This document presents three secret sharing schemes based on the Chinese Remainder Theorem (CRT).
The first scheme uses three large prime numbers p, q, and r to construct shares for t share holders. The secret S is split into t parts using the CRT and distributed to the shares, and all t shares are needed to reconstruct the secret.
The second scheme is proved using a lemma showing there exist integers that allow reconstructing the secret from three shares defined using the primes.
The third scheme provides an example of secret sharing using quadratic polynomials to generate shares such that combining any two shares does not reveal the secret.
Testing of Matrices Multiplication Methods on Different ProcessorsEditor IJMTER
There are many algorithms we found for matrices multiplication. Until now it has been
found that complexity of matrix multiplication is O(n3). Though Further research found that this
complexity can be decreased. This paper focus on the algorithm and its complexity of matrices
multiplication methods.
k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
Slides for Introductory session on K Means Clustering.
simple and good. ppt
Could be used for taking classes for MCA students on Clustering Algorithms for Data mining.
Prepared By K.T.Thomas HOD of Computer Science, Santhigiri College Vazhithala
Hierarchical Clustering | Hierarchical Clustering in R |Hierarchical Clusteri...Simplilearn
This presentation about hierarchical clustering will help you understand what is clustering, what is hierarchical clustering, how does hierarchical clustering work, what is distance measure, what is agglomerative clustering, what is divisive clustering and you will also see a demo on how to group states based on their sales using clustering method. Clustering is the method of dividing the objects into clusters which are similar between them and are dissimilar to the objects belonging to another cluster. It is used to find data clusters such that each cluster has the most closely matched data. Prototype-based clustering, hierarchical clustering, and density-based clustering are the three types of clustering algorithms. Lets us discuss hierarchical clustering in this video. In simple terms, Hierarchical clustering is separating data into different groups based on some measure of similarity.
Below topics are explained in this "Hierarchical Clustering" presentation:
1. What is clustering?
2. What is hierarchical clustering
3. How hierarchical clustering works?
4. Distance measure
5. What is agglomerative clustering
6. What is divisive clustering
7. Demo: to group states based on their sales
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive Bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
We recommend this Machine Learning training course for the following professionals in particular:
1. Developers aspiring to be a data scientist or Machine Learning engineer
2. Information architects who want to gain expertise in Machine Learning algorithms
3. Analytics professionals who want to work in Machine Learning or artificial intelligence
4. Graduates looking to build a career in data science and Machine Learning
Learn more at www.simplilearn.com
A Novel Design Architecture of Secure Communication System with Reduced-Order...ijtsrd
In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced-order linear receiver is developed to guarantee the global exponential stability of the resulting error signals. Besides, the guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Novel Design Architecture of Secure Communication System with Reduced-Order Linear Receiver" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20212.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20212/a-novel-design-architecture-of-secure-communication-system-with-reduced-order-linear-receiver/yeong-jeu-sun
Testing of Matrices Multiplication Methods on Different ProcessorsEditor IJMTER
There are many algorithms we found for matrices multiplication. Until now it has been
found that complexity of matrix multiplication is O(n3). Though Further research found that this
complexity can be decreased. This paper focus on the algorithm and its complexity of matrices
multiplication methods.
k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
Slides for Introductory session on K Means Clustering.
simple and good. ppt
Could be used for taking classes for MCA students on Clustering Algorithms for Data mining.
Prepared By K.T.Thomas HOD of Computer Science, Santhigiri College Vazhithala
Hierarchical Clustering | Hierarchical Clustering in R |Hierarchical Clusteri...Simplilearn
This presentation about hierarchical clustering will help you understand what is clustering, what is hierarchical clustering, how does hierarchical clustering work, what is distance measure, what is agglomerative clustering, what is divisive clustering and you will also see a demo on how to group states based on their sales using clustering method. Clustering is the method of dividing the objects into clusters which are similar between them and are dissimilar to the objects belonging to another cluster. It is used to find data clusters such that each cluster has the most closely matched data. Prototype-based clustering, hierarchical clustering, and density-based clustering are the three types of clustering algorithms. Lets us discuss hierarchical clustering in this video. In simple terms, Hierarchical clustering is separating data into different groups based on some measure of similarity.
Below topics are explained in this "Hierarchical Clustering" presentation:
1. What is clustering?
2. What is hierarchical clustering
3. How hierarchical clustering works?
4. Distance measure
5. What is agglomerative clustering
6. What is divisive clustering
7. Demo: to group states based on their sales
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive Bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
We recommend this Machine Learning training course for the following professionals in particular:
1. Developers aspiring to be a data scientist or Machine Learning engineer
2. Information architects who want to gain expertise in Machine Learning algorithms
3. Analytics professionals who want to work in Machine Learning or artificial intelligence
4. Graduates looking to build a career in data science and Machine Learning
Learn more at www.simplilearn.com
A Novel Design Architecture of Secure Communication System with Reduced-Order...ijtsrd
In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced-order linear receiver is developed to guarantee the global exponential stability of the resulting error signals. Besides, the guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Novel Design Architecture of Secure Communication System with Reduced-Order Linear Receiver" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20212.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20212/a-novel-design-architecture-of-secure-communication-system-with-reduced-order-linear-receiver/yeong-jeu-sun
LITTLE DRAGON TWO: AN EFFICIENT MULTIVARIATE PUBLIC KEY CRYPTOSYSTEMIJNSA Journal
In 1998 [8], Patarin proposed an efficient cryptosystem called Little Dragon which was a variant a variant of Matsumoto Imai cryptosystem C*. However Patarin latter found that Little Dragon cryptosystem is not secure [8], [3]. In this paper we propose a cryptosystem Little Dragon Two which is as efficient as Little Dragon cryptosystem but secure against all the known attacks. Like Little Dragon cryptosystem the public key of Little Dragon Two is mixed type that is quadratic in plaintext and cipher text variables. So the public key size of Little Dragon Two is equal to Little Dragon Cryptosystem. Our public key algorithm is bijective and can be used for both encryption and signatures.
Hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis which seeks to build a hierarchy of clusters. It is an algorithm that groups similar objects into groups called clusters. The endpoint is a set of clusters, where each cluster is distinct from each other cluster, and the objects within each cluster are broadly similar to each other. Let’s try to understand how exactly Hierarchical clustering works.
K-means Clustering Algorithm with Matlab Source codegokulprasath06
K-means algorithm
The most common method to classify unlabeled data.
Also Checkout: http://bit.ly/2Mub6xP
Any Queries, Call us@ +91 9884412301 / 9600112302
Backtracking based integer factorisation, primality testing and square root c...csandit
Breaking a big integer into two factors is a famous problem in the field of Mathematics and
Cryptography for years. Many crypto-systems use such a big number as their key or part of a
key with the assumption - it is too big that the fastest factorisation algorithms running on the
fastest computers would take impractically long period of time to factorise. Hence, many efforts
have been provided to break those crypto-systems by finding two factors of an integer for
decades. In this paper, a new factorisation technique is proposed which is based on the concept
of backtracking. Binary bit by bit operations are performed to find two factors of a given
integer. This proposed solution can be applied in computing square root, primality test, finding
prime factors of integer numbers etc. If the proposed solution is proven to be efficient enough, it
may break the security of many crypto-systems. Implementation and performance comparison of
the technique is kept for future research.
K Means Clustering Algorithm | K Means Example in Python | Machine Learning A...Edureka!
** Python Training for Data Science: https://www.edureka.co/python **
This Edureka Machine Learning tutorial (Machine Learning Tutorial with Python Blog: https://goo.gl/fe7ykh ) series presents another video on "K-Means Clustering Algorithm". Within the video you will learn the concepts of K-Means clustering and its implementation using python. Below are the topics covered in today's session:
1. What is Clustering?
2. Types of Clustering
3. What is K-Means Clustering?
4. How does a K-Means Algorithm works?
5. K-Means Clustering Using Python
Machine Learning Tutorial Playlist: https://goo.gl/UxjTxm
Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm.CSCJournals
In this paper, methods of generating self-invertible matrix for Hill Cipher algorithm have been proposed. The inverse of the matrix used for encrypting the plaintext does not always exist. So, if the matrix is not invertible, the encrypted text cannot be decrypted. In the self-invertible matrix generation method, the matrix used for the encryption is itself self-invertible. So, at the time of decryption, we need not to find inverse of the matrix. Moreover, this method eliminates the computational complexity involved in finding inverse of the matrix while decryption.
LITTLE DRAGON TWO: AN EFFICIENT MULTIVARIATE PUBLIC KEY CRYPTOSYSTEMIJNSA Journal
In 1998 [8], Patarin proposed an efficient cryptosystem called Little Dragon which was a variant a variant of Matsumoto Imai cryptosystem C*. However Patarin latter found that Little Dragon cryptosystem is not secure [8], [3]. In this paper we propose a cryptosystem Little Dragon Two which is as efficient as Little Dragon cryptosystem but secure against all the known attacks. Like Little Dragon cryptosystem the public key of Little Dragon Two is mixed type that is quadratic in plaintext and cipher text variables. So the public key size of Little Dragon Two is equal to Little Dragon Cryptosystem. Our public key algorithm is bijective and can be used for both encryption and signatures.
Hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis which seeks to build a hierarchy of clusters. It is an algorithm that groups similar objects into groups called clusters. The endpoint is a set of clusters, where each cluster is distinct from each other cluster, and the objects within each cluster are broadly similar to each other. Let’s try to understand how exactly Hierarchical clustering works.
K-means Clustering Algorithm with Matlab Source codegokulprasath06
K-means algorithm
The most common method to classify unlabeled data.
Also Checkout: http://bit.ly/2Mub6xP
Any Queries, Call us@ +91 9884412301 / 9600112302
Backtracking based integer factorisation, primality testing and square root c...csandit
Breaking a big integer into two factors is a famous problem in the field of Mathematics and
Cryptography for years. Many crypto-systems use such a big number as their key or part of a
key with the assumption - it is too big that the fastest factorisation algorithms running on the
fastest computers would take impractically long period of time to factorise. Hence, many efforts
have been provided to break those crypto-systems by finding two factors of an integer for
decades. In this paper, a new factorisation technique is proposed which is based on the concept
of backtracking. Binary bit by bit operations are performed to find two factors of a given
integer. This proposed solution can be applied in computing square root, primality test, finding
prime factors of integer numbers etc. If the proposed solution is proven to be efficient enough, it
may break the security of many crypto-systems. Implementation and performance comparison of
the technique is kept for future research.
K Means Clustering Algorithm | K Means Example in Python | Machine Learning A...Edureka!
** Python Training for Data Science: https://www.edureka.co/python **
This Edureka Machine Learning tutorial (Machine Learning Tutorial with Python Blog: https://goo.gl/fe7ykh ) series presents another video on "K-Means Clustering Algorithm". Within the video you will learn the concepts of K-Means clustering and its implementation using python. Below are the topics covered in today's session:
1. What is Clustering?
2. Types of Clustering
3. What is K-Means Clustering?
4. How does a K-Means Algorithm works?
5. K-Means Clustering Using Python
Machine Learning Tutorial Playlist: https://goo.gl/UxjTxm
Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm.CSCJournals
In this paper, methods of generating self-invertible matrix for Hill Cipher algorithm have been proposed. The inverse of the matrix used for encrypting the plaintext does not always exist. So, if the matrix is not invertible, the encrypted text cannot be decrypted. In the self-invertible matrix generation method, the matrix used for the encryption is itself self-invertible. So, at the time of decryption, we need not to find inverse of the matrix. Moreover, this method eliminates the computational complexity involved in finding inverse of the matrix while decryption.
A SECURE DIGITAL SIGNATURE SCHEME WITH FAULT TOLERANCE BASED ON THE IMPROVED ...csandit
Fault tolerance and data security are two important issues in modern communication systems.
In this paper, we propose a secure and efficient digital signature scheme with fault tolerance
based on the improved RSA system. The proposed scheme for the RSA cryptosystem contains
three prime numbers and overcome several attacks possible on RSA. By using the Chinese
Reminder Theorem (CRT) the proposed scheme has a speed improvement on the RSA decryption
side and it provides high security also.
A SECURE DIGITAL SIGNATURE SCHEME WITH FAULT TOLERANCE BASED ON THE IMPROVED ...cscpconf
Fault tolerance and data security are two important issues in modern communication systems.
In this paper, we propose a secure and efficient digital signature scheme with fault tolerance
based on the improved RSA system. The proposed scheme for the RSA cryptosystem contains
three prime numbers and overcome several attacks possible on RSA. By using the Chinese
Reminder Theorem (CRT) the proposed scheme has a speed improvement on the RSA decryption
side and it provides high security also.
CASCADE BLOCK CIPHER USING BRAIDING/ENTANGLEMENT OF SPIN MATRICES AND BIT ROT...IJNSA Journal
Secure communication of the sensitive information in disguised form to the genuine recipient so that an intended recipient alone can remove the disguise and recover the original message is the essence of Cryptography. Encrypting the message two or more times with different encryption techniques and with different keys increases the security levels than the single encryption. A cascade cipher is stronger than the first component. This paper presents multiple encryption schemes using different encryption techniques Braiding/Entanglement of Pauli Spin 3/2 matrices and Rotation of the bits with independent secret keys.
THE KEY EXCHANGE CRYPTOSYSTEM USED WITH HIGHER ORDER DIOPHANTINE EQUATIONSIJNSA Journal
One-way functions are widely used for encrypting the secret in public key cryptography, although they are regarded as plausibly one-way but have not been proven so. Here we discuss the public key cryptosystem based on the system of higher order Diophantine equations. In this system those Diophantine equations are used as public keys for sender and recipient, and both sender and recipient can obtain the shared secret through a trapdoor, while attackers must solve those Diophantine equations without trapdoor. Thus the scheme of this cryptosystem might be considered to represent a possible one-way function. We also discuss the problem on implementation, which is caused from additional complexity necessary for constructing Diophantine equations in order to prevent from attacking by tamperers.
Enhancement and Analysis of Chaotic Image Encryption Algorithms cscpconf
The focus of this paper is to improve the level of security and secrecy provided by the chaotic
map based image encryption.An encryption algorithm based on the Logistic and the Henon
maps is proposed. The algorithm uses chaotic iteration to generate the encryption keys, and
then carries out the XOR and cyclic shift operations on the plain text to change the values of
image pixels. Chaotic Map Lattice based image encryption algorithm suggested by Pisarchik is
also examined which is based on Logistic map alone. In experiments, the corresponding results
showed the proposed method is a promising scheme for image encryption in terms of security
and secrecy. At the end, we show the results of a security analysis and a comparison of both
schemes
2. Systems Science & Control Engineering: An Open Access Journal, 2015
Vol. 3, 106–112, http://dx.doi.org/10.1080/21642583.2014.985803
Authenticated key distribution using given set of primes for secret sharing
N. Chandramowliswarana∗
, S. Srinivasanb
and P. Muralikrishnab
aDepartment of Applied Sciences, ITM University, Gurgaon-122017, Haryana, India; bSchool of Advanced Sciences, VIT University,
Vellore – 632014, India
(Received 16 November 2013; accepted 1 November 2014)
In recent years, Chinese remainder theorem (CRT)-based function sharing schemes are proposed in the literature. In this
paper, we study systems of two or more linear congruences. When the moduli are pairwise coprime, the main theorem is
known as the CRT, because special cases of the theorem were known to the ancient Chinese. In modern algebra the CRT
is a powerful tool in a variety of applications, such as cryptography, error control coding, fault-tolerant systems and certain
aspects of signal processing. Threshold schemes enable a group of users to share a secret by providing each user with a
share. The scheme has a threshold t + 1 if any subset with cardinality t + 1 of the shares enables the secret to be recovered.
In this paper, we are considering 2t prime numbers to construct t share holders. Using the t share holders, we split the secret
S into t parts and all the t shares are needed to reconstruct the secret using CRT.
Keywords: key distribution; Chinese remainder theorem; Pell’s equation; graceful labeling
AMS Classification: 94A60; 94A62; 05C78
1. Introduction
A threshold scheme enables a secret to be shared among a
group of members providing each member with a share.
The scheme has a threshold t + 1 if any subset with car-
dinality t + 1 out of the shares enables the secret to
be recovered. We will use the notation (t + 1, ) to refer
to such a scheme. Ideally, in a (t + 1) threshold scheme,
t shares should not give any information on the secret.
We will discuss later how to express this information.
In the 1980s, several algebraic constructions of (t + 1, )
threshold schemes were proposed.
Key distribution is a central problem in cryptographic
systems, one of the nicest ones is the idea of secret shar-
ing, originally suggested by Blakley (1979). Somewhat
surprisingly, Shamir was able to construct a very efficient
such scheme for any n and t without relying on any cryp-
tographic assumptions. Such schemes are called t out of
n secret sharing schemes. An n out of n schemes is a
scheme where all n shares are needed to reconstruct, and
if even one share is missing then there is absolutely no
information about the secret. Secret sharing was invented
independently by Shamir (1979) and Blakley (1979).
A number of common mathematical techniques in sig-
nal processing and data transmission have as their common
basis an earliest number-theoretic theorem known as the
Chinese remainder theorem (CRT). The scope of problems
to which this applies is very wide. It includes cryptogra-
phy, error control coding, fault-tolerant systems and certain
∗Corresponding author. Email: ncmowli@hotmail.com
aspects of signal processing. In this paper, we present three
new centralized group key management protocols based
on the CRT. By shifting more computing load onto the
key server we optimize the number of re-key broadcast
messages, user-side key computation, and number of key
storages. It is attracted much attention in the research com-
munity and a number of schemes have been proposed,
including many encryption schemes and signature schemes
(Lu & Li, 2013).
The CRT can also be used in secret sharing, there are
two secret sharing schemes that make use of the CRT,
Mignotte’s and Asmuth-Bloom’s Schemes see in Mignotte
(1983) and Asmuth and Bloom (1983). They are threshold
secret sharing schemes, in which the shares are gener-
ated by reduction modulo the integers mi, and the secret is
recovered by essentially solving the system of congruences
using the CRT (Apostol (1976)).
THEOREM 1.1 (CRT) Suppose that m1, m2, . . . , mr are
pairwise relatively prime positive integers, and let
a1, a2, . . . , ar be integers. Then the system of congruences,
x ≡ ai (mod mi) for 1 ≤ i ≤ r, has a unique solution mod-
ulo M = m1 × m2 × . . . × mr, which is given by: x ≡
a1M1y1 + a2M2y2 + . . . + arMryr (mod M), where Mi =
M/mi and yi ≡ (Mi)−1
(mod mi) for 1 ≤ i ≤ r.
All types of secret key sharing considered in this
paper mainly uses factorization difficulty and discrete log
c 2015 The Author(s). Published by Taylor & Francis.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/Licenses/by/4.0/), which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Downloadedby[14.99.174.78]at03:4324May2016
3. Systems Science & Control Engineering: An Open Access Journal 107
problem difficulty. Here, we propose three secret sharing
scheme among t shares. The motivation for the use of
secret key sharing scheme is that, it gives confidence to the
source node or the owner about the genuinely participating
shares in the network. Here, a key is transmitted or shared
among the multiple share holders in the network that are
under the process of encryption and decryption. The objec-
tive is to maintain the genuineness of the nodes that are
present in the network. Here, the shares are properly dis-
tributed by choosing 2t prime numbers and then it is shared
to their corresponding nodes for which it is generated.
2. Main result
In this section we give key distribution theorem and algo-
rithms. The proposed system involves a design of a pre-
distribution algorithm using a deterministic approach. A
key pre-distribution algorithm using number theory with
high connectivity, high resilience and memory require-
ments is being designed by implementing a deterministic
approach. Most of the related technical terms and defi-
nitions appear in Mignotte (1983), Muralikrishna, Srini-
vasan, and Chandramowliswaran (2013), Okamoto and
Tanaka (1989) and Muralikrishna et al. (2013). The oth-
ers can be found in text books such as Apostol (1976),
Berlekamp (1968), Blakley (1979), and Koblitz (1994).
In this section, we give three distinct novel secret shar-
ing schemes. Consider the three very large odd primes p, q
and r with (qr−1
+ rq−1
) ≡ 0 (mod p), (rp−1
+ pr−1
) ≡
0 (mod q) and (pq−1
+ qp−1
) ≡ 0 (mod r). To accomplish
our first secret key sharing scheme, we adopt the following
theorem.
THEOREM 2.1 Let S be the given secret and N = pqr
where p, q and r are distinct large odd primes. Define
three secret shareholders Y1, Y2, Y3 as follows: Y1 ≡
(−Sk1p(qr−1
+ rq−1
)) (mod N), Y2 ≡ (−Sk2q(pr−1
+ rp−1
))
(mod N) and Y3 ≡ (−S(k3r(pq−1
+ qp−1
) + 1)) (mod N)
then S = Y1 + Y2 + Y3 (mod N)
In order to prove the proposed theorem, we regard a
Lemma 2.2, as the secret key information.
LEMMA 2.2 Let p, q and r be three given distinct odd
primes. Then there exist integers k1, k2 and k3 such that
k1p(qr−1
+ rq−1
) + k2q(pr−1
+ rq−1
) + k3r(pq−1
+ qp−1
)
+ 2 ≡ 0 (mod pqr).
Proof Define: X = (pq−1
+ qp−1
) + (pr−1
+ rp−1
) +
(qr−1
+ rq−1
) − 2. Then
X ≡ (qr−1
+ rq−1
) (mod p)
X ≡ (pr−1
+ rp−1
) (mod q) and
X ≡ (pq−1
+ qp−1
) (mod r).
By CRT, the above system of congruences has exactly one
solution modulo the product pqr.
Define M = pqr then Mp = M/p = qr, Mq = M/q =
pr and Mr = M/r = pq.
Since (Mp , p) = 1, then there is a unique Mp such that
Mp Mp ≡ 1 (mod p).
Similarly there are unique Mq and Mr such that
MqMq ≡ 1 (mod q) and MrMr ≡ 1 (mod r).
Consider
X ≡ ((pq−1
+ qp−1
)MrMr + (pr−1
+ rp−1
)MqMq
+ (qr−1
+ rq−1
)Mp Mp ) (mod pqr)
that is,
pq−1
+ qp−1
+ pr−1
+ rp−1
+ qr−1
+ rq−1
− 2
≡ ((pq−1
+ qp−1
)MrMr + (pr−1
+ rp−1
)MqMq
+ (qr−1
+ rq−1
)Mp Mp ) (mod pqr)
− 2 ≡ ((pq−1
+ qp−1
)(MrMr − 1) + (pr−1
+ rp−1
)
× (MqMq − 1) + (qr−1
+ rq−1
)(Mp Mp − 1))
× (mod pqr).
Thus
k1p(qr−1
+ rq−1
) + k2q(pr−1
+ rq−1
) + k3r(pq−1
+ qp−1
)
+ 2 ≡ 0 (mod pqr).
Proof of Theorem 2.1 By the above Lemma 2.2, we have
k1p(qr−1
+ rq−1
) + k2q(pr−1
+ rq−1
) + k3r(pq−1
+ qp−1
)
+ 2 ≡ 0 (mod N).
1 ≡ (−(k1p(qr−1
+ rq−1
)) − (k2q(pr−1
+ rq−1
))
− (k3r(pq−1
+ qp−1
) + 1)) (mod N).
Thus S = Y1 + Y2 + Y3 (mod N).
The following three examples motivating us to write
nice secret sharing algorithms
Example 1 Secret Key Sharing using Quadratic Polyno-
mials
Step 1 Define P(x) = 1x2
+ 2x + 3 (secret) where
i ∈ Z+
, i ∈ {1, 2, 3}
Let λ be a positive integer with P(λ) = 1λ2
+
2λ + 3 = μ (say)
Step 2 Define Q(x) = P(x) − μ then Q(λ) = 0
Step 3 Let s is the given secret. Find integers
a, b, c, d, e, f , g, , h, r satisfying 1x2
+ 2x +
( 3 − μ + s) = α[a(1 + x)2
+ b(1 + x) + c] +
Downloadedby[14.99.174.78]at03:4324May2016
4. 108 N. Chandramowliswaran et al.
β[d(1 + x)2
+ e(1 + x) + f ] + γ [g(1 + x)2
+
h(1 + x) + r] with
a d g
2a + b 2d + e 2g + h
a + b + c d + e + f g + h + r
= ±1.
Step 4 Compare the coefficients on both sides we get,
αa + βd + γ g = 1
α(2a + b) + β(2d + e) + γ (2g + h) = 2
α(a + b + c) + β(d + e + f ) + γ (g + h + r)
= 3 − μ + s.
Step 5
⎛
⎝
a d g
2a + b 2d + e 2g + h
a + b + c d + e + f g + h + r
⎞
⎠
⎛
⎝
α
β
γ
⎞
⎠
=
⎛
⎝
1
2
3 − μ + s
⎞
⎠ ,
where
⎛
⎝
a d g
2a + b 2d + e 2g + h
a + b + c d + e + f g + h + r
⎞
⎠
∈ GL3(Z),
where GL3(Z) be the set of all 3 × 3 matrices
of integer coefficients with determinant is ±1
Step 6
⎛
⎝
α
β
γ
⎞
⎠ =
⎛
⎝
a d g
2a + b 2d + e 2g + h
a + b + c d + e + f g + h + r
⎞
⎠
−1
×
⎛
⎝
1
2
3 − μ + s
⎞
⎠ ,
where α, β and γ are uniquely solved by the
above information
Step 7 Select three secret share holders P1, P2 and P3
P1 ←→ ax2
+ (2a + b)x + (a + b + c) = P1(x)
P2 ←→ dx2
+ (2d + e)x + (d + e + f ) = P2(x)
and
P3 ←→ gx2
+ (2g + h)x + (g + h + r) = P3(x)
Example 2 Secret Key Sharing using Finite Groups
Step 1 Let P = 2pr
+ 1 and Q = 2qs
+ 1, where
P, Q, p and q are very large odd primes (which
is kept secret).
Step 2 Let N = PQ
Step 3 Define G = {1 ≤ x ≤ N | (x, N) = 1}
Step 4 Let ×N be the multiplication modulo N.
Clearly (G, ×N ) forms a finite group with
O(G) = φ(N) = 4pr
qs
Step 5 Let s (given secret) be the element of G
Step 6 From finite group theory, any map , g −→
gm
is always an automorphism of G, if
(m, O(G)) = 1
Step 7 Let m = 1 + 2 + · · · + t.
Consider s = xm
s = x 1+ 2+···+ t
s = x 1
x 2
· · · x t
s = y1y2 · · · yt,
where yi = x i
(mod N), 1 ≤ i ≤ t be the indi-
vidual share holders.
Example 3 Secret Key Sharing using affine number theo-
retic functions
Step 1 Let S = {ak | 1 ≤ k ≤ N} be the given set of
distinct positive integers
Step 2 N
k=1 ak = P, where P is very large odd prime
Step 3 Clearly, ( N
j =1 aj , P) = 1 and (aj , P − aj ) =
1, ∀j , 1 ≤ j ≤ N
Step 5 Denote {0, 1, 2, . . . , N
j =1 aj − 1} = [0, N
j =1
aj − 1], then
Define fP : [0, N
j =1 aj − 1]
1−1
−−→
onto
[0, N
j =1 aj −
1] such that for each x ∈ [0, N
j =1 aj − 1],
fP(x) = Px + t (mod N
j =1 aj ) where t ∈ [0,
N
j =1 aj − 1]
Step 6 Define faj
: [0, P − aj − 1]
1−1
−−→
onto
[0, P − aj − 1]
such that for each y ∈ [0, P − aj − 1] faj
(y) =
aj y + bj (mod P − aj ) where bj ∈ [0, P −
aj − 1]
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5. Systems Science & Control Engineering: An Open Access Journal 109
Step 7 Define gaj
: [0, aj − 1]
1−1
−−→
onto
[0, aj − 1] such
that for each z ∈ [0, aj − 1] gaj
(z) = (P −
aj )z + cj (mod aj ) where cj ∈ [0, aj − 1]
Step 8 Define Yj = gaj
(z) = (P − aj )w + dj (mod aj ),
w ∈ [0, aj − 1] and ∀j , j {1, 2, . . . , N} with
(ar, as) = 1, ∀s, r ∈ {1, 2, . . . , N}.
Solve w uniquely mod N
j =1 aj
Step 9 Let S = fP(w) = Pw + t (mod N
j =1 aj ) be the
given secret
3. Algorithms
ALGORITHM 1 By means of our first secret key sharing
scheme, we execute the following hierarchy.
Step 1 Consider {pi, qi : i ∈ {1, 2, . . . , t}} be the given
distinct secrete odd primes
Step 2 Let Ni = piqi
Step 3 Pick ai such that (ai, Ni) = 1
Step 4 Choose the positive integers ei such that
(ei, (pi − 1)(qi − 1)) = 1
Step 5 Select a common secret S such that (S, Ni) =
1, i ∈ {1, 2, . . . , t}
Step 6 Define xi, i ∈ {1, 2, . . . , t} by Niy2
i + 1 = x2
i
where xi, yi be the least positive integer solution
of Niy2
+ 1 = x2
Step 7 For each i, 1 ≤ i ≤ t then construct xi ≡
aiSei
(mod Ni)
Step 8 Solve S uniquely under (mod Ni) i ∈
{1, 2 . . . , t} using CRT
Step 9 S is the common secret shared by the each share
holder xi, i ∈ {1, 2, . . . , t}
The following proposition asserts that algorithm 2 is a
nontrivial secret share holders.
PROPOSITION Let P, Q be given very large odd primes
with the following conditions
(i) P does not divides x2 and y2
(ii) Q does not divides x1 and y1
(iii) 2y2
1 ≡ −1 (mod Q) and 2y2
2 ≡ −1 (mod P)
where x1, y1, x2, y2, x3 and y3 satisfy y2
1 − Px2
1 = 1 y2
2 −
Qx2
2 = 1 y2
3 − PQx2
3 = 1 and 1 ≡ ((y1y2y3)2
+ (−P(x1
y2y3)2
) + (−Q(x2y1y3)2
)) (mod PQ) gives non-degenerate
key sharing.
ALGORITHM 2 Construction of Secret sharing by two
odd primes P and Q
Step 1 Let P, Q be given very large odd primes
Step 2 Define N = PQ
Step 3 Consider the following Pell’s equations
Px2
+ 1 = y2
(1)
Qx2
+ 1 = y2
(2) and
PQx2
+ 1 = y2
(3).
Step 4 Let (x1, y1), (x2, y2) and (x3, y3) be the least pos-
itive integral solution of (1), (2) and (3) (i.e.)
Px2
1 + 1 = y2
1 , Qx2
2 + 1 = y2
2 and PQx2
3 + 1 =
y2
3
y2
1 − Px2
1 = 1 (1)
y2
2 − Qx2
2 = 1 (2)
y2
3 − PQx2
3 = 1 (3) .
Step 5 1 = (y2
1 − Px2
1)(y2
2 − Qx2
2)(y2
3 − PQx2
3)
1 ≡ (y2
1 − Px2
1)(y2
2 − Qx2
2)y2
3 (mod PQ)
1 ≡ (y2
1 y2
2 − Px2
1y2
2 − Qx2
2y2
1 )y2
3 (mod PQ)
1 ≡ ((y1y2y3)2
− P(x1y2y3)2
− Q(x2y1y3)2
)
× (mod PQ)
1 ≡ ((y1y2y3)2
+ (−P(x1y2y3)2
)
+ (−Q(x2y1y3)2
)) (mod PQ).
Step 6 Select a secret S such that (S, PQ) = 1
Step 7
S = (S(y1y2y3)2
+ (−PS(x1y2y3)2
)
+ (−QS(x2y1y3)2
)) (mod PQ).
Step 8 Y1, Y2 and Y3 are secret share holders, where
Y1 = S(y1y2y3)2
(mod PQ),
Y2 = (−PS(x1y2y3)2
) (mod PQ) and
Y3 = (−QS(x2y1y3)2
) (mod PQ).
ALGORITHM 3 Extension of Algorithm 2 for three odd
primes P, Q and R
Step 1 Let P, Q and R be given very large odd primes
Step 2 Consider the following Pell’s equations
Px2
+ 1 = y2
(1)
Qx2
+ 1 = y2
(2)
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7. Systems Science & Control Engineering: An Open Access Journal 111
(1) There are 10 Login ID and 15 users in the given
network
(2) Any two Login IDs can be utilized by at most one
user
(3) Every Login ID is used by exactly three users
(4) Represent the Login IDs by the nodes (vertices)
of the graph G
(5) If there is a user-j using Login IDs Log IDr and
Log IDs, then join them by an edge
(6) If the two users have a common Login ID then
they are conflict users, otherwise non-conflict
users. For example, Conflict users: user-1, user-2
and user-7, they have common Login ID Log ID1
and Non-Conflict users: user-2, user-5 and user-9
(7) Define V(G) = {vi = Log IDi | 1 ≤ i ≤ 10}
Define E(G) = {k = user k | 1 ≤ k ≤ 15}
(8) Define f (vi) = f (Log IDi) = σ(i), where σ is a
permutation on the set of numbers {1, 2, . . . , 10}.
This σ(i) is given for each Log IDi
(9) Now define the graceful labeling g on the set
{σ(1), σ(2), . . . , σ(10)}
g : {σ(i) : 1 ≤ i ≤ 10} −→ {0, 1, 2, . . . , q − 1, q}.
Suppose
g[user j ] =| g(σ(r)) − g(σ(s)) |∈ {1, 2, . . . , q}
where 1 ≤ r, s ≤ 10, r = s
(10) g : E(G) −→ {1, 2, . . . , q}
(11) g is kept secret, but g[user j ] is given for each
user j
(12) g[user j ] is called user-ID
(σ(r), σ(s)) are two Login IDs for the user j
(13) Entire Network is kept secret
(14) P : V(G) −→ {p1, p2, . . . , p10} where pi, 1 ≤ i ≤
10 are distinct odd primes with q < min{pi}, 1 ≤
i ≤ 10, q < pj ∀j (P is kept secret)
g[user j ] is known 1 ≤ j ≤ 15
(15) Define ej : (ej , (pr − 1)(ps − 1)) = 1 (ej kept
secret)
(16) Define mj ≡ (g[user j ])ej
(mod prps) P[LogIDr] =
pr, P[Log IDs] = ps, 1 ≤ r, s ≤ 10, r = s
(17) Decompose the user (edges) into subset of Non-
Conflict users (set of Independent Edges)
(18)
A = {user-2, user-5, user-9, user-11, user-13} :
user-2 ←→ {Log ID1, Log ID5}
user-5 ←→ {Log ID2, Log ID3}
user-9 ←→ {Log ID4, Log ID8}
user-11 ←→ {Log ID6, Log ID9}
user-13 ←→ {Log ID7, Log ID10}
B = {user-1, user-3, user-12, user-14} :
user-1 ←→ {Log ID1, Log ID2}
user-3 ←→ {Log ID5, Log ID4}
user-12 ←→ {Log ID6, Log ID8}
user-14 ←→ {Log ID7, Log ID9}
C = {user-4, user-7, user-8, user-15} :
user-4 ←→ {Log ID3, Log ID4}
user-7 ←→ {Log ID1, Log ID6}
user-8 ←→ {Log ID5, Log ID7}
user-15 ←→ {Log ID8, Log ID10}
D = {user-6, user-10} :
user-6 ←→ {Log ID2, Log ID10}
user-10 ←→ {Log ID3, Log ID9}
(19) Define congruences equations for the set A, B, C
and D as follows
x ≡ m2 (mod p1p5)
x ≡ m5 (mod p2p3)
x ≡ m9 (mod p4p8)
x ≡ m11 (mod p6p9)
x ≡ m13 (mod p7p10)
x has a unique solution (mod p1p2 · · · p10)
Thus x is the common secret shared by the
group A Non-Conflict users
y ≡ m1 (mod p1p2)
y ≡ m3 (mod p4p5)
y ≡ m12 (mod p6p8)
y ≡ m14 (mod p7p9)
y has a unique solution (mod p1p2p4p5p6p7p8p9)
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8. 112 N. Chandramowliswaran et al.
Thus y is the common secret shared by the
group B Non-Conflict users
z ≡ m4 (mod p3p4)
z ≡ m7 (mod p1p6)
z ≡ m8 (mod p5p7)
z ≡ m15 (mod p8p10)
z has a unique solution (mod p1p3p4p5p6p7p8p10)
Thus z is the common secret shared by the group
C Non-Conflict users
w ≡ m6 (mod p2p10)
w ≡ m10 (mod p3p9)
w has a unique solution (mod p2p3p9p10)
Thus w is the common secret shared by the
group D Non-Conflict users
4. Conclusion
In the proposed system we only focused on protecting the
group key information broadcasted from the Dealer to all
the share holders in the group and the group guarantees the
confidentiality authentication of the key generated. This
confirms that the protocol is secure for both inside and
outside attack. In this paper, an algorithm is proposed for
secure key sharing. This method can be used for factor-
ization of positive integer N. The proposed tool is more
efficient key distribution algorithm used for a secret code,
since it involves more number of prime numbers. The tech-
nique used in this paper for secret sharing is to split the
secret into different primes and send it to the participat-
ing share holders in the network. Also it is not able to
decode the secret without the knowledge of all shares and
any attacker cannot identify if any one share is missing.
Hence forth one can use it for various network protocols
and it leads a opening of new developments in the field of
cryptosystems
Disclosure statement
No potential conflict of interest was reported by the authors.
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