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.

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{26}& =3\times 7+5\\ \multicolumn{1}{c}{7}& =1\times 5+2\\ \multicolumn{1}{c}{5}& =2\times 2+1\end{array}}$

Then working backwards we can write 1 as a multiple of 26 plus a multiple of 7.

${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{1}& =5-2\times 2=5-2(7-5)=3\times 5-2\times 7\\ \multicolumn{1}{c}{}& =3(26-3\times 7)-2\times 7=3\times 26-11\times 7\end{array}}$

Hence $-11\times 7\equiv 15\times 7\equiv 1mod26$ showing that 15 is the multiplicative inverse of 7 (mod 26). The additive inverse of 17 is 9 because $17+9\equiv 0mod26$

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