This document summarizes an article that proposes three new secret key sharing schemes based on the Chinese Remainder Theorem (CRT). It begins by providing background on CRT and secret sharing schemes. It then presents the main result, which is three theorems and algorithms for authenticated key distribution using a given set of primes. The first theorem describes how to construct three secret shares from a secret S such that combining the shares recovers S. It proves this using a lemma about finding integers that satisfy a system of congruences. The next sections provide examples and algorithms to motivate the secret sharing schemes. In summary, the document presents new methods for secret sharing based on number theory and the CRT.
This document proposes using a genetic algorithm approach to parallelize cryptographic algorithms and identify encryption keys. It describes generating random numbers using a linear congruential equation, then applying crossover and mutation operators from genetic algorithms to the numbers. The encrypted data and key are transmitted over the network. Decryption reverses the encryption process. Testing on different core machines showed the parallelized encryption had faster execution times than serial encryption, with greater speedups on more cores. The authors conclude the genetic algorithm operators improve performance and security compared to other algorithms.
BREAKING MIGNOTTE’S SEQUENCE BASED SECRET SHARING SCHEME USING SMT SOLVERijcsit
This document summarizes a research paper that proposes a new method for reconstructing secrets from secret sharing schemes using an SMT solver with one less than the threshold number of shares. It introduces Mignotte's sequence-based secret sharing, describes how shares are generated and distributed, and how an SMT solver is used to check the satisfiability of logical formulas representing constraints on the shares to reconstruct the secret with one less than the threshold. The method is demonstrated with an example and results showing the secret is successfully reconstructed from two shares out of a threshold of three.
Image encryption using chaotic sequence and its cryptanalysisIOSR Journals
1) The document analyzes an image encryption algorithm that uses chaotic sequences. It finds that the algorithm can be broken with only a small number of known or chosen plaintexts using two attacks.
2) A chosen plaintext attack is described that requires only one known plaintext and two chosen plaintexts to reveal the secret chaotic sequences and encryption keys.
3) A known plaintext attack is also introduced that requires two known plaintext-ciphertext pairs to determine the secret parameters and completely break the encryption scheme.
The Architecture of Cloud Storage Model Based On Confusion Theoryinventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document proposes a secure hash function for fingerprints that achieves a balance between security guarantees and matching accuracy. The hash function applies an off-the-shelf cryptographic hash to triplets of "minutia triangles" extracted from fingerprints. This exploits the geometry of fingerprints while maintaining security. However, initial experiments showed the scheme was not secure against brute-force attacks. The authors overcome this by hashing triplets of triangles instead of individual triangles, increasing the search space. They analyze the security of the updated scheme and evaluate its matching performance on standard fingerprint datasets.
This document compares the k-means and grid density clustering algorithms. K-means partitions data into k clusters based on minimizing distances between points and cluster centroids. It works well with numerical data but can be affected by outliers. Grid density determines dense grids based on neighbor densities and can handle different shaped and multi-density clusters without knowing the number of clusters beforehand. It has advantages over k-means in that it can handle categorical data, noise and arbitrary shaped clusters.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
Extended pso algorithm for improvement problems k means clustering algorithmIJMIT JOURNAL
The clustering is a without monitoring process and one of the most common data mining techniques. The
purpose of clustering is grouping similar data together in a group, so were most similar to each other in a
cluster and the difference with most other instances in the cluster are. In this paper we focus on clustering
partition k-means, due to ease of implementation and high-speed performance of large data sets, After 30
year it is still very popular among the developed clustering algorithm and then for improvement problem of
placing of k-means algorithm in local optimal, we pose extended PSO algorithm, that its name is ECPSO.
Our new algorithm is able to be cause of exit from local optimal and with high percent produce the
problem’s optimal answer. The probe of results show that mooted algorithm have better performance
regards as other clustering algorithms specially in two index, the carefulness of clustering and the quality
of clustering.
This document proposes using a genetic algorithm approach to parallelize cryptographic algorithms and identify encryption keys. It describes generating random numbers using a linear congruential equation, then applying crossover and mutation operators from genetic algorithms to the numbers. The encrypted data and key are transmitted over the network. Decryption reverses the encryption process. Testing on different core machines showed the parallelized encryption had faster execution times than serial encryption, with greater speedups on more cores. The authors conclude the genetic algorithm operators improve performance and security compared to other algorithms.
BREAKING MIGNOTTE’S SEQUENCE BASED SECRET SHARING SCHEME USING SMT SOLVERijcsit
This document summarizes a research paper that proposes a new method for reconstructing secrets from secret sharing schemes using an SMT solver with one less than the threshold number of shares. It introduces Mignotte's sequence-based secret sharing, describes how shares are generated and distributed, and how an SMT solver is used to check the satisfiability of logical formulas representing constraints on the shares to reconstruct the secret with one less than the threshold. The method is demonstrated with an example and results showing the secret is successfully reconstructed from two shares out of a threshold of three.
Image encryption using chaotic sequence and its cryptanalysisIOSR Journals
1) The document analyzes an image encryption algorithm that uses chaotic sequences. It finds that the algorithm can be broken with only a small number of known or chosen plaintexts using two attacks.
2) A chosen plaintext attack is described that requires only one known plaintext and two chosen plaintexts to reveal the secret chaotic sequences and encryption keys.
3) A known plaintext attack is also introduced that requires two known plaintext-ciphertext pairs to determine the secret parameters and completely break the encryption scheme.
The Architecture of Cloud Storage Model Based On Confusion Theoryinventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document proposes a secure hash function for fingerprints that achieves a balance between security guarantees and matching accuracy. The hash function applies an off-the-shelf cryptographic hash to triplets of "minutia triangles" extracted from fingerprints. This exploits the geometry of fingerprints while maintaining security. However, initial experiments showed the scheme was not secure against brute-force attacks. The authors overcome this by hashing triplets of triangles instead of individual triangles, increasing the search space. They analyze the security of the updated scheme and evaluate its matching performance on standard fingerprint datasets.
This document compares the k-means and grid density clustering algorithms. K-means partitions data into k clusters based on minimizing distances between points and cluster centroids. It works well with numerical data but can be affected by outliers. Grid density determines dense grids based on neighbor densities and can handle different shaped and multi-density clusters without knowing the number of clusters beforehand. It has advantages over k-means in that it can handle categorical data, noise and arbitrary shaped clusters.
Comparison Between Clustering Algorithms for Microarray Data AnalysisIOSR Journals
Currently, there are two techniques used for large-scale gene-expression profiling; microarray and
RNA-Sequence (RNA-Seq).This paper is intended to study and compare different clustering algorithms that used
in microarray data analysis. Microarray is a DNA molecules array which allows multiple hybridization
experiments to be carried out simultaneously and trace expression levels of thousands of genes. It is a highthroughput
technology for gene expression analysis and becomes an effective tool for biomedical research.
Microarray analysis aims to interpret the data produced from experiments on DNA, RNA, and protein
microarrays, which enable researchers to investigate the expression state of a large number of genes. Data
clustering represents the first and main process in microarray data analysis. The k-means, fuzzy c-mean, selforganizing
map, and hierarchical clustering algorithms are under investigation in this paper. These algorithms
are compared based on their clustering model.
Extended pso algorithm for improvement problems k means clustering algorithmIJMIT JOURNAL
The clustering is a without monitoring process and one of the most common data mining techniques. The
purpose of clustering is grouping similar data together in a group, so were most similar to each other in a
cluster and the difference with most other instances in the cluster are. In this paper we focus on clustering
partition k-means, due to ease of implementation and high-speed performance of large data sets, After 30
year it is still very popular among the developed clustering algorithm and then for improvement problem of
placing of k-means algorithm in local optimal, we pose extended PSO algorithm, that its name is ECPSO.
Our new algorithm is able to be cause of exit from local optimal and with high percent produce the
problem’s optimal answer. The probe of results show that mooted algorithm have better performance
regards as other clustering algorithms specially in two index, the carefulness of clustering and the quality
of clustering.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Encryption is a technique that transforms a code from an understandable into an incomprehensible code. Many methods can be applied to an encryption process. One such method is RSA. RSA works by appointing on byte values. The value is obtained from character conversion to ASCII code. This algorithm is based on the multiplication of two relatively large primes. Applications of the RSA algorithm can be used in data security. This research provides RSA algorithm application on data security system that can guarantee data confidentiality. RSA algorithm is known as a very secure algorithm. This algorithm works with the number of bits in the search for prime numbers. The larger the bits, the less chance of ciphertext can be solved. The weakness of this method is the amount of ciphertext capacity will be floating in line with the number of prime numbers used. Also, to perform the process of encryption and decryption, RSA requires a relatively long time than other algorithms. The advantage of RSA is that complicated ciphertext is solved into plaintext.
DATA SECURITY USING PRIVATE KEY ENCRYPTION SYSTEM BASED ON ARITHMETIC CODINGIJNSA Journal
Problem faced by today’s communicators is not only security but also the speed of communication and size of content.In the present paper, a scheme has been proposed which uses the concept of compression and data encryption. In first phase the focus has been made on data compression and cryptography. In the next phase we have emphasized on compression cryptosystem. Finally, proposed technique has been discussed which used the concept of data compression and encryption. In this first data is compressed to reduce the size of the data and increase the data transfer rate. Thereafter compress data is encrypted to provide security. Hence our proposed technique is effective that can reduce data size, increase data transfer rate and provide the security during communication.
This presentation introduces clustering analysis and the k-means clustering technique. It defines clustering as an unsupervised method to segment data into groups with similar traits. The presentation outlines different clustering types (hard vs soft), techniques (partitioning, hierarchical, etc.), and describes the k-means algorithm in detail through multiple steps. It discusses requirements for clustering, provides examples of applications, and reviews advantages and disadvantages of k-means clustering.
Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm.CSCJournals
The document proposes novel methods for generating self-invertible matrices for use in the Hill cipher encryption algorithm. It discusses how the Hill cipher works and the issue that the encryption matrix may not be invertible, preventing decryption. It then presents three methods for generating self-invertible matrices of sizes 2x2, 3x3, and 4x4 that can be used as the encryption key matrix in Hill cipher to allow for decryption without needing to calculate the inverse matrix.
A mathematical model of access control in big data using confidence interval ...csandit
Nowadays, the concept of big data grows incessantly
; recent researches proved that 90% of the
whole data existed on the web had been created in l
ast two years. However, this growing
bumped by many critical challenges resides generall
y in security level; the users care about
how could providers protect their privacy on their
data. Access control, cryptography, and de-
identification are the main search areas grouped un
der a specific domain known as Privacy
Preserving Data Publishing. In this paper, we bring
in suggestion a new model for access
control over big data using digital signature and c
onfidence interval; we first introduce our
work by presenting some general concepts used to bu
ild our approach then presenting the idea
of this report and finally we evaluate our system b
y conducting several experiments and
showing and discussing the results that we got
Highly secure scalable compression of encrypted imageseSAT Journals
Abstract A highly secure scalable compression method for stream cipher encrypted images is described in this journal. The input image first undergoes encryption and then shuffling. This shuffling in the image pixels enhances the security. For shuffling, Henon map is used. There are two layers for the scalable compression namely base layer and enhancement layer. Base layer bits are produced by coding a series of non-overlapping patches of uniformly down sampled version of encrypted image. In the enhancement layer pixels are selected by random permutation and then coded. From all the available pixel samples an iterative multi scale technique is used to reconstruct the image and finally performs decryption. The proposed method has high security. Key Words: Encryption, Decryption, Shuffling, Scalable compression
FAST DETECTION OF DDOS ATTACKS USING NON-ADAPTIVE GROUP TESTINGIJNSA Journal
This document proposes a method for fast detection of DDoS attacks using non-adaptive group testing (NAGT). It begins with background on DDoS attacks and group testing techniques. It then describes using a strongly explicit d-disjunct matrix in NAGT to map IP addresses to "tests" performed by routers. The router counters would indicate potential hot items (attackers or victims). Two decoding algorithms are presented to identify the hot items from the test results with poly-log time complexity meeting data stream requirements. The method aims to provide early warning of DDoS attacks through efficient group testing of IP packets.
Symmetric-Key Based Privacy-Preserving Scheme For Mining Support Countsacijjournal
In this paper we study the problem of mining support counts using symmetric-key crypto which is more
efficient than previous work. Consider a scenario that each user has an option (like or unlike) of the
specified product, and a third party wants to obtain the popularity of this product. We design a much more
efficient privacy-preserving scheme for users to prevent the loss of the personal interests. Unlike most
previous works, we do not use any exponential or modular algorithms, but we provide a symmetric-key
based method which can also protect the information. Specifically, our protocol uses a third party that
generates a number of matrixes as each user’s key. Then user uses these key to encrypt their data which is
more efficient to obtain the support counts of a given pattern.
A Survey on Privacy-Preserving Data Aggregation Without Secure ChannelIRJET Journal
This document summarizes a research paper on privacy-preserving data aggregation without a secure channel. It discusses two models for aggregating private data from multiple participants: one with an external aggregator and one where participants calculate the aggregation jointly. The paper proposes protocols for the aggregator or participants to calculate the sum and product of the private data in a way that preserves the privacy of each participant's data, without requiring secure pairwise channels between participants. The protocols are based on the computational hardness of solving certain cryptographic problems like the discrete logarithm problem.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
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.
This document proposes a secure scheme for secret sharing and key distribution using Pell's equation. It begins with an introduction to key distribution problems and secret sharing schemes. It then presents an algorithm that uses Pell's equation to generate a secret exponent S, which is used to distribute shares of a key to users. Each user's share allows them to collectively compute the key when a minimum number of users combine their shares, but individual shares reveal no information about the key. The algorithm provides perfectly secure secret sharing against coalitions of k or fewer users.
This document proposes a secure scheme for secret sharing and key distribution using Pell's equation. It begins with an introduction to key distribution problems and secret sharing schemes. It then presents an algorithm that uses Pell's equation to generate a secret exponent S, which is used to distribute shares of a key to users. Each user's share allows them to collectively compute the key when a minimum number of users combine their shares, but individual shares reveal no information about the key. The algorithm provides perfectly secure secret sharing against coalitions of k or fewer users.
The secret sharing schemes are the important tools in cryptography that are used as building blocks in many secured protocols. It is a method used for distributing a secret among the participants in a manner that only the threshold number of participants together can recover the secret and the remaining set of participants cannot get any information about the secret. Secret sharing schemes are absolute for storing highly sensitive and important information. In a secret sharing scheme, a secret is divided into several shares. These shares are then distributed to the participants’ one each and thus only the threshold (t) number of participants can recover the secret. In this paper we have used Mignotte’s Sequence based Secret Sharing for distribution of shares to the participants. A (k, m) Mignotte's sequence is a sequence of pair wise co-prime positive integers. We have proposed a new method for reconstruction of secret even with t-1 shares using the SMT solver.
The secret sharing schemes are the important tools in cryptography that are used as building blocks in many secured protocols. It is a method used for distributing a secret among the participants in a manner that only the threshold number of participants together can recover the secret and the remaining set of participants cannot get any information about the secret. Secret sharing schemes are absolute for storing highly sensitive and important information. In a secret sharing scheme, a secret is divided into several shares. These shares are then distributed to the participants’ one each and thus only the threshold (t) number of participants can recover the secret. In this paper we have used Mignotte’s Sequence based Secret Sharing for distribution of shares to the participants. A (k, m) Mignotte's sequence is a sequence of pair wise co-prime positive integers. We have proposed a new method for reconstruction of secret even with t-1 shares using the SMT solver.
A MATHEMATICAL MODEL OF ACCESS CONTROL IN BIG DATA USING CONFIDENCE INTERVAL ...cscpconf
- The document proposes a new model for access control over big data using digital signatures and confidence intervals. It involves a multi-step process of 1) identifying users hierarchically, 2) normalizing identities, 3) computing confidence intervals for each group, 4) computing digital signatures for each user, and 5) defining an access control matrix based on these computations.
- The model utilizes mathematical concepts such as standard deviation, confidence intervals, and primitive roots. Standard deviations are used to compute confidence intervals for each group's identity range. Primitive roots are used to uniquely generate digital signatures for each user.
- The goal is to provide access control while preserving user privacy over large datasets where direct control is lost, by bas
To allot secrecy-safe association rules mining schema using FP treeUvaraj Shan
This document proposes a secure frequent-pattern tree (FP-tree) based scheme to preserve private information while doing collaborative association rules mining between multiple parties. The scheme uses attribute-based encryption to create a global FP-tree for each party and homomorphic encryption to merge the FP-trees to obtain the final global association rules results without revealing individual transaction data. The scheme is proven to be secure and collusion-resistant against up to n-1 colluding parties attempting to learn honest respondents' private data or responses.
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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Encryption is a technique that transforms a code from an understandable into an incomprehensible code. Many methods can be applied to an encryption process. One such method is RSA. RSA works by appointing on byte values. The value is obtained from character conversion to ASCII code. This algorithm is based on the multiplication of two relatively large primes. Applications of the RSA algorithm can be used in data security. This research provides RSA algorithm application on data security system that can guarantee data confidentiality. RSA algorithm is known as a very secure algorithm. This algorithm works with the number of bits in the search for prime numbers. The larger the bits, the less chance of ciphertext can be solved. The weakness of this method is the amount of ciphertext capacity will be floating in line with the number of prime numbers used. Also, to perform the process of encryption and decryption, RSA requires a relatively long time than other algorithms. The advantage of RSA is that complicated ciphertext is solved into plaintext.
DATA SECURITY USING PRIVATE KEY ENCRYPTION SYSTEM BASED ON ARITHMETIC CODINGIJNSA Journal
Problem faced by today’s communicators is not only security but also the speed of communication and size of content.In the present paper, a scheme has been proposed which uses the concept of compression and data encryption. In first phase the focus has been made on data compression and cryptography. In the next phase we have emphasized on compression cryptosystem. Finally, proposed technique has been discussed which used the concept of data compression and encryption. In this first data is compressed to reduce the size of the data and increase the data transfer rate. Thereafter compress data is encrypted to provide security. Hence our proposed technique is effective that can reduce data size, increase data transfer rate and provide the security during communication.
This presentation introduces clustering analysis and the k-means clustering technique. It defines clustering as an unsupervised method to segment data into groups with similar traits. The presentation outlines different clustering types (hard vs soft), techniques (partitioning, hierarchical, etc.), and describes the k-means algorithm in detail through multiple steps. It discusses requirements for clustering, provides examples of applications, and reviews advantages and disadvantages of k-means clustering.
Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm.CSCJournals
The document proposes novel methods for generating self-invertible matrices for use in the Hill cipher encryption algorithm. It discusses how the Hill cipher works and the issue that the encryption matrix may not be invertible, preventing decryption. It then presents three methods for generating self-invertible matrices of sizes 2x2, 3x3, and 4x4 that can be used as the encryption key matrix in Hill cipher to allow for decryption without needing to calculate the inverse matrix.
A mathematical model of access control in big data using confidence interval ...csandit
Nowadays, the concept of big data grows incessantly
; recent researches proved that 90% of the
whole data existed on the web had been created in l
ast two years. However, this growing
bumped by many critical challenges resides generall
y in security level; the users care about
how could providers protect their privacy on their
data. Access control, cryptography, and de-
identification are the main search areas grouped un
der a specific domain known as Privacy
Preserving Data Publishing. In this paper, we bring
in suggestion a new model for access
control over big data using digital signature and c
onfidence interval; we first introduce our
work by presenting some general concepts used to bu
ild our approach then presenting the idea
of this report and finally we evaluate our system b
y conducting several experiments and
showing and discussing the results that we got
Highly secure scalable compression of encrypted imageseSAT Journals
Abstract A highly secure scalable compression method for stream cipher encrypted images is described in this journal. The input image first undergoes encryption and then shuffling. This shuffling in the image pixels enhances the security. For shuffling, Henon map is used. There are two layers for the scalable compression namely base layer and enhancement layer. Base layer bits are produced by coding a series of non-overlapping patches of uniformly down sampled version of encrypted image. In the enhancement layer pixels are selected by random permutation and then coded. From all the available pixel samples an iterative multi scale technique is used to reconstruct the image and finally performs decryption. The proposed method has high security. Key Words: Encryption, Decryption, Shuffling, Scalable compression
FAST DETECTION OF DDOS ATTACKS USING NON-ADAPTIVE GROUP TESTINGIJNSA Journal
This document proposes a method for fast detection of DDoS attacks using non-adaptive group testing (NAGT). It begins with background on DDoS attacks and group testing techniques. It then describes using a strongly explicit d-disjunct matrix in NAGT to map IP addresses to "tests" performed by routers. The router counters would indicate potential hot items (attackers or victims). Two decoding algorithms are presented to identify the hot items from the test results with poly-log time complexity meeting data stream requirements. The method aims to provide early warning of DDoS attacks through efficient group testing of IP packets.
Symmetric-Key Based Privacy-Preserving Scheme For Mining Support Countsacijjournal
In this paper we study the problem of mining support counts using symmetric-key crypto which is more
efficient than previous work. Consider a scenario that each user has an option (like or unlike) of the
specified product, and a third party wants to obtain the popularity of this product. We design a much more
efficient privacy-preserving scheme for users to prevent the loss of the personal interests. Unlike most
previous works, we do not use any exponential or modular algorithms, but we provide a symmetric-key
based method which can also protect the information. Specifically, our protocol uses a third party that
generates a number of matrixes as each user’s key. Then user uses these key to encrypt their data which is
more efficient to obtain the support counts of a given pattern.
A Survey on Privacy-Preserving Data Aggregation Without Secure ChannelIRJET Journal
This document summarizes a research paper on privacy-preserving data aggregation without a secure channel. It discusses two models for aggregating private data from multiple participants: one with an external aggregator and one where participants calculate the aggregation jointly. The paper proposes protocols for the aggregator or participants to calculate the sum and product of the private data in a way that preserves the privacy of each participant's data, without requiring secure pairwise channels between participants. The protocols are based on the computational hardness of solving certain cryptographic problems like the discrete logarithm problem.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
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.
This document proposes a secure scheme for secret sharing and key distribution using Pell's equation. It begins with an introduction to key distribution problems and secret sharing schemes. It then presents an algorithm that uses Pell's equation to generate a secret exponent S, which is used to distribute shares of a key to users. Each user's share allows them to collectively compute the key when a minimum number of users combine their shares, but individual shares reveal no information about the key. The algorithm provides perfectly secure secret sharing against coalitions of k or fewer users.
This document proposes a secure scheme for secret sharing and key distribution using Pell's equation. It begins with an introduction to key distribution problems and secret sharing schemes. It then presents an algorithm that uses Pell's equation to generate a secret exponent S, which is used to distribute shares of a key to users. Each user's share allows them to collectively compute the key when a minimum number of users combine their shares, but individual shares reveal no information about the key. The algorithm provides perfectly secure secret sharing against coalitions of k or fewer users.
The secret sharing schemes are the important tools in cryptography that are used as building blocks in many secured protocols. It is a method used for distributing a secret among the participants in a manner that only the threshold number of participants together can recover the secret and the remaining set of participants cannot get any information about the secret. Secret sharing schemes are absolute for storing highly sensitive and important information. In a secret sharing scheme, a secret is divided into several shares. These shares are then distributed to the participants’ one each and thus only the threshold (t) number of participants can recover the secret. In this paper we have used Mignotte’s Sequence based Secret Sharing for distribution of shares to the participants. A (k, m) Mignotte's sequence is a sequence of pair wise co-prime positive integers. We have proposed a new method for reconstruction of secret even with t-1 shares using the SMT solver.
The secret sharing schemes are the important tools in cryptography that are used as building blocks in many secured protocols. It is a method used for distributing a secret among the participants in a manner that only the threshold number of participants together can recover the secret and the remaining set of participants cannot get any information about the secret. Secret sharing schemes are absolute for storing highly sensitive and important information. In a secret sharing scheme, a secret is divided into several shares. These shares are then distributed to the participants’ one each and thus only the threshold (t) number of participants can recover the secret. In this paper we have used Mignotte’s Sequence based Secret Sharing for distribution of shares to the participants. A (k, m) Mignotte's sequence is a sequence of pair wise co-prime positive integers. We have proposed a new method for reconstruction of secret even with t-1 shares using the SMT solver.
A MATHEMATICAL MODEL OF ACCESS CONTROL IN BIG DATA USING CONFIDENCE INTERVAL ...cscpconf
- The document proposes a new model for access control over big data using digital signatures and confidence intervals. It involves a multi-step process of 1) identifying users hierarchically, 2) normalizing identities, 3) computing confidence intervals for each group, 4) computing digital signatures for each user, and 5) defining an access control matrix based on these computations.
- The model utilizes mathematical concepts such as standard deviation, confidence intervals, and primitive roots. Standard deviations are used to compute confidence intervals for each group's identity range. Primitive roots are used to uniquely generate digital signatures for each user.
- The goal is to provide access control while preserving user privacy over large datasets where direct control is lost, by bas
To allot secrecy-safe association rules mining schema using FP treeUvaraj Shan
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This document summarizes a paper that proposes novel secret sharing and key distribution schemes based on number theory. It introduces generalized secret sharing schemes and key distribution schemes that are useful for multi-party systems. The paper then presents two main results: 1) A lemma showing that there exist integers that satisfy a particular congruence relation involving three distinct primes. 2) A theorem demonstrating how to generate secret shares of a secret S among three shareholders such that combining the shares recovers S. The scheme is based on the lemma and uses modular arithmetic with a modulus composed of three large primes.
This document summarizes a paper that proposes novel secret sharing and key distribution schemes based on number theory. It introduces generalized secret sharing schemes and key distribution schemes that are useful for multi-party systems. The paper then presents two main results: 1) A lemma showing that there exist integers that satisfy a particular congruence relation involving three distinct primes. 2) A theorem demonstrating how to generate secret shares of a secret S among three shareholders such that combining the shares recovers S. The scheme is based on the lemma and uses modular arithmetic with a modulus composed of three large primes.
This document provides the schedule for a 7-day summer course on basic engineering mathematics, discrete mathematics, and graph theory held from July 6-13, 2009 at VIT University. The course schedule lists the daily topics to be covered from 9:45am-4:15pm, including lectures on graph theory, advanced calculus, formal languages and automata, probability and random processes, mathematical modeling, and programming fundamentals. Various professors from VIT University and other institutions are listed as lecturers for the different topics. The course is coordinated by Dr. N. Chandramowliswaran and overseen by the director Dr. K. Sathiyanarayanan.
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3) Defining the public key α using Y0, X0, R, and the Euler totient function φ(n).
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Authenticated key distribution using given set of
primes for secret sharing
N. Chandramowliswaran
a
, S. Srinivasan
b
& P. Muralikrishna
b
a
Department of Applied Sciences, ITM University, Gurgaon-122017, Haryana, India
b
School of Advanced Sciences, VIT University, Vellore – 632014, India
Accepted author version posted online: 16 Dec 2014.Published online: 14 Jan 2015.
To cite this article: N. Chandramowliswaran, S. Srinivasan & P. Muralikrishna (2015) Authenticated key distribution using
given set of primes for secret sharing, Systems Science & Control Engineering: An Open Access Journal, 3:1, 106-112, DOI:
10.1080/21642583.2014.985803
To link to this article: http://dx.doi.org/10.1080/21642583.2014.985803
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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.
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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[117.193.201.172]at12:1623January2015
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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