Encription Based On The Josephus Permutation

Encryption based on the Josephus permutation.
Author: Engineer Geologist Sergio G. Pérez Rodríguez.
email: tigleon@yahoo.com
Abstract
An alternative approach is proposed to develop a variant of the JP (as an encryption tool).
A fundamental algorithm tackles how to perform a JP based on the operation of cyclic permutations on a list of elements. To that end a subroutine is used for thesequential extraction of every element from the list according to an initial rule. The suggested approach modifies the fixed initial rule for functions or operations that change the cyclic permutations This provides an efficient way to create encryption codes as intricate as required.
The examples provided for the change in the cyclic permutations include the use of list convolution, conditionalprobability and some arithmetical functions and operations.
The algorithms are programmed in RPN, taking advantage of the HP 48-49 calculator series built-in functions and facilities for the permutation of stacked elements.

Introduction

Provided two positive integers m and n (m < n), lets arrange n elements in a circle. Starting with a random first element, lets proceed around the circle,removing every mth element. After each element is removed, counting continues around the circle until each one is removed. The sequential order of elimination is the (m,n)-Josephus permutation.
Cyclic permutations offers the core for a straightforward way to model the Josephus Permutation. The length of the cycle would be m-1, since the mth element is extracted from the list . Be sv the listcomprising the elements to be permutated. If we count as r (1≤r≤n) the number of times a m-1 length cycle is performed, then sv[r] corresponds to the list at the r-instance the JP has been applied.
Be Z[sv[r]] the size (number of elements) of sv[r]. Lets h (1≤h≤ Z[sv[r]]) indicate the position, from left to right, of an element of sv[r]. Then E[h,r] provides the elements of the list at the r-instanceat the hth place. Here bring in m[r] as the value of m corresponding to the r-instance the cyclic permutation has been applied. In the original Josephus Permutation m[r] =constant. The suggested option is to change m[r] to alter the cyclic permutation length.
As we shall see as a result of the suggested change , a wide range of variants of the JP where m[r] is obtained or not from sv[r]and/or E[h,r] are proposed.

Algorithmic approach

The flow chart corresponding to the usual modelling of the JP via cyclic permutations is as follows:
1.Enter the initial list size (n=Number of elements) and the cyclic permutation length (m)
2. Build a list sv of n elements and an empty list ct.
2.Make an outer loop of n iterations
3. Make an inner loop of m-1 iterations
4.For every iteration atthe inner loop a single cyclic permutation to the left is applied to sv. As a result of the
Complete internal Operation, m-1 elements would be re-placed sequentially at the tail of sv.
5.Once sv is out of the inner loop, extract from sv the first element to the left . Reassign it to a cumulative list ct.
6. Follow the outer loop an perform again the inner loop until its n iterations areaccomplished.
7. Once the outer loop is completed, the list ct provides the JP of the initial list.
As the stated procedure is analysed, is outstanding the fact the arranged sequence of the list of elements depends on the value of m (Steps 3-4). Having indicated this essential fact, the purpose of the next section is to show how to exploit the dependence on m of the building order of the list ct.Programming the JP and its Variants

Permutations on lists or sets of integers can be performed with the aid of compact RPN programs in fine coupling with the design of the HP 48-49 calculator series stack manipulation and list handling facilities. Their built-in functions for handling a stack of number registers coupled with RPN based programming is a tailor made mean for permutation programming....