dihedralcheck

Implementation of checksum algorithms based on the dihedral group for the serial numbers of the Deutsche Mark (DM).

How it works

Lets check if the serial number DS5885114Y8 is valid:

1. Substitute all letters with numbers using following table:

   A	D	G	K	L	N	S	U	Y	Z
   0	1	2	3	4	5	6	7	8	9

   Now we receive 16588511488.

2. Now permute all digit using following permutation (given as table). Row corresponds to index of the digit and columns to digit itself.

   These permutations can be generated by $p = (1 ~ 5 ~ 7 ~ 6 ~ 2 ~ 8 ~ 3 ~ 0 ~ 9 ~ 4)$.

   Having $p_i = p^i$ with exception of $p_{11} = \text{id}$.

   	0	1	2	3	4	5	6	7	8	9
   p1	1	5	7	6	2	8	3	0	9	4
   p2	5	8	0	3	7	9	6	1	4	2
   p3	8	9	1	6	0	4	3	5	2	7
   p4	9	4	5	3	1	2	6	8	7	0
   p5	4	2	8	6	5	7	3	9	0	1
   p6	2	7	9	3	8	0	5	4	1	5
   p7	7	0	4	6	9	1	3	2	5	8
   p8	0	1	2	3	4	5	6	7	8	9
   p9	1	5	7	6	2	8	3	0	9	4
   p10	5	8	0	3	7	9	6	1	4	2
   p11	0	1	2	3	4	5	6	7	8	9

   Now we receive 56470001248

3. Let's simplify this expression by treating all of its digits as group elements of the dihedral group $D_5$. Here we got the Cayley Table of $D_5$.

   	0	1	2	3	4	5	6	7	8	9
   0	0	1	2	3	4	5	6	7	8	9
   1	1	2	3	4	0	6	7	8	9	5
   2	2	3	4	0	1	7	8	9	5	6
   3	3	4	0	1	2	8	9	5	6	7
   4	4	0	1	2	3	9	5	6	7	8
   5	5	9	8	7	6	0	4	3	2	1
   6	6	5	9	8	7	1	0	4	3	2
   7	7	6	5	9	8	2	1	0	4	3
   8	8	7	6	5	9	3	2	1	0	4
   9	9	8	7	6	5	4	3	2	1	0

   We can evaluate the digits as following using the law of associativity:

   $(5 6)(4 7)(0 0)(0 1)(2 4)8 = (46)(01)(18) = (51)9 = (99) = 0$

4. If the result of your evaluation is the neutral element 0 the serial number is valid, else invalid. So our example is a valid serial number.