I have encoded a message, but I forgot how I encoded it! I drew a diagram, but I'm not sure I understand it...
Here's the message:
170, 358, 247, 137, 289, 175, 333, 233, 116, 263, 166, 305, 200, 344, 234, 119
And here's the diagram:
Hint:
Think: hexadecimal...
The message is
Congradulations!
(Note the 'd' instead of 't' seems intentional, perhaps a pun!)
How it works:
The first number given is 170, which is 'AA' in hexadecimal.
Now we already have an 'AA' given in the chart, so this is our starting point! This also means this is a rolling decryption.
So, following the chart, if the square represents '$C_i$', our current character number in ASCII, and the blue 'c' represents '$C_{i-1}$', our previous number, then we can see we we have $C_i + (FF-C_{i-1})$ in the middle.
'AA' is fed in at the start to give the first value, and then everything is fed into $\text{mod}\ FF$ at the bottom. Rearranging and inverting to reverse the process, we can find the characters using the formula:
$$FF - ( (C_i-(C_{i-1}\ \text{mod}\ FF))\ \text{mod}\ FF)$$ We can convert all of this to base 10 by noting 'FF' is 255, so we are working in mod 255.
So lets decrypt!
Starting with 170, our next number is 358:
1. $255 - (358-(170\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 188 = 67$
2. $255 - (247-(358\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 144 = 111$
3. $255 - (137-(247\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 145 = 110$
4. $255 - (289-(137\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 152 = 103$
5. $255 - (175-(289\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 141 = 114$
6. $255 - (333-(175\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 158 = 97$
7. $255 - (233-(333\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 155 = 100$
8. $255 - (116-(233\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 138 = 117$
9. $255 - (263-(116\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 147 = 108$
10. $255 - (166-(263\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 158 = 97$
11. $255 - (305-(166\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 139 = 116$
12. $255 - (200-(305\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 150 = 105$
13. $255 - (344-(200\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 144 = 111$
14. $255 - (234-(344\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 145 = 110$
15. $255 - (119-(234\ \text{mod}\ 255)\ \text{mod}\ 255) = 255 - 140 = 115$
So now we have the new string of numbers
$67,111,110,103,114,97,100,117,108,97,116,105,111,110,115$
Which we can decode
Back into text via ASCII to get the final answer:
'Congradulations'!
Some thoughts on how to interpret the diagram:
- With the hint, it is evident that that
AAandFFrepresent hexadecimal numbers 0xAA = 170 and 0xFF = 255 respectively.- The cyclic nature of the diagram suggests that there is a loop of some sort, and it seems plausible to assume that
AAis the initial seed value used for this loop.
With the above assumptions in mind, one can read off the encoding operations as:
$$\boxed{\begin{aligned} d_0 &= \texttt{0xAA} \\ d_i &= d_{i - 1} + (\texttt{0xFF} - c_i) \\ e_i &= c_i + (d_i \bmod \texttt{0xFF}) \end{aligned}}$$ where
- $c_i$ is the original, un-encoded character,
- $d_i$ is the loop state, carried from one iteration to the next, and
- $e_i$ is the encoded character.
With the encoding algorithm documented, the next step is to figure out how to reverse it into a decoding algorithm:
A crucial observation is that $e_i$ contains both a $c_i$ term and a $(\cdots - c_i) \bmod \mathtt{0xFF}$ term: $$\begin{aligned} e_i &= c_i + (d_i \bmod \texttt{0xFF}) \\ &= c_i + ((d_{i - 1} + \texttt{0xFF} - c_i) \bmod \texttt{0xFF}) \end{aligned}$$ By applying $\bmod \mathtt{0xFF}$ to both sides, the $c_i$ term can be eliminated using the properties of modulo: $$\begin{aligned} e_i \bmod \mathtt{0xFF} &= (c_i + ((d_{i - 1} + \texttt{0xFF} - c_i) \bmod \texttt{0xFF})) \bmod \mathtt{0xFF} \\ &= (c_i + d_{i - 1} + \texttt{0xFF} - c_i) \bmod \mathtt{0xFF} \\ &= d_{i - 1} \bmod \mathtt{0xFF} \end{aligned}$$ Substituting $i \mapsto i + 1$: $$e_{i + 1} \bmod \mathtt{0xFF} = d_i \bmod \mathtt{0xFF}$$ This result can then be plugged back into the definition of $e_i$ and solved for $c_i$: $$\begin{aligned} e_i &= c_i + (d_i \bmod \texttt{0xFF}) \\ &= c_i + (e_{i + 1} \bmod \mathtt{0xFF}) \\ e_i - (e_{i + 1} \bmod \mathtt{0xFF}) &= c_i \end{aligned}$$ This yields the decoding algorithm: $$\boxed{c_i = e_i - (e_{i + 1} \bmod \mathtt{0xFF})}$$
With the decoding algorithm in hand, it can be applied to the encoded message to recover the original:
For example, the first character is given by: $$\begin{aligned} c_1 &= e_1 - (e_2 \bmod \mathtt{0xFF}) \\ &= 170 - (358 \bmod 255) \\ &= 170 - 103 \\ &= 67 \end{aligned}$$ One may guess that the original message was in ASCII format, so a numerical value such as 67 can be interpreted as the ASCII character
C. The remaining characters can be decoded as follows:i e[i] c[i] ASCII ---------------------- 1 170 67 C 2 358 111 o 3 247 110 n 4 137 103 g 5 289 114 r 6 175 97 a 7 333 100 d 8 233 117 u 9 116 108 l 10 263 97 a 11 166 116 t 12 305 105 i 13 200 111 o 14 344 110 n 15 234 115 s 119
