To find the multiplicative inverse of 7 (mod 26) we can use Euclid's algorithm to write the number 1
in terms of a multiple of 26 plus a multiple of 7.
Then working backwards we can write 1 as a multiple of 26 plus a multiple of 7.
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= 5−2×2 = 5−2(7−5) = 3×5 − 2×7 |
| |
| =3(26−3×7)−2×7 = 3×26 − 11×7 |
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Hence −11×7 ≡ 15×7 ≡ 1 mod 26 showing that 15 is the multiplicative
inverse of 7 (mod 26).
The additive inverse of 17 is 9 because 17+9 ≡ 0 mod 26
This gives the decoding formula as P=15(C+9) so the numbers 0 to 25 decode as follows:
xxx
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
|
C
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
|
P
5 |
20 |
9 |
24 |
13 |
2 |
17 |
6 |
21 |
10 |
25 |
14 |
3 |
18 |
7 |
22 |
11 |
0 |
15 |
4 |
19 |
8 |
23 |
12 |
1 |
16 |
f |
u |
j |
y |
n |
c |
r |
g |
v |
k |
z |
o |
d |
s |
h |
w |
l |
a |
p |
e |
t |
i |
x |
m |
b |
q |
20 14 19 23 11 13 20 21 4 5 11 23 18 6 19 14 19 4 13 21 24 16 19 20 14 21 4 7 17 24 11 1 20 20 14 19 15 11 6 16 12 21 13 20 14 17 20 21 20 21 13 5 11 23 18 6 19 14 19 4 13 21 24 16 19
The message is a comment by Einstein:
'The most incomprehensible thing about the world is that it is comprehensible'.
The letters of the alphabet are encoded as follows
xxx
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
xxx |
|
a |
b |
c |
d |
e |
f |
g |
h |
i |
j |
k |
l |
m |
n |
o |
p |
q |
r |
s |
t |
u |
v |
w |
x |
y |
z |
P
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 |
C
17 |
24 |
5 |
12 |
19 |
0 |
7 |
14 |
21 |
2 |
9 |
16 |
23 |
4 |
11 |
18 |
25 |
6 |
13 |
20 |
1 |
8 |
15 |
22 |
3 |
10 |