28, 31, 30, 17, 16, 19, 18, 21, 20, 23, 22],
[11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4, 27, 26, 25, 24, 31, 30, 29, 28, 19, 18, 17, 16, 23, 22, 21, 20],
[21, 20, 23, 22, 17, 16, 19, 18, 29, 28, 31, 30, 25, 24, 27, 26, 5, 4, 7, 6, 1, 0, 3, 2, 13, 12, 15, 14, 9, 8, 11, 10],
[23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8],
[17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14],
[19, 18, 17, 16, 23, 22, 21, 20, 27, 26, 25, 24, 31, 30, 29, 28, 3, 2, 1, 0, 7, 6, 5, 4, 11, 10, 9, 8, 15, 14, 13, 12],
[29, 28, 31, 30, 25, 24, 27, 26, 21, 20, 23, 22, 17, 16, 19, 18, 13, 12, 15, 14, 9, 8, 11, 10, 5, 4, 7, 6, 1, 0, 3, 2],
[31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0],
[25, 24, 27, 26, 29, 28, 31, 30, 17, 16, 19, 18, 21, 20, 23, 22, 9, 8, 11, 10, 13, 12, 15, 14, 1, 0, 3, 2, 5, 4, 7, 6],
[27, 26, 25, 24, 31, 30, 29, 28, 19, 18, 17, 16, 23, 22, 21, 20, 11, 10, 9, 8, 15, 14, 13, 12, 3, 2, 1, 0, 7, 6, 5, 4]]
下面是非常糟糕的Haskell代码。它可能适用于其他两种力量。 main中的参数5是2 ^ 5。
module Main where
import Data.Char
import Data.List
xs +. ys = simplify (xs ++ ys)
xs *. ys
= simplify $
do x <- xs
y <- ys
return (x + y)
simplify xs
= (reverse . map head . filter (odd . length) . group . sort) xs
subseqs [] = [[]]
subseqs (x : xs) = let xss = subseqs xs in xss ++ map (x :) xss
polys n = subseqs [n, n - 1 .. 0]
reduce [] ys = ys
reduce xs [] = []
reduce xs@(x : _) ys@(y : _)
= if x > y then ys else reduce xs (map ((y - x) +) xs +. ys)
irred [] = False
irred ys@(y : _)
= let xss = polys (y `div` 2) \\ [[0]] in
(not . any null . map (flip reduce ys)) xss
irreds n = filter irred (polys n)
ip n = (head . filter irred . map (n :)) (polys (n - 1))
eval xs = (sum . map (2 ^)) xs
timesTable n
= let ms = ip n
zs = polys (n - 1) !! 2
in
do xs <- polys (n - 1)
return $
do ys <- polys (n - 1)
return (reduce ms ((zs *. xs) +. ys))
verify t
= all ((1 ==) . length . filter id) $
zipWith (zipWith (==)) t (transpose t)
main = print $ map (map eval) $ timesTable 5
答案 2 :(得分：2)
如果你有一个TA拟群，你可以简单地重新排列列，使得0在主对角线上。然后，拟群（通常）是WTA拟群，可以用于Damm算法。我已经为订单32执行了此操作，请参阅下面的结果，除了2和6之外，每个订单都可以这样做。
我认为维基百科中可以找到10阶的拟群，是由Damm论文的引理5.2构建的。这是因为在重新排列列后它仍应检测语音错误，因此需要重命名元素，并且需要相应地重新排列行。
最后，这是Damm算法的32阶WTA准群：
00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 03 01 07 05 11 09 15 13 19 17 23 21 27 25 31 29
02 00 06 04 10 08 14 12 18 16 22 20 26 24 30 28 01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31
04 06 00 02 12 14 08 10 20 22 16 18 28 30 24 26 07 05 03 01 15 13 11 09 23 21 19 17 31 29 27 25
06 04 02 00 14 12 10 08 22 20 18 16 30 28 26 24 05 07 01 03 13 15 09 11 21 23 17 19 29 31 25 27
08 10 12 14 00 02 04 06 24 26 28 30 16 18 20 22 11 09 15 13 03 01 07 05 27 25 31 29 19 17 23 21
10 08 14 12 02 00 06 04 26 24 30 28 18 16 22 20 09 11 13 15 01 03 05 07 25 27 29 31 17 19 21 23
12 14 08 10 04 06 00 02 28 30 24 26 20 22 16 18 15 13 11 09 07 05 03 01 31 29 27 25 23 21 19 17
14 12 10 08 06 04 02 00 30 28 26 24 22 20 18 16 13 15 09 11 05 07 01 03 29 31 25 27 21 23 17 19
16 18 20 22 24 26 28 30 00 02 04 06 08 10 12 14 19 17 23 21 27 25 31 29 03 01 07 05 11 09 15 13
18 16 22 20 26 24 30 28 02 00 06 04 10 08 14 12 17 19 21 23 25 27 29 31 01 03 05 07 09 11 13 15
20 22 16 18 28 30 24 26 04 06 00 02 12 14 08 10 23 21 19 17 31 29 27 25 07 05 03 01 15 13 11 09
22 20 18 16 30 28 26 24 06 04 02 00 14 12 10 08 21 23 17 19 29 31 25 27 05 07 01 03 13 15 09 11
24 26 28 30 16 18 20 22 08 10 12 14 00 02 04 06 27 25 31 29 19 17 23 21 11 09 15 13 03 01 07 05
26 24 30 28 18 16 22 20 10 08 14 12 02 00 06 04 25 27 29 31 17 19 21 23 09 11 13 15 01 03 05 07
28 30 24 26 20 22 16 18 12 14 08 10 04 06 00 02 31 29 27 25 23 21 19 17 15 13 11 09 07 05 03 01
30 28 26 24 22 20 18 16 14 12 10 08 06 04 02 00 29 31 25 27 21 23 17 19 13 15 09 11 05 07 01 03
03 01 07 05 11 09 15 13 19 17 23 21 27 25 31 29 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30
01 03 05 07 09 11 13 15 17 19 21 23 25 27 29 31 02 00 06 04 10 08 14 12 18 16 22 20 26 24 30 28
07 05 03 01 15 13 11 09 23 21 19 17 31 29 27 25 04 06 00 02 12 14 08 10 20 22 16 18 28 30 24 26
05 07 01 03 13 15 09 11 21 23 17 19 29 31 25 27 06 04 02 00 14 12 10 08 22 20 18 16 30 28 26 24
11 09 15 13 03 01 07 05 27 25 31 29 19 17 23 21 08 10 12 14 00 02 04 06 24 26 28 30 16 18 20 22
09 11 13 15 01 03 05 07 25 27 29 31 17 19 21 23 10 08 14 12 02 00 06 04 26 24 30 28 18 16 22 20
15 13 11 09 07 05 03 01 31 29 27 25 23 21 19 17 12 14 08 10 04 06 00 02 28 30 24 26 20 22 16 18
13 15 09 11 05 07 01 03 29 31 25 27 21 23 17 19 14 12 10 08 06 04 02 00 30 28 26 24 22 20 18 16
19 17 23 21 27 25 31 29 03 01 07 05 11 09 15 13 16 18 20 22 24 26 28 30 00 02 04 06 08 10 12 14
17 19 21 23 25 27 29 31 01 03 05 07 09 11 13 15 18 16 22 20 26 24 30 28 02 00 06 04 10 08 14 12
23 21 19 17 31 29 27 25 07 05 03 01 15 13 11 09 20 22 16 18 28 30 24 26 04 06 00 02 12 14 08 10
21 23 17 19 29 31 25 27 05 07 01 03 13 15 09 11 22 20 18 16 30 28 26 24 06 04 02 00 14 12 10 08
27 25 31 29 19 17 23 21 11 09 15 13 03 01 07 05 24 26 28 30 16 18 20 22 08 10 12 14 00 02 04 06
25 27 29 31 17 19 21 23 09 11 13 15 01 03 05 07 26 24 30 28 18 16 22 20 10 08 14 12 02 00 06 04
31 29 27 25 23 21 19 17 15 13 11 09 07 05 03 01 28 30 24 26 20 22 16 18 12 14 08 10 04 06 00 02
29 31 25 27 21 23 17 19 13 15 09 11 05 07 01 03 30 28 26 24 22 20 18 16 14 12 10 08 06 04 02 00