An Algorithm for Accelerated Acquirement of Minimal Representation of Super-large Numbers

An algorithm for accelerated acquirement of minimal representation of super-large numbers was presented. The algorithm considers a new form of arithmetic, which was called arithmetic of q- representation of large range integers; which was based on the numbers of the generalized sequence of Fibonacci. The estimations of the complexity of the offered algorithms are presented. p p p p i p p p p p l p i p l w p l w w ϕ ϕ , Where ( ) n p ϕ is the n th Fibonacci р-number. Fibonacci р-numbers for positive n are calculated on the basis of the recurrent relationship


INTRODUCTION
Storage of large volume of information and their protection against an unauthorized access are considered as one of the most important problems facing the designers of modern computer systems. Often in solving such problems, quite different approaches are applied. The results that have been found by many researchers [1][2][3][4][5][6] are based only on the application of the original method of super-large integer numbers representation (1024 binary digits and more) along with the arithmetic of such numbers. In this case, for compression of information, the block of digital data of any length will be considered as super-large integer positive number, which is presented as a set of +2 small numbers [3] . In this kind of representation of numbers, which is called linear form, is the basis of efficient algorithms, for performing computations over super-large numbers [2,4] required for the solution of problems dealing with cryptographic protection of information.
Super-large numbers representation is based on the optimizing properties of Fibonacci -numbers. Anisimov et al. [1] presented an algorithm for obtaining Fibonacci linear form for particular case =1. However, Luzhetsky and Al-Maitah [3] described an algorithm for determining any positive integer . The shortcomings of these algorithms are the huge volume of computations, in addition to the necessity of the multiplication of super-large numbers.
The proposed algorithm, in the current work, makes less complexity of computations because of its dependence on of the addition and subtraction operations only. The proposed algorithm depends on the linear form representation.
Linear form representation: In order to represent integer numbers in linear form, let us consider the integer number z as one of the integer number elements series { } l p w , , which is generated by the recurrent relationship as follows: For determining the values of the initial elements: It has been proved by [3] that for the given integer numbers the following relationship can be used as follows: Proceeding from this assumption, any integer number z is represented in the following form: Where i q is an integer number (representing the coordinates Rounded off to the nearest integer number using p α , the word length of which is equal to the word length of the number z. Taking the calculated products as initial elements of the series: If computations are performed j times, then Mpresentation index will become j. In this case, the coordinate's representation (Eq.2) will be as follows: .  Addition of the similar coordinates may lead to the case, that the sums of coordinates will not possess the properties of minimally. In this case, the coordinates must be minimized by performing a certain number of the following transformations: ; ; . ; ; Algorithms of accelerated achievement Mrepresentation: It is known [5] that any integer positive number z can be represented in the form: The following algorithm is used to determine the values of l a .

Algorithm 1: [Traditional]
Step 1: Determine the element in Fibonacci p-numbers series; which satisfies the condition ( ) , Step 2: Step 3: Take remainder r as an initial number and pass to step 1.
Steps 1-3 are to be executed until remainder r equals zero.
The non-unity values of l a have zero-values.
The set 1 2 1 a a a a n n − is called Fibonacci p-code of the number z [5] .
M-representation corresponds to number ( ) ( ) By substituting (Eq.6) into (Eq.5), the number, z, can be written as follows: Stakhov and Luzhetsky [7] have shown that the formation of increasing series of Fibonacci p-numbers requires the storage of +1 current element values of the series and computation by the formula (1). On the other hand, such formation of the series is not efficient for determination of l a figures according to the algorithm 1. Since number z, being transformed and further remainder r, are compared with Fibonacci pnumbers, which have the greatest values, the storage of the whole p-numbers series will be required. Taking into consideration this fact, it is suggested to form diminishing series of Fibonacci p-numbers, using the recurrent relationship: In this case, in order to realize algorithm 1 it is sufficient to store only p+1 current value of ( ) l p ϕ elements and successively compare number z (remainder r) being transformed with Fibonacci pnumbers, using for this purpose the operation of subtraction.
Each accumulating addition of M-representations is not expedient to perform in four stages as it is mentioned early. It is sufficient to compare indices and realize addition of the similar coordinates and the stages of minimization of sum coordinates are performed after the execution of all additions, (i.e. intermediate sums will have q-representation and final sum -Mrepresentation).
All these three procedures, therefore, necessary for obtaining M-representation of z number can be performed using the operations of addition and subtraction. In this case the initial data is integer positive number z, being transformed and the set of Taking into account the above-mentioned peculiarities of realization each of the procedures, the next algorithm of accelerated achievement of M-representation is proposed.
Step 6: Perform addition of representations ( ) Step 7: Decrease by 1 value l .
Step 8: If 1 + > p l then go to step 1.
Step 11: Increase by 1 value j and go to step 9.
Thus M-representation of number 45 has the form ( ) In order to obtain a general evaluation of the considered algorithm, the complexity of each of its three procedures is to be evaluated. As a conventional unit of complexity will be used the operation of addition or subtraction of m-digit binary codes.
The word-length of binary codes, being processed while realization of the algorithm is determined proceeding from maximal value of Fibonacci pnumbers, ( )  The proposed algorithm will be more efficient than the known one, if . Table 1 illustrates given values η for difference . For instance, if p=3 the complexity of suggested algorithm is two times less complicated than the complexity of the known algorithm.

CONCLUSION
The new algorithm of accelerated acquirement of minimal representation of super-large numbers is performed. The complexity of the given algorithm is less than the complexity of the known algorithm. Implementation of such an algorithm in practical aspects should have no shortcomings. The suggested algorithm is proven valid and it is expected to be used as a base of a new generation of arithmetic units.