Trying to Develop a Decryption Scheme for a Given Encryption

[BWElbert]

Hello PF friends!

Earlier this week a friend of mine (both of us are in the same mathematics department) posed an encryption mapping to me and I have thus far not been able to solve it. Here's the map of the k+1 layer:

$\lambda_n^{k+1} = (\sum_{i=1}^{n} \lambda_i^{k})\; mod\; 27,$

where $\lambda_i^{k}$ is the numerical representation (A=0,B=1,...,' '=27) of the i^th letter of the k^th layer.

Clearly this encryption is punctuation-free and does not act on numbers. Let me show you what the process looks like:

Plaintext: H-E-L-L-O T-H-E-R-E
# Repr. : 7-4-11-11-14-26-19-7-4-17-4
Encrypt : 7-11-22-6-20-19-11-18-22-12-16

So far, I have shown that because the first number in the code never changes, determining the second letter reduces to solving a modular equation if we know how deeply encrypted the data is (a requisite for decrypting this in full, I think).

The first part I am working on is trying to show if it is Uniquely Decipherable or not--I haven't found a counter-example to it, but am also not sure how to apply the theorems of Sardinas and Kraft to this code.

Finally, if it is uniquely decipherable, I wonder if this code is at best probabilistically decipherable.

Any insight or thoughts would be great...I don't want to work on writing an encryption algorithm if I can't find a way to decrypt it!

 

[fresh_42]

I doubt it is uniquely decodable as $\mathbb{Z}_{27}$ has zero divisors. This should lead to problems.
First write the scheme in matrix form over this ring, which gives us a tool to deal with the problem. It looks as if it is a power of an upper triangular matrix. Decoding then means to invert this matrix, which is in general not possible in my opinion.