Abstract
We all know that every positive integer has a unique Fibonacci representation, but some positive integers have multiple Gopala Hemachandra (GH) representations, or some positive integers haven't any GH representation. Here, the authors found the first k-positive integer k=(3 2^((m-1))-1) for which there is no Zeckendorf's representation for Gopala Hemachandra sequence whose order m. Thus, the authors formulated the first positive integer whose Zeckendorf's representation can't be found in terms of its order. The authors also described the fourth, the fifth, and the sixth order GH representation of positive integers and obtained the fifth and the sixth order GH representations of the first 26 positive integers uniformly according to a certain rule with a table. Finally, the authors used these GH representations in symmetric cryptography, and the authors made some applications with a method which they construct similar to Nalli and Ozyilmaz.
TopIntroduction
The Fibonacci sequence, 
, is a sequence of numbers, beginning with the integer couple 0 and 1, in which the value of any element is computed by taking the summation of the two antecedent numbers. If so, for k≥2, 
(Koshy, 2001). There have been many studies in the literature dealing with the quadratic number sequences. One of the studied areas of the Fibonacci sequence is the representations and codes of this sequence.
A universal code transforms positive integers representing source messages into code words of different lengths. There are various universal codes such as the Elias codes, the Fibonacci universal code, Narayana code and non-universal codes such as Rice coding, Huffman coding and Golomb coding (Thomas, 2007), (Platos et., 2007), (Buschmann and Bystrykh, 2013),(Kirthi and Kak, 2016). The best known of them is the Fibonacci code and the Fibonacci code is more useful in comparison with other universal codes. Because Fibonacci universal code is a prefix code of variable size, it is uniquely decodable binary code. Also, this code easily fixs data from damaged parts of code words (Kirthi and Kak, 2016). Fibonacci and Gopala Hemachandra universal codes encode positive integers with binary representations and these code words are obtained based on Zeckendorf representation. Each positive integer has one and only one representation as the summation of non-sequential Fibonacci numbers by Zeckendorf’s theorem (Zeckendorf, 1972). Fibonacci sequences and codes can also be defined from higher orders.
for
k>0 and the boundary conditions
and
(-
m<
k<0) (Klein and Ben-Nissan, 2010).
One representation can be obtained for each positive integer A with a binary string of length t, 
, such that 
. The representation is one and only if one uses algorithm to find it as follows: When it is given the integer A, it is detected the largest Fibonacci number 
equivalent or smaller to A; after that it is continued repeating with 
(Klein and Ben-Nissan, 2010). For instance 17=1+3+13, hence its Fibonacci representation is 101001.
According to above algorithm, Fibonacci numbers aren’t used consecutively in any of these summations, that is, in the binary representation, there are no contiguous 1 bits. When generalizing this procedure to higher orders, the same operations are realized as above. Additionally, it is appended (m–1) 1 bits to the mth order variant of Fibonacci representation of k to build the mth order variant of Fibonacci code of any positive integer k. But, unlike the Fibonacci representation, there are no adjacent of m 1 bits in the statement (Klein and Ben-Nissan, 2010).
Key Terms in this Chapter
Symmetric cryptography: Symmetric encryption is a type of encryption where only one key (a secret key) is used to both encrypt and decrypt electronic information.
Fibonacci Sequence: It is a recurrence sequence beginning with the integer couple 0 and 1, in which the value of any element is computed by taking the summation of the two antecedent numbers.
Gopala Hemachandra Sequence: It is a sequence which generalizes the Fibonacci sequence by allowing the same recursive construction of the Fibonacci sequence to be used with arbitrary starting terms.
Zeckendorf’s Representation: The Zeckendorf representation of an integer is the unique way of representing that integer as a sum of non-consecutive Fibonacci numbers.
Fibonacci Code: It is a universal code which encodes positive integers into binary code words.
Recurrence Sequences: Recurrence sequence is a sequence of numbers indexed by an integer and generated by solving a recurrence equation.
Universal Code: It is a code transforms positive integers representing source messages into code words of different lengths.
Gopala Hemachandra Code: It is a code which is a variation of the Fibonacci universal code and has applications in cryptography and data compression.
Cryptography: It is a process of protecting information and communications such that the only one for whom the information is intended can understand.