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From: Risto Lankinen on 3 May 2010 04:31

Hi!

Take an arbitrarily large two dimensional table and place pegs onto a
finite number of its cells while observing the following rules:

1. Cell (0,0) contains a peg.
2. Cell (0,n) contains a peg if and only if Cell (n,0) contains a peg.
3. Cell (p,q) contains a peg if and only if both Cell (p,0) and Cell
(0,q) contain a peg.

The resultant peg configuration necessarily consists of a square (n^2)
number of pegs having a symmetric pattern wrt/ the main diagonal (e.g.
cells (0,0), (1,1), (2,2), …, (n,n)) .

Next, assign values to the cells so that Cell (p,q) == Re((i-1)^(p+q))
where Re means real part, and i is the imaginary unit. Examples of
some cell values follow:

Cell (0,0) == 1
Cell (0,1) == Cell (1,0) == -1
Cell (0,2) == Cell (1,1) == Cell (2,0) == 0
Cell (0,3) == Cell (1,2) == Cell (2,1) == Cell (3,0) == 2
etc… the sequence continues
1,-1,0,2,-4,4,0,-8,16,-16,0,32,-64,64,0,-128,256,…

See also A009116 in The On-Line Encyclopedia of Integer Sequences:
http://www.research.att.com/~njas/sequences/A009116

The sum of the values of all pegged cells is some integer, and the sum
of the values of pegged cells on the first row is the arithmetic
average of (one pair of) its integer factors (including negatives,
e.g. 45 = 5*9 = -5*-9 ). Knowing N (the grand sum) and A (the first
row sum) it is easy to factorize N .

- - -

Explain the underlying mechanism.

- Risto -
From: adacrypt on 3 May 2010 06:20
On May 3, 9:31 am, Risto Lankinen <rlank...(a)gmail.com> wrote:
> Hi!
>
> Take an arbitrarily large two dimensional table and place pegs onto a
> finite number of its cells while observing the following rules:
>
> 1. Cell (0,0) contains a peg.
> 2. Cell (0,n) contains a peg if and only if Cell (n,0) contains a peg.
> 3. Cell (p,q) contains a peg if and only if both Cell (p,0) and Cell
> (0,q) contain a peg.
>
> The resultant peg configuration necessarily consists of a square (n^2)
> number of pegs having a symmetric pattern wrt/ the main diagonal (e.g.
> cells (0,0), (1,1), (2,2), …, (n,n)) .
>
> Next, assign values to the cells so that Cell (p,q) == Re((i-1)^(p+q))
> where Re means real part, and i is the imaginary unit.  Examples of
> some cell values follow:
>
> Cell (0,0) == 1
> Cell (0,1) == Cell (1,0) == -1
> Cell (0,2) == Cell (1,1) == Cell (2,0) == 0
> Cell (0,3) == Cell (1,2) == Cell (2,1) == Cell (3,0) == 2
> etc… the sequence continues
> 1,-1,0,2,-4,4,0,-8,16,-16,0,32,-64,64,0,-128,256,…
>
> See also A009116 in The On-Line Encyclopedia of Integer Sequences:http://www.research.att.com/~njas/sequences/A009116
>
> The sum of the values of all pegged cells is some integer, and the sum
> of the values of pegged cells on the first row is the arithmetic
> average of (one pair of) its integer factors (including negatives,
> e.g. 45 = 5*9 = -5*-9 ).  Knowing N (the grand sum) and A (the first
> row sum) it is easy to factorize N .
>
>  - - -
>
> Explain the underlying mechanism.
>
>  - Risto -

Hi ,

On first inspection this seems a bit like a Vigenere Cipher that uses
a square in which the plaintext is valid for a given key if and only
if the plaintext in hand for the key in hand is also the key
elsewhere for the plaintext elsewhere. This means that key/plaintext
combinations are pegged inseperably to each other. This convolution
of the Vigenere square sounds a bit tortuous and unnecessary as an
embellishment of a well known existing cipher. It could cause much
difficulty of encryption without any apparent increase in security but
don't let me discourage from trying something new - Good Luck with
everything. - adacrypt
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