=Paper=
{{Paper
|id=Vol-2740/20200102
|storemode=property
|title=Non-Binary Constant Weight Coding Technique
|pdfUrl=https://ceur-ws.org/Vol-2740/20200102.pdf
|volume=Vol-2740
|authors=Alexandr Kuznetsov,Anastasiia Kiian,Tetiana Kuznetsova,Alexey Smirnov
|dblpUrl=https://dblp.org/rec/conf/icteri/KuznetsovKKS20
}}
==Non-Binary Constant Weight Coding Technique==
https://ceur-ws.org/Vol-2740/20200102.pdf
Non-Binary Constant Weight Coding Technique
Alexandr Kuznetsov 1[0000-0003-2331-6326], Anastasiia Kiian 1[0000-0003-2110-010X],
Tetiana Kuznetsova 1[0000-0002-5605-9293] and Oleksii Smirnov 2[0000-0001-9543-874X]
1
V. N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
kuznetsov@karazin.ua, nastyak931@gmail.com,
kuznetsova.tatiana17@gmail.com
2
Central Ukrainian National Technical University, avenue University, 8, Kropivnitskiy, 25006,
Ukraine, dr.smirnovoa@gmail.com
Abstract. This paper presents research results of mixed base number systems
using a binomial representation of numbers and nonlinear coding techniques by
constant weight codes, which are based on a binomial count. Also we propose a
technique of non-binary constant weight coding based on a generalized binomi-
al-positional representation, which allows to generalize the known approach to
the non-binary case and practically implement computational algorithms for
generating non-binary sequences of constant weight. The proposed technique of
non-binary constant weight coding can be useful for improving post-quantum
code-based cryptographic algorithms.
Keywords: constant weight coding, mixed base number systems, binomial
count, code-based cryptographic algorithms.
1 Introduction
The computational efficiency of arithmetic operations directly depends on the tech-
nique of representing numbers on which operations are performed, i.e. on the applied
number system [1–4]. The most common is the positional number system in which
the same numerical sign (digit) in the number record has different meanings, depend-
ing on the position where it is located [3, 4]. Among these systems is the modern
decimal number system, the occurrence of which is associated with a finger count, the
binary number system used in modern computers, etc.
A mixed base number system is a generalization of the positional system, its base
is an increasing number sequence and each represented number is expressed through a
linear combination of base elements [3–6]. Mixed base number systems include the
Fibonacci number system, factorial, binomial and other systems [6–8].
It should be noted that many applications are based on the binomial number sys-
tem, including the so-called binomial codes that belong to the class of nonlinear bina-
ry redundancy codes used to increase the noise immunity of binary asymmetric data
transmission channels [3, 6–8]. The main property of binomial codes, an equal Ham-
ming weight (the number of nonzero elements) of all codewords, is used to effectively
detect asymmetric distortions of transmitted sequences. In this case, a change of
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
Hamming weight of the sequence is an adequate criterion for detecting errors in
asymmetric binary data channels.
Another, equally demanded application of constant weight codes consists in con-
structing provably robust cryptosystems [9–12], the security of which is justified by
the reducability of the private key calculation task to the solution of the theoretical
complexity problem of syndrome decoding [13, 14]. So, for example, in [15–17],
provable secure encryption systems are considered. In works [18–20], code-based
electronic signature schemes are considered, their parameters and basic properties are
investigated. In articles [21, 22], pseudorandom number generators based on codes
are studied, as well as promising stream encryption schemes [23–26]. Papers [27–29]
are devoted to the study of zero-knowledge proof schemes. The design of fast and
secure code-based hash functions is investigated in [25, 30–32].
Thus, code-based cryptography has many important applications. These crypto
primitives are post-quantum security algorithms, i.e. they will be strong even under
the conditions of possible quantum cryptanalysis [9–12].
Constant weight codes are used as one of the elements of code-based cryptosys-
tems. In particular, in encryption systems, sequences with constant weights are used
as a session key [16, 33, 34]. To generate a digital signature using the decoding algo-
rithm, it is necessary to find a sequence with constant weight [18–20, 35]. When gen-
erating pseudo-random numbers, constant weight codes are used in the feedback cir-
cuit of the generator [21–23]. The input data for a code-based hash function is first
converted to a constant weight sequence and then processed by a compression func-
tion [30, 32].
It should be noted that all listed applications only use binary constant weight codes.
The discrete mathematics literature also describes only binary algorithms [1, 2, 4]. We
offer non-binary constant weight coding technique. This will greatly expand the area
of possible use of code-based cryptosystems. In particular, the practical implementa-
tion of non-binary (for example, over finite field GF (2m ) ) codes can work faster and
more efficiently, while providing a high level of strength of code-based cryptosys-
tems. This determines the relevance of this article, a purpose of which is to develop a
technique of non-binary constant weight coding based on a generalized binomial-
positional representation.
2 Positional and Mixed Base Number Systems. Binary
technique and Algorithm of Constant Weight Coding
The positional number system is based on a positional numbering, i.e. on a local value
of digits, and is determined by a certain number b 1 (a base of the number system)
such that the b units in each category are combined into one unit of the next highest
rank. The number system is also called b -positional system [1, 2, 13].
A number x in a b -ary positional number system is represented as a linear combi-
nation of powers of number b
n 1
x ai bi ,
i 0
where ai are integers called numbers and they satisfy an inequality:
0 ai b ,
i is a discharge sequence number, starting from zero, n is a number of bits (length)
of the position code.
Each degree b i in such record is called a rank, the certain rank and the correspond-
ing numbers is determined by the value of the indicator i . Usually, for a nonzero
number x , the leading digit an 1 in the b -ary representation is also required to be
nonzero.
If there are no discrepancies (for example, when all the numbers are presented in
the form of unique written characters), the number is recorded as a sequence of its b -
ary digits, listed in ascending order of rank from left to right:
x a0 a1 ... an 1 .
A mixed base number system is a generalization of the b -ary system and also often
refers to positional number systems. The basis of the mixed base system is an increas-
ing numbers sequence
b0 , b1 , b2 ,...
and each number is represented as a linear combination:
n 1
x ai bi ,
i 0
where some code restrictions are imposed on the coefficients ai .
Numeral x in a mixed base number system is an enumeration of its numbers in de-
creasing order of the index i , starting with the first nonzero. If for some bi bi , then
the mixed base number system coincides with the b - ary positional number system.
The binomial number system is based on the representation of numbers through an
increasing sequence of binomial coefficients
u u u
b0 1 , b1 2 ,..., bn 1 n ,
1 2 n
u ui 1 !
bi i 1 ,
i 1 i 1! ui 1 i 1!
n!
0 u1 u2 ... un ,
w! n w!
where w is number of non-zero elements of binomial code.
A number x in the binomial system is represented as a linear combination:
n 1 n 1
u
x ai bi ai i 1 ,
i 0 i 0 i 1
where coefficients ai 0,1 .
In the case when there are no discrepancies in the calculation of binomial coeffi-
u
cients , bi i 1 , i.e. when a rule is established for the formation of a numbers set
i 1
0 u1 u2 ... un , the number x is written in increasing order of ranks ai from left
to right:
x a0 a1 ... an 1 .
The considered binomial number system is used to construct binary constant
weight codes consisting of a set of binary sequences with a fixed number of nonzero
elements in each sequence (a constant Hamming weight).
We introduce the following notation:
n is a length of the constant weight code, i.e. the number of elements (bits) of code
sequences (code words);
С С0 , C1 ,..., CM 1 is a set of code words for the constant weight code, where
C j C j0 C j1
... C jn1 C , C ji 0,1 ,
j 0,1,..., M 1 , i 0,1,..., n 1 .
For all vectors Cj , j 0,1,..., M 1 we have equal weight Hamming:
j : w C j const w , where
w C j0 C j1
... C jn1 # C j0 C j1 ... C jn1 C ji 0
,
# C j0 C j1 ... C jn1 C ji 0
is a number C j , i 0,1,..., n 1 , where C j 0 .
i i
The power of the constant weight code (the number of elements in the set С ) is
determined by the number of binary vectors of length n and weight w :
n!
С М .
w! n w!
The well-known binomial (binary constant weight) coding technique [6, 7, 13] is
based on the presentation of information data in the form of a numerical equivalent
(denoted by a number A ) with further decomposition into a linear combination of
binomial coefficients. Such system of code restrictions on the length of sequences of
constant weight n , codeword weights w and code cardinality М are satisfied:
j : w C j const w;
0 A M;
0 w n.
The number A is presented as binary sequence CA CA0 CA1 ... CAn1 of
n 1
n i 1
constant weight and A C Ani1 bi , where bi , l is a nonzero element
i 0 wl
number in C A , l 0,1,..., w .
i 1
Nonzero element is a C A
n i 1
, for which bi C Anm1 bm .
m0
Obviously, the sum on the right side of the expression is equal to the sum of only
bm those for which the corresponding elements of the vector C A are not equal to zero (
CA 0 ).
nm1
It should be noted that the considered technique does not imply the formation of
non-binary sequences with constant weight (vectors C A with C ji 0,1,..., q 1 ,
q 2 ) and thus does not allow the implementation of non-binary constant weight
coding.
The article proposes a new technique of non-binary constant weight coding based
on a generalized binomial-positional representation, which allows us to generalize the
above approach to the non-binary case and practically implement computational algo-
rithms for generating non-binary sequences of constant weight.
3 Proposed Technique of Non-Binary Constant Weight Coding
To generalize the considered approach of generating sequences of constant weight to
the non-binary case, a new form of a generalized binomial-positional representation
of numbers is proposed. The proposed number system belongs to the class of mixed
base systems and is based on the representation of numbers through an increasing
sequence of binomial coefficients, each of which is encoded by positional numbering,
i.e. the representation of digits with binomial coefficients is based on the local value
of the digits.
Consider the number x in the proposed generalized binomial-positional number
n 1
system: x ai bi .
i 0
Let’s introduce the following notation and equalities:
ui 1 ui 1 !
ai 0,1,..., q 1 , bi ,
i 1 i 1! ui 1 i 1!
n!
0 u1 u2 ... un ,
w! n w!
where w is a number of nonzero elements of the generalized binomial-positional code.
Then the number x is represented through an increasing sequence of binomial co-
efficients b0 , b1 ,..., bn 1 , and corresponding sequence a0 , a1 ,..., an 1 .
Let's consider nonzero elements ai 1 , i 0,1,..., n 1 , sequences a0 , a1 ,..., an 1
and renumber them, i.e. we denote them as elements of the sequence a0 , a1 ,..., aw1 ,
l 0,1,..., w 1 and l : al 1,..., q 1 .
A sequence a0 , a1 ,..., aw1 and all its elements al (nonzero elements of a sequence
a0 , a1 ,..., an 1 numbered according to the increasing digits order) are formed using a
positional number system on the base q 1 , i.e. q 1 units in each rank are combined
into one unit of the next highest rank. A set of nonzero elements sets al ,
l 0,1,..., w 1 defines the number xP that is represented in the positional system as
follows:
w 1
xP al 1 hl ,
l 0
where h q 1 is a base of used positional system, 1 al q .
The increasing sequence of binomial coefficients b0 , b1 ,..., bn 1 sets a number xB ,
n 1
which is represented in the binomial number system as xB aBi bi , where coeffi-
i 0
cients aBi 0,1 .
The number x in the proposed generalized binomial-positional number system sat-
isfies the following equality:
x xB q-1 xP ,
w
which sets a main code restriction on the elements of the generalized binomial-
positional code.
Thus, the number x in the proposed system of generalized binomial-positional
counting is represented as a linear combination:
n 1
x ai bi xB q-1 xP
w
i 0
n 1 w 1
q-1 aBi bi al 1 q 1 .
w l
i 0 l 0
The proposed generalization of the binomial-positional way of representing num-
bers consists in the complex use of the positional number system and the binomial
counting system: the first term on the right side of the equality, through the increasing
sequence of binomial coefficients, determines the placement of nonzero elements of
the generalized binomial-positional code, the second term defines the values of non-
zero elements of the sequence in positional code.
The proposed technique of representing numbers is based on the technique of non-
binary constant weight coding. For the abstract definition of a non-binary constant
weight code, we introduce the following formal notation:
n is a code length;
С С0 ,C1 ,...,CM 1 is a set of code words,
C j C j0 C j1
... C jn1 C, C ji 0,1,..., q 1 ,
j 0,1,..., M 1, i 0,1,..., n 1, j : w C j const w .
The cardinality of a non-binary constant weight code defined in this way is deter-
mined by the number of length n and weight w vectors with elements from the set
0,1,..., q 1 :
n!
С М q 1
w
.
w! n w!
The proposed technique is based on the presentation of information data in the
form of a numerical equivalent A with further decomposition into a linear combina-
tion of binomial coefficients, each of which is encoded by positional numbering so
that a system of code restrictions on the length of n sequences of constant weight, the
weight w of code words and code cardinality М is satisfied:
j : w C j const w;
0 A M;
0 w n;
0 C ji q.
The number A is presented as non-binary sequence of constant weight
n 1
CA CA0 CA1 ... CAn1 and A AB q-1 AP ,
w
where AB aBi bi ,
i 0
n i 1 w 1
bi , AP al 1 h , h q 1 .
l
w l l 0
The process of generating a non-binary sequence of constant weight is presented in
four stages.
1. Representation of number A in the form of numbers AB and AP :
A
AB , AP A mod q-1 ,
w
q-1
w
where y is the integer part of number y .
The uniqueness of the representation of number A in the form of numbers AB and
AP is justified by the Chinese remainder theorem [13].
M
The number AB lies within 0 AB and can be represented in a binomial
q 1
w
number system with code restrictions:
j : w c j const w;
n!
0 AB ;
w ! n w !
0 w n.
The number AP lies within 0 AP q 1 and, accordingly, can be represented
w
in the positional number system on the basis of h q 1 .
1. Representation of numbers in a binomial number system:
n 1
n i 1
AB aBi bi , bi .
i 0 wl
2. Representation of a number AP in a positional number system:
w 1
AP al 1 hl , h q 1 .
l 0
Generating of sequence CA CA0 CA1
... CAn1 C :
CAi al aBi , i 0,1,..., n 1 , l 0,1,..., w 1 ,
i.e.,
if we have aBi 0 for some i 0,1,..., n 1 in the AB representation, then we
get CA 0 ; i
if aBi 1 , then we get CA al , i.e. the required element is equal to the corre-
i
sponding nonzero element in the view AP .
Example. Let’s n 3 , w 1 , q 4 . The result of the proposed algorithm in the
form of the obtained correspondence of all numbers A , their binary representations
I A I A0 I A1 ... I Ak 1 in a positional binary code of length
k log 2 M log 2 9 4 , numbers AB and AP , corresponding vectors
a B0 aB1 aB2 and a and generated non-binary vectors C C
0 A A0 CA1 CA2
of constant weight are shown in table 1.
Table 1. An example of the formation of non-binary sequences of constant weight
A IA AB aB0 aB1 aB2 AP a0 C A0 CA1 CA2
0 (0000) 0 (100) 0 (1) (100)
1 (1000) 0 (100) 1 (2) (200)
2 (0100) 0 (100) 2 (3) (300)
3 (1100) 1 (010) 0 (1) (010)
4 (0010) 1 (010) 1 (2) (020)
5 (1010) 1 (010) 2 (3) (030)
6 (0110) 2 (001) 0 (1) (001)
7 (1110) 2 (001) 1 (2) (002)
8 (0001) 2 (001) 2 (3) (003)
The formed set of 9 non-binary vectors CA CA0 CA1 CA2 of constant weight
forms a non-binary constant weight code C C0 , C1 ,..., C8 .
Thus, the proposed number system based on the generalized binomial-positional
representation of numbers allows complex use of both the local value of the digits of
the code sequence and the values of binomial coefficients specified by the placement
of nonzero elements of the sequence. Application of the developed number system
allows us to build effective techniques and algorithms of non-binary constant weight
coding for their use in various practical applications. For example, the proposed tech-
nique can be useful for improving post-quantum code-based cryptographic algorithms
(ciphers, electronic digital signatures, key encapsulation schemes, pseudorandom
number generators, etc.) [19, 21, 23, 31, 32, 36]. The use of constant weight codes is
also one of the basic transformations for the functioning of code-based cryptosystems.
In particular, constant weight codes are used in the McEliece [33] and Niederreiter
[17] public key encryption schemes, as well as in electronic signature schemes and
code-based pseudorandom numbers generators. It is worth noting that at the moment,
code-based cryptography is considered one of the most optimal ways of developing
post-quantum cryptography. A good example of the use of constant weight codes in
cryptosystems is a Niederreiter encryption process, which consists of several basic
steps. First, user generates key and system parameters. Next, an information sequence
e that needs to be encrypted is converted using constant-weight codes into a se-
quence of fixed length n and constant weight t . Then vector syndrome s e H xT
must be calculated.
This vector is a ciphertext, i.e. an encrypted message that can be decrypted only by
the user who has a secret key. Decryption consists in removing the action of masking
matrices, after which the sequence is decoded using a fast algebraic technique. The
found error vector is a sequence of constant weight e . To restore the information
sequence, the conversion is used, the inverse of what was used. As you can see, each
possible information sequence should be uniquely associated with the corresponding
constant weight sequence, i.e. to build a code-based cryptosystem, it is necessary to
implement an algebraic rule of this correspondence, implemented through an constant
weight account system.
Thus, it is necessary to realize a one-to-one correspondence between all possible
information sequences and various constant weight vectors. In the other words, for an
arbitrary countable set (in advance of a given power), it is required to implement a
new recording technique (number system) in the form of a set of constant weight
vectors. At the moment, resistance versions of the constructions of code-based cryp-
tosystems are based on the use of binary Goppa codes. In this simplest case, it is pos-
sible to use the well-known binomial count to convert information sequences into
constant weight binary vectors. In the general case, code-based cryptosystems can be
built for arbitrary base. And for this general case, we offer a generalized binomial-
positional number system and a new technique of non-binary constant weight coding
(binomial-positional counting). This will not only increase the strength of code-based
cryptosystems, but also provide additional useful properties, such as the control of
non-binary errors during transmission.
In addition, this study can be useful in other practical applications, for example, to
improve channel coding techniques, to prevent interference in telecommunication
networks, etc.
4 Conclusions
As a result of the research, a new number system based on the generalized binomial-
positional representation of numbers is proposed. It is that the complex use of the
binomial counting system (through the increasing binomial coefficients sequence
defines a position of nonzero element) and the positional number system (the values
of nonzero elements are specified through the local value of numbers).
For the first time, a technique of non-binary constant weight coding based on a
generalized binomial-positional representation of numbers is proposed, which allows
us to generalize the well-known approach to the non-binary case and practically im-
plement computational algorithms for generating non-binary sequences of constant
weight.
The proposed non-binary constant weight coding technique is a generalization of
the well-known binary case. In fact, a well-known binary code is used to represent a
number in the binomial number system. Nonzero elements of the resulting binomial
sequence are additionally encoded with a positional code. Thus, the complexity of the
implementation of the proposed technique is defined as the total complexity of the
known techniques of binomial and positional coding.
The developed technique can be used in various practical applications, for exam-
ple, in code-based cryptosystems: ciphers, electronic digital signatures, key encapsu-
lation schemes, pseudorandom number generators, etc. These cryptosystems are ex-
pected to be safe even in the conditions of the possible application of quantum cryp-
tographic analysis techniques, i.e. focused on the post-quantum period.
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