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.
This document presents an improved asymmetric key encryption algorithm using MATLAB. It begins with an introduction to asymmetric key cryptography and the RSA cryptosystem. It then describes a modified RSA algorithm using multiple public and private keys to increase security. Next, it explains how to implement RSA using the Chinese Remainder Theorem to reduce computational time. The document implements the original, modified, and CRT-based RSA algorithms in MATLAB and analyzes computation time versus number of prime numbers. It concludes the modified and CRT-based approaches provide more security than the original RSA algorithm with reduced computational time.
BLIND SIGNATURE SCHEME BASED ON CHEBYSHEV POLYNOMIALSIJNSA Journal
A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair.This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms of Chebyshev polynomials.
This document summarizes a research paper that proposes a new public key cryptosystem based on the difficulty of inverting the function F(x) = (a × x)Mod(2p)Div(2q). The cryptosystem includes a key exchange algorithm, public key encryption algorithm, and digital signature algorithm. The document analyzes the efficiency and security of the cryptosystem, showing it has O(n) faster time complexity than RSA and Diffie-Hellman. It also reduces breaking the cryptosystem to solving difficult SAT instances or sets of multivariate polynomial equations over F(2). Python implementations of the key exchange and signature algorithms are provided in appendices.
This document discusses public key cryptography and the RSA encryption algorithm. It provides an overview of public key cryptography, how the RSA algorithm works using a public and private key pair, and some of its applications. The RSA algorithm is based on the difficulty of factoring large prime numbers and allows for secure communication without needing to share secret keys. Some advantages are convenience and enabling message authentication and non-repudiation using digital signatures, while disadvantages include slower performance and the need to authenticate public keys.
The document discusses encryption and decryption techniques including symmetric and asymmetric cryptosystems. It describes the goals of cryptography including confidentiality, integrity, authentication, and non-repudiation. The document outlines the RSA cryptosystem including key generation, encryption, and digital signatures. It also discusses hashing, the discrete logarithm problem, and how elliptic curves can be used in cryptography.
This document proposes a method for hiding secret data in encrypted and compressed video bitstreams for privacy protection. It involves encrypting the video using standard encryption keys, compressing it using H.264 encoding, and then embedding additional encrypted data into the encrypted video stream using bit wrapping without accessing the original video content. At the receiver end, the hidden data can be extracted from the encrypted or decrypted video stream. The performance is evaluated based on mean square error, peak signal-to-noise ratio, and correlation coefficient between the original and embedded video streams. The method aims to protect video privacy during transmission and storage while allowing hidden data to be embedded and extracted.
This document discusses ring-based homomorphic encryption schemes and compares the efficiency of four schemes: BGV, FV, NTRU, and YASHE. The schemes are analyzed by measuring ciphertext size under varying parameters like plaintext modulus size and circuit depth. For small plaintext sizes, YASHE is most efficient, but BGV generally performs best as plaintext size increases. The analysis provides a starting point for comparing ring-based schemes but could be improved with a stricter security analysis.
Lattice Based Cryptography - GGH CryptosystemVarun Janga
This document discusses lattice-based cryptography and the GGH cryptosystem. It provides an overview of lattices and their properties. The GGH cryptosystem is based on the closest vector problem in lattices. The private key is a good basis for a lattice, while the public key is a bad basis for the same lattice. The document describes the key generation process and analyzes attacks on the GGH cryptosystem such as the embedding attack and Nguyen's attack based on leaking remainders. It also discusses advantages and disadvantages of lattice-based cryptography.
This document presents an improved asymmetric key encryption algorithm using MATLAB. It begins with an introduction to asymmetric key cryptography and the RSA cryptosystem. It then describes a modified RSA algorithm using multiple public and private keys to increase security. Next, it explains how to implement RSA using the Chinese Remainder Theorem to reduce computational time. The document implements the original, modified, and CRT-based RSA algorithms in MATLAB and analyzes computation time versus number of prime numbers. It concludes the modified and CRT-based approaches provide more security than the original RSA algorithm with reduced computational time.
BLIND SIGNATURE SCHEME BASED ON CHEBYSHEV POLYNOMIALSIJNSA Journal
A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair.This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms of Chebyshev polynomials.
This document summarizes a research paper that proposes a new public key cryptosystem based on the difficulty of inverting the function F(x) = (a × x)Mod(2p)Div(2q). The cryptosystem includes a key exchange algorithm, public key encryption algorithm, and digital signature algorithm. The document analyzes the efficiency and security of the cryptosystem, showing it has O(n) faster time complexity than RSA and Diffie-Hellman. It also reduces breaking the cryptosystem to solving difficult SAT instances or sets of multivariate polynomial equations over F(2). Python implementations of the key exchange and signature algorithms are provided in appendices.
This document discusses public key cryptography and the RSA encryption algorithm. It provides an overview of public key cryptography, how the RSA algorithm works using a public and private key pair, and some of its applications. The RSA algorithm is based on the difficulty of factoring large prime numbers and allows for secure communication without needing to share secret keys. Some advantages are convenience and enabling message authentication and non-repudiation using digital signatures, while disadvantages include slower performance and the need to authenticate public keys.
The document discusses encryption and decryption techniques including symmetric and asymmetric cryptosystems. It describes the goals of cryptography including confidentiality, integrity, authentication, and non-repudiation. The document outlines the RSA cryptosystem including key generation, encryption, and digital signatures. It also discusses hashing, the discrete logarithm problem, and how elliptic curves can be used in cryptography.
This document proposes a method for hiding secret data in encrypted and compressed video bitstreams for privacy protection. It involves encrypting the video using standard encryption keys, compressing it using H.264 encoding, and then embedding additional encrypted data into the encrypted video stream using bit wrapping without accessing the original video content. At the receiver end, the hidden data can be extracted from the encrypted or decrypted video stream. The performance is evaluated based on mean square error, peak signal-to-noise ratio, and correlation coefficient between the original and embedded video streams. The method aims to protect video privacy during transmission and storage while allowing hidden data to be embedded and extracted.
This document discusses ring-based homomorphic encryption schemes and compares the efficiency of four schemes: BGV, FV, NTRU, and YASHE. The schemes are analyzed by measuring ciphertext size under varying parameters like plaintext modulus size and circuit depth. For small plaintext sizes, YASHE is most efficient, but BGV generally performs best as plaintext size increases. The analysis provides a starting point for comparing ring-based schemes but could be improved with a stricter security analysis.
Lattice Based Cryptography - GGH CryptosystemVarun Janga
This document discusses lattice-based cryptography and the GGH cryptosystem. It provides an overview of lattices and their properties. The GGH cryptosystem is based on the closest vector problem in lattices. The private key is a good basis for a lattice, while the public key is a bad basis for the same lattice. The document describes the key generation process and analyzes attacks on the GGH cryptosystem such as the embedding attack and Nguyen's attack based on leaking remainders. It also discusses advantages and disadvantages of lattice-based cryptography.
Apresentação sobre Criptografia baseada em reticulados (lattices), realizada no contexto da disciplina de Post-Quantum Cryptography do PPGCC da UFSC.
Versão odp: http://coenc.td.utfpr.edu.br/~giron/presentations/aula_lattice.odp
Information and network security 33 rsa algorithmVaibhav Khanna
RSA algorithm is asymmetric cryptography algorithm. Asymmetric actually means that it works on two different keys i.e. Public Key and Private Key. As the name describes that the Public Key is given to everyone and Private key is kept private
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.
Cloud computing is an ever-growing field in today‘s era.With the accumulation of data and the
advancement of technology,a large amount of data is generated everyday.Storage, availability and security of
the data form major concerns in the field of cloud computing.This paper focuses on homomorphic encryption,
which is largely used for security of data in the cloud.Homomorphic encryption is defined as the technique of
encryption in which specific operations can be carried out on the encrypted data.The data is stored on a remote
server.The task here is operating on the encrypted data.There are two types of homomorphic encryption, Fully
homomorphic encryption and patially homomorphic encryption.Fully homomorphic encryption allow arbitrary
computation on the ciphertext in a ring, while the partially homomorphic encryption is the one in which
addition or multiplication operations can be carried out on the normal ciphertext.Homomorphic encryption
plays a vital role in cloud computing as the encrypted data of companies is stored in a public cloud, thus taking
advantage of the cloud provider‘s services.Various algorithms and methods of homomorphic encryption that
have been proposed are discussed in this paper
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
This document provides an overview of cyclic redundancy checks (CRC). It defines CRC as treating bit strings as polynomial coefficients to detect errors. The process involves generating a message polynomial from the data and a generator polynomial. A CRC code is calculated by dividing the message polynomial by the generator polynomial. This CRC code is sent along with the data. On receipt, the data and CRC code are divided by the generator polynomial again. If the remainder is zero, no error is detected. The document gives an example of calculating CRC and discusses the properties of CRC, including its ability to detect single, double, and short burst errors.
Image encryption using elliptical curve cryptosytem with hill cipherkarthik kedarisetti
IMAGE ENCRYPTION-BTECH FINAL YEAR PROJECT ZEROTH REVIEW.
Image encryption is rapidly increased recently by the increasing use of the internet and communication
media. Sharing important images over unsecured channels is liable for attacking and stealing. Encryption
techniques are the suitable methods to protect images from attacks. Hill cipher algorithm is one of the
symmetric techniques, it has a simple structure and fast computations, but weak security because sender
and receiver need to use and share the same private key within a non-secure channels. A new image
encryption technique that combines Elliptic Curve Cryptosystem with Hill Cipher (ECCHC) has been proposed
in this paper to convert Hill cipher from symmetric technique to asymmetric one and increase its
security and efficiency and resist the hackers. Self-invertible key matrix is used to generate encryption
and decryption secret key. So, no need to find the inverse key matrix in the decryption process. A secret
key matrix with dimensions 4 4 will be used as an example in this study. Entropy, Peak Signal to Noise
Ratio (PSNR), and Unified Average Changing Intensity (UACI) will be used to assess the grayscale image
encryption efficiency and compare the encrypted image with the original image to evaluate the performance
of the proposed encryption technique.
Information security is one of the most important issues in the
recent times. Elliptic Curve Cryptography (ECC) is one of the most
efficient public key cryptosystems that is secured against adversaries
because it is hard for them to find the secret key and solve
the elliptic curve discrete logarithm problem. Its strengthened
security also comes from the small key size that is used in it with
the same level of safety compared to the other cryptosystems like RSA(Rivest–Shamir–Adleman))
IRJET- Data Analysis for Braking System in Time Domain for Fault DiagnosisIRJET Journal
This document discusses securing cloud data when encryption keys are exposed. It presents a system that distributes encrypted data across multiple cloud storage servers administered by different entities. This is done to limit an attacker's access even if they acquire the encryption key, as they would not be able to compromise every server. The paper studies data confidentiality against an adversary who knows the encryption key and has access to most but not all of the encrypted data blocks. It proposes a new security definition to capture this threat and evaluates a prototype implementation to measure performance overhead.
Cryptosystem An Implementation of RSA Using Verilogijcncs
This document describes an implementation of the RSA cryptosystem using Verilog for an FPGA. It presents the design of modules for key generation, encryption, and decryption. For key generation, it generates random prime numbers using an LFSR and primality tester, then calculates the public and private keys. Encryption and decryption are performed through modular exponentiation implemented with a right-to-left binary method. The modules are coded in Verilog and synthesized for an FPGA to provide a secure cryptosystem.
Secure E-voting System by Utilizing Homomorphic Properties of the Encryption ...TELKOMNIKA JOURNAL
The use of cryptography in the e-voting system to secure data is a must to ensure the authenticity
of the data. In contrast to common encryption algorithms, homomorphic encryption algorithms had unique
properties that can perform mathematical operations against ciphertext. This paper proposed the use of
the Paillier and Okamoto-Uchiyama algorithms as two homomorphic encryption algorithms that have the
additional properties so that it can calculate the results of voting data that has been encrypted without
having to be decrypted first. The main purpose is to avoid manipulation and data falsification during vote
tallying process by comparing the advantages and disadvantages of each algorithm.
This document discusses using threshold cryptography and maximum distance separable (MDS) codes for key management in mobile ad hoc networks (MANETs). It begins with an introduction to MANETs and the need for distributed key management approaches. It then provides background on threshold cryptography and MDS codes. The document proposes using threshold cryptography combined with MDS codes to create a distributed cooperative key management system for MANETs that generates and distributes encryption keys among network nodes in a secure and fault-tolerant manner.
The document discusses error detection techniques used in direct link networks, focusing on cyclic redundancy checks (CRCs) and internet checksum algorithms. It provides an overview of CRCs, describing how they make use of polynomial arithmetic modulo 2 to generate a redundant checksum that is sent along with messages. The document also gives examples of how CRCs can be used to detect errors by dividing the received polynomial by the generator polynomial and checking if the remainder is zero. Finally, it compares error detection, which requires retransmissions, to error correction, which has overhead on all messages.
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
This document contains a Java program that implements the Cyclic Redundancy Check (CRC) algorithm for error detection in data transmission. The CRC algorithm uses polynomial arithmetic to compute a checksum over transmitted data by dividing the data by a fixed generator polynomial. The checksum is appended to the data before transmission. On the receiving end, the data and checksum are divided again by the generator polynomial. If the remainder is non-zero, then an error is detected in the received data. The Java program takes input data and generator polynomial from the user, computes the CRC checksum, transmits the data with checksum, and checks for errors on receiving end.
A comparative analysis of the possible attacks on rsa cryptosystemIAEME Publication
The document summarizes various attacks on the RSA cryptosystem. It discusses two main categories of attacks: 1) Mathematical attacks that exploit weaknesses in the mathematical structure of RSA, such as factoring the modulus or using a small private key. 2) Implementation attacks that take advantage of flaws in how RSA is implemented, like timing attacks. It then proposes a new attack algorithm that claims to be able to recover the private exponent d and factor the modulus n given only the public key (e,n), potentially breaking RSA without factoring.
Development of anonymous networks based on cryptographyMd. Hasibur Rashid
1. The document presents Enhanced Symmetric Key Encryption based Mixnet (ESEBM), an anonymous network proposed to address drawbacks of existing anonymous networks like Mixnet and DC-net.
2. ESEBM consists of a concealing pattern generator and anonymous channel. The generator provides encryption keys to senders and the channel routes encrypted messages anonymously.
3. ESEBM uses symmetric key encryption, which provides faster processing than asymmetric schemes used in other networks. Evaluation shows ESEBM can handle over 36 times the message volume of Mixnet.
A New hybrid method in watermarking using DCT and AESIJERD Editor
In this paper I'm trying to make a combination between the encryption by using one of the most
powerful algorithm called Advanced Encryption Standard (AES) to encrypt a secret message another word logo
and then embed it in the digital image in frequency domain by using the Discrete Cosine Transform (DCT) in
low frequency to increase the robustness and then applying some attacks to check it.
Lightweight Cryptography for Distributed PKI Based MANETSIJCNCJournal
This document proposes a lightweight cryptography solution for secure communication in mobile ad hoc networks (MANETs). It describes creating a distributed public key infrastructure (PKI) using Shamir's secret sharing to decentralize the certificate authority role among MANET nodes. Each node holds a share of the private key. It then proposes using Tiny Encryption Algorithm (TEA), an efficient symmetric-key cipher, along with elliptic curve Diffie-Hellman key exchange to establish secure communication between nodes with limited resources. The system initializes by having founding MANET nodes act as dealers to distribute secret shares. Nodes then use Diffie-Hellman to independently derive a secret key to encrypt communications.
A Signature Algorithm Based On Chaotic Maps And Factoring ProblemsSandra Long
This document describes a new digital signature algorithm based on chaotic maps and factorization problems. It consists of three main phases:
1) System initialization which defines parameters like cryptographic hash function, large prime numbers p and q, element a of order n in GF(p), and multiplicative group G generated by a.
2) Key generation where the signer selects private keys d, x and computes public keys e, y using chaotic maps and modular arithmetic.
3) Signature generation where the signer selects a random number r, computes intermediate values using chaotic maps and factorization, and outputs the signature (v1, v2, S) for the hashed message. The security relies on the difficulty of simultaneously solving
Apresentação sobre Criptografia baseada em reticulados (lattices), realizada no contexto da disciplina de Post-Quantum Cryptography do PPGCC da UFSC.
Versão odp: http://coenc.td.utfpr.edu.br/~giron/presentations/aula_lattice.odp
Information and network security 33 rsa algorithmVaibhav Khanna
RSA algorithm is asymmetric cryptography algorithm. Asymmetric actually means that it works on two different keys i.e. Public Key and Private Key. As the name describes that the Public Key is given to everyone and Private key is kept private
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.
Cloud computing is an ever-growing field in today‘s era.With the accumulation of data and the
advancement of technology,a large amount of data is generated everyday.Storage, availability and security of
the data form major concerns in the field of cloud computing.This paper focuses on homomorphic encryption,
which is largely used for security of data in the cloud.Homomorphic encryption is defined as the technique of
encryption in which specific operations can be carried out on the encrypted data.The data is stored on a remote
server.The task here is operating on the encrypted data.There are two types of homomorphic encryption, Fully
homomorphic encryption and patially homomorphic encryption.Fully homomorphic encryption allow arbitrary
computation on the ciphertext in a ring, while the partially homomorphic encryption is the one in which
addition or multiplication operations can be carried out on the normal ciphertext.Homomorphic encryption
plays a vital role in cloud computing as the encrypted data of companies is stored in a public cloud, thus taking
advantage of the cloud provider‘s services.Various algorithms and methods of homomorphic encryption that
have been proposed are discussed in this paper
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
This document provides an overview of cyclic redundancy checks (CRC). It defines CRC as treating bit strings as polynomial coefficients to detect errors. The process involves generating a message polynomial from the data and a generator polynomial. A CRC code is calculated by dividing the message polynomial by the generator polynomial. This CRC code is sent along with the data. On receipt, the data and CRC code are divided by the generator polynomial again. If the remainder is zero, no error is detected. The document gives an example of calculating CRC and discusses the properties of CRC, including its ability to detect single, double, and short burst errors.
Image encryption using elliptical curve cryptosytem with hill cipherkarthik kedarisetti
IMAGE ENCRYPTION-BTECH FINAL YEAR PROJECT ZEROTH REVIEW.
Image encryption is rapidly increased recently by the increasing use of the internet and communication
media. Sharing important images over unsecured channels is liable for attacking and stealing. Encryption
techniques are the suitable methods to protect images from attacks. Hill cipher algorithm is one of the
symmetric techniques, it has a simple structure and fast computations, but weak security because sender
and receiver need to use and share the same private key within a non-secure channels. A new image
encryption technique that combines Elliptic Curve Cryptosystem with Hill Cipher (ECCHC) has been proposed
in this paper to convert Hill cipher from symmetric technique to asymmetric one and increase its
security and efficiency and resist the hackers. Self-invertible key matrix is used to generate encryption
and decryption secret key. So, no need to find the inverse key matrix in the decryption process. A secret
key matrix with dimensions 4 4 will be used as an example in this study. Entropy, Peak Signal to Noise
Ratio (PSNR), and Unified Average Changing Intensity (UACI) will be used to assess the grayscale image
encryption efficiency and compare the encrypted image with the original image to evaluate the performance
of the proposed encryption technique.
Information security is one of the most important issues in the
recent times. Elliptic Curve Cryptography (ECC) is one of the most
efficient public key cryptosystems that is secured against adversaries
because it is hard for them to find the secret key and solve
the elliptic curve discrete logarithm problem. Its strengthened
security also comes from the small key size that is used in it with
the same level of safety compared to the other cryptosystems like RSA(Rivest–Shamir–Adleman))
IRJET- Data Analysis for Braking System in Time Domain for Fault DiagnosisIRJET Journal
This document discusses securing cloud data when encryption keys are exposed. It presents a system that distributes encrypted data across multiple cloud storage servers administered by different entities. This is done to limit an attacker's access even if they acquire the encryption key, as they would not be able to compromise every server. The paper studies data confidentiality against an adversary who knows the encryption key and has access to most but not all of the encrypted data blocks. It proposes a new security definition to capture this threat and evaluates a prototype implementation to measure performance overhead.
Cryptosystem An Implementation of RSA Using Verilogijcncs
This document describes an implementation of the RSA cryptosystem using Verilog for an FPGA. It presents the design of modules for key generation, encryption, and decryption. For key generation, it generates random prime numbers using an LFSR and primality tester, then calculates the public and private keys. Encryption and decryption are performed through modular exponentiation implemented with a right-to-left binary method. The modules are coded in Verilog and synthesized for an FPGA to provide a secure cryptosystem.
Secure E-voting System by Utilizing Homomorphic Properties of the Encryption ...TELKOMNIKA JOURNAL
The use of cryptography in the e-voting system to secure data is a must to ensure the authenticity
of the data. In contrast to common encryption algorithms, homomorphic encryption algorithms had unique
properties that can perform mathematical operations against ciphertext. This paper proposed the use of
the Paillier and Okamoto-Uchiyama algorithms as two homomorphic encryption algorithms that have the
additional properties so that it can calculate the results of voting data that has been encrypted without
having to be decrypted first. The main purpose is to avoid manipulation and data falsification during vote
tallying process by comparing the advantages and disadvantages of each algorithm.
This document discusses using threshold cryptography and maximum distance separable (MDS) codes for key management in mobile ad hoc networks (MANETs). It begins with an introduction to MANETs and the need for distributed key management approaches. It then provides background on threshold cryptography and MDS codes. The document proposes using threshold cryptography combined with MDS codes to create a distributed cooperative key management system for MANETs that generates and distributes encryption keys among network nodes in a secure and fault-tolerant manner.
The document discusses error detection techniques used in direct link networks, focusing on cyclic redundancy checks (CRCs) and internet checksum algorithms. It provides an overview of CRCs, describing how they make use of polynomial arithmetic modulo 2 to generate a redundant checksum that is sent along with messages. The document also gives examples of how CRCs can be used to detect errors by dividing the received polynomial by the generator polynomial and checking if the remainder is zero. Finally, it compares error detection, which requires retransmissions, to error correction, which has overhead on all messages.
Introduction to Public key Cryptosystems with block diagrams
Reference : Cryptography and Network Security Principles and Practice , Sixth Edition , William Stalling
This document contains a Java program that implements the Cyclic Redundancy Check (CRC) algorithm for error detection in data transmission. The CRC algorithm uses polynomial arithmetic to compute a checksum over transmitted data by dividing the data by a fixed generator polynomial. The checksum is appended to the data before transmission. On the receiving end, the data and checksum are divided again by the generator polynomial. If the remainder is non-zero, then an error is detected in the received data. The Java program takes input data and generator polynomial from the user, computes the CRC checksum, transmits the data with checksum, and checks for errors on receiving end.
A comparative analysis of the possible attacks on rsa cryptosystemIAEME Publication
The document summarizes various attacks on the RSA cryptosystem. It discusses two main categories of attacks: 1) Mathematical attacks that exploit weaknesses in the mathematical structure of RSA, such as factoring the modulus or using a small private key. 2) Implementation attacks that take advantage of flaws in how RSA is implemented, like timing attacks. It then proposes a new attack algorithm that claims to be able to recover the private exponent d and factor the modulus n given only the public key (e,n), potentially breaking RSA without factoring.
Development of anonymous networks based on cryptographyMd. Hasibur Rashid
1. The document presents Enhanced Symmetric Key Encryption based Mixnet (ESEBM), an anonymous network proposed to address drawbacks of existing anonymous networks like Mixnet and DC-net.
2. ESEBM consists of a concealing pattern generator and anonymous channel. The generator provides encryption keys to senders and the channel routes encrypted messages anonymously.
3. ESEBM uses symmetric key encryption, which provides faster processing than asymmetric schemes used in other networks. Evaluation shows ESEBM can handle over 36 times the message volume of Mixnet.
A New hybrid method in watermarking using DCT and AESIJERD Editor
In this paper I'm trying to make a combination between the encryption by using one of the most
powerful algorithm called Advanced Encryption Standard (AES) to encrypt a secret message another word logo
and then embed it in the digital image in frequency domain by using the Discrete Cosine Transform (DCT) in
low frequency to increase the robustness and then applying some attacks to check it.
Lightweight Cryptography for Distributed PKI Based MANETSIJCNCJournal
This document proposes a lightweight cryptography solution for secure communication in mobile ad hoc networks (MANETs). It describes creating a distributed public key infrastructure (PKI) using Shamir's secret sharing to decentralize the certificate authority role among MANET nodes. Each node holds a share of the private key. It then proposes using Tiny Encryption Algorithm (TEA), an efficient symmetric-key cipher, along with elliptic curve Diffie-Hellman key exchange to establish secure communication between nodes with limited resources. The system initializes by having founding MANET nodes act as dealers to distribute secret shares. Nodes then use Diffie-Hellman to independently derive a secret key to encrypt communications.
A Signature Algorithm Based On Chaotic Maps And Factoring ProblemsSandra Long
This document describes a new digital signature algorithm based on chaotic maps and factorization problems. It consists of three main phases:
1) System initialization which defines parameters like cryptographic hash function, large prime numbers p and q, element a of order n in GF(p), and multiplicative group G generated by a.
2) Key generation where the signer selects private keys d, x and computes public keys e, y using chaotic maps and modular arithmetic.
3) Signature generation where the signer selects a random number r, computes intermediate values using chaotic maps and factorization, and outputs the signature (v1, v2, S) for the hashed message. The security relies on the difficulty of simultaneously solving
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
An Advance Approach of Image Encryption using AES, Genetic Algorithm and RSA ...IJEACS
In current scenario the entire world is moving towards digital communication for fast and better communication. But in this a problem rises with security i.e. when we have to store information (either data or image) at any casual location or transmit information through internet. As internet is an open transmission medium, security of data becomes very important. To defend our information from piracy or from hacking we use a technique and i.e. known as Encryption Technique. In this paper, we use image as information and use an advance approach of well-known encryption techniques like AES, Genetic Algorithm, and RSA algorithm to encrypt it and keep our information safe from hackers or intruders making it highly difficult and time consuming to decipher the image without using the key.
Distributed Beamforming in Sensor NetworksDaniel Tai
The document discusses two papers on distributed beamforming techniques in sensor networks. The first technique computes beamforming vectors iteratively using feedback from the receiver to maximize SNR. The second technique uses an adaptive least squares algorithm by the cluster head to minimize error between the received and reference signals. Key differences are that the first method is fully distributed while the second has a cluster head, and the first uses feedback while the second uses a reference signal. Simulation results show convergence over time for the first method and the impact of factors like node density for the second.
Identity-based threshold group signature scheme based on multiple hard number...IJECEIAES
This document presents a new identity-based threshold group signature scheme based on multiple hard number theoretic problems - residuosity and discrete logarithm. The scheme relies on the difficulty of simultaneously solving both problems, improving security over schemes based on a single hard problem. The scheme is described, including system setup, key generation, signature generation, and signature verification. It is argued that the scheme is secure against common attacks on signature schemes like trying to derive secret keys from public parameters, as solving either the residuosity or discrete logarithm problems alone would not be enough to break the system. The performance and efficiency of the proposed scheme is also analyzed.
This document discusses using genetic algorithms to cryptanalyze the RSA cryptosystem. It first provides an overview of the RSA algorithm and how it works. It then discusses the Karatsuba algorithm, which can be used to efficiently multiply very large numbers. The document goes on to explain the basic concepts behind genetic algorithms, including representation, selection, crossover, and mutation operators. The authors propose applying genetic algorithms to generate new number pairs p and q that could be used to factor the RSA modulus N and break the cryptosystem. Specifically, they suggest using genetic algorithm operators to generate a new population of p and q values to use in cryptanalyzing RSA.
This document proposes an adaptive steganography technique for hiding secret data in digital images. The technique uses variable bit length embedding based on wavelet coefficients of the cover image. A logistic map is used to generate a secret key, which determines the RGB color planes and blocks used for data hiding. Wavelet coefficients are classified into ranges, and the number of bits hidden corresponds to the coefficient value range. Extraction involves selecting the same coefficients based on the key and calculating the hidden bits. The technique aims to improve security, capacity and imperceptibility compared to existing constant bit length methods. Evaluation shows the proposed method provides good security since variable bits are hidden randomly using the secret key.
Utilizing distributed storage administrations, clients can store their Information in the cloud to maintain a strategic distance from
the consumption of neighborhood information stockpiling support. To guarantee the uprightness of the information put away in
the Cloud, numerous information, honesty examining plans have been proposed. A client needs to Utilize his private key to
produce the information authenticators for Understanding the information respectability reviewing. In this way, the client needs to
have an equipment token to store his private Key and retain a secret phrase to enact this private key. In the event that this
Equipment token is lost or this secret phrase is overlooked, the majority of the Current information, trustworthiness inspecting
plans would be notable work. We propose another worldview Called information uprightness inspecting without private key
stockpiling and Plan such a plan. In this plan, we use biometric informationas the client's fluffy private key to Abstain from
utilizing the equipment token. In the interim, the plan can at present Successfully complete the information respectability
auditing. We use a direct Sketch with coding and blunder revision procedures to affirm The personality of the client. We use
another mark Conspire which supports blacklist certainty. The security evidence and the Execution examination demonstrates that our proposed plan accomplishes Attractive security andeffectiveness.
Secure Image Transmission for Cloud Storage System Using Hybrid SchemeIJERD Editor
- Data over the cloud is transferred or transmitted between servers and users. Privacy of that
data is very important as it belongs to personal information. If data get hacked by the hacker, can be
used to defame a person’s social data. Sometimes delay are held during data transmission. i.e. Mobile
communication, bandwidth is low. Hence compression algorithms are proposed for fast and efficient
transmission, encryption is used for security purposes and blurring is used by providing additional
layers of security. These algorithms are hybridized for having a robust and efficient security and
transmission over cloud storage system.
This document proposes a new digital image encryption technique based on multi-scroll chaotic delay differential equations (DDEs). The technique uses a XOR operation between separated binary planes of a grayscale image and a shuffled attractor image from a DDE. Security keys include DDE parameters like initial conditions, time constants, and simulation time. Experimental results using a 512x512 Lena image in MATLAB demonstrate the DDE dynamics, encryption/decryption security through histograms, power spectrums, and image correlations. Wrong key decryption is also shown. The technique offers potential for simple yet secure image transmission applications.
Today among various medium of data transmission or storage our sensitive data
are not secured with a third-party, that we used to take help of. Cryptography plays an
important role in securing our data from malicious attack. This paper present a partial
image encryption based on bit-planes permutation using Peter De Jong chaotic map for
secure image transmission and storage. The proposed partial image encryption is a raw data
encryption method where bits of some bit-planes are shuffled among other bit-planes based
on chaotic maps proposed by Peter De Jong. By using the chaotic behavior of the Peter De
Jong map the position of all the bit-planes are permuted. The result of the several
experimental, correlation analysis and sensitivity test shows that the proposed image
encryption scheme provides an efficient and secure way for real-time image encryption and
decryption.
Public Key Cryptography uses two keys - a public key that can encrypt messages and verify signatures, and a private key that can decrypt messages and create signatures. The RSA algorithm, the most widely used public key algorithm, is based on the mathematical difficulty of factoring large prime numbers. It works by having users generate a public/private key pair using two large prime numbers and performing modular exponentiation. The security of RSA relies on the fact that it is computationally infeasible to derive the private key from the public key and modulus.
Chaotic systems with pseudorandom number generate to protect the transmitted...nooriasukmaningtyas
Communication techniques have witnessed rapid development in recent years, especially the internet and the mobile network, which led to rapid data transmission. The latest developments, in turn, have come out with advanced decisions to secure information from eavesdropping. Myriad in-depth studies in cryptography. It was implemented with the intention of proposing a revolutionary solution to protect data by encryption techniques, tend maps, and logistics. This work had proposed a new design to the generator of pseudo-random numbers (GPRN) which had utilized multi chaotic systems. Synchronization of Multi-parameters chaotic arises in many applications, in natural or industrial systems. Many methods have been introduced for using a chaotic system in the encryption of data. Analysis of security of chaotic system had been executed on key sensitivity and key space.
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In this paper, simulated annealing guided traingularized encryption using multilayer perceptron generated session key (SATMLP) has been proposed for secured wireless ommunication. Bothsender and receiver station uses identical multilayer perceptron and depending on the final output of the both side multilayer perceptron, weights vector of hidden layer get tuned in bothends. After this tunning step both perceptrons generates identical weight vectors which is consider as an one time session key. In the 1st level of encryption process traingularized sub key1 is use to encrypt the plain text. In 2nd level of encryption simulated annealing method helps to generates sub key 2 for further encryption of 1st level generated traingularized encrypted text. Finally multilayer perceptron generated one time session key is used to perform
3rd level of encryption of 2nd level generated encrypted text. Recipient will use same identical multilayer perceptron guided session key for performing deciphering process for getting the simulated annealing guided intermediate cipher text. Then using sub key 2 deciphering technique is performed to get traingularized encrypted text. Finally sub key 1 is used to generate the plain text. In this proposed technique session key is not transmitted over public channel apart from few data transfers are needed for weight simulation process. Because after synchronization process both multilayer perceptron generates identical weight vector which acts as a session key. Parametric tests are done and results are compared in terms of ChiSquare test, response time in transmission with some existing classical techniques, which shows comparable results for the proposed system.
Secured wireless communication through simulated annealing guided traingulari...csandit
In this paper, simulated annealing guided traingularized encryption using multilayer perceptron
generated session key (SATMLP) has been proposed for secured wireless communication. Both
sender and receiver station uses identical multilayer perceptron and depending on the final
output of the both side multilayer perceptron, weights vector of hidden layer get tuned in both
ends. After this tunning step both perceptrons generates identical weight vectors which is
consider as an one time session key. In the 1st level of encryption process traingularized sub
key1 is use to encrypt the plain text. In 2nd level of encryption simulated annealing method
helps to generates sub key 2 for further encryption of 1st level generated traingularized
encrypted text. Finally multilayer perceptron generated one time session key is used to perform
3rd level of encryption of 2nd level generated encrypted text. Recipient will use same identical
multilayer perceptron guided session key for performing deciphering process for getting the
simulated annealing guided intermediate cipher text. Then using sub key 2 deciphering
technique is performed to get traingularized encrypted text. Finally sub key 1 is used to
generate the plain text. In this proposed technique session key is not transmitted over public
channel apart from few data transfers are needed for weight simulation process. Because after
synchronization process both multilayer perceptron generates identical weight vector which
acts as a session key. Parametric tests are done and results are compared in terms of Chi-
Square test, response time in transmission with some existing classical techniques, which shows
comparable results for the proposed system.
This document provides an overview of fully homomorphic encryption (FHE). It discusses how FHE allows computations to be performed on encrypted data without decrypting it first. The document covers different types of homomorphic encryption schemes, including partially homomorphic encryption which supports only addition or multiplication, and fully homomorphic encryption which supports arbitrary computations. It also discusses Gentry's construction of the first FHE scheme and techniques like bootstrapping which are used to reduce noise in ciphertexts to allow unlimited computations. The document surveys potentials and applications of FHE for securing data in cloud computing environments.
A Good Performance OTP Encryption Image based on DCT-DWT SteganographyTELKOMNIKA JOURNAL
The security aspect is very important in data transmission. One way to secure data is with
steganography and cryptography. Surely research on this should continue to be developed to improve
security. In this paper, we proposed a combination of steganographic and cryptographic algorithms for
double protection during data transmission. The selected steganographic algorithm is the use of a
combination of DCT and DWT domain transformations. Because the Imperceptibility aspect is a very
important aspect of steganographic techniques, this aspect needs to be greatly improved. In the proposed
method of DCT transformation first, proceed with DWT transformation. From the experimental results
obtained better imperceptibility quality, compared with existing methods. To add OTP message security
applied algorithm to encrypt the message image, before it is inserted. This is evidenced by experiments
conducted on 20 grayscale images measuring 512x512 with performance tests using MSE, PSNR, and
NC. Experimental results prove that DCT-DWT-OTP generates PNSR more than 50 dB, and NC of all
images is 1.
Public-key cryptography uses two keys: a public key for encryption and digital signatures, and a private key for decryption and signature verification. RSA is the most widely used public-key cryptosystem, using large prime factorization and modular exponentiation. It allows secure communication without prior key exchange. While brute force attacks on RSA are infeasible due to large key sizes, its security relies on the difficulty of factoring large numbers.
EFFICIENT SCRAMBLING-SUBSTITUTION IMAGE SECURITY SCHEME USING CHAOTIC ARNOLD-...IJCNCJournal
This paper introduces an efficient scrambling-substitution image security scheme using chaotic Arnold and
Logistic (Arnold-Logistic) maps in the discrete cosine transform (DCT). The Arnold map is employed as a
scrambling stage while the Logistic map is employed as a substitution stage. The hybrid Arnold-Logistic
mapping is performed in the DCT. The encipherment phase of the introduced DCT-based Arnold-Logistic
security scheme begins by applying the DCT to the plainimage and the resulted DCT coefficient of the
plainimage are scrambled for m iterations using the Arnold transformation. Then, the Arnold-based
transformed DCT coefficients are substituted for n iterations using the Logistic map and the inverse of
DCT (IDCT) is employed to produce the cipherimage. The decipherment phase of the introduced DCTbased Arnold-Logistic security scheme is the inverse of the encryption stage and begins by applying the
DCT to the cipherimage. The resulted DCT coefficient of the cipherimage is inversely substituted for n
iterations using the inverse Logistic map. Then, the inverse Logistic-based transformed DCT coefficients
are inversely scrambled for m iterations using the inverse Arnold map and the IDCT is employed to
produce the decrypted image. A series of test experiments are applied to investigate the introduced DCTbased Arnold-Logistic security scheme. The outcome results demonstrated the superiority of the introduced
DCT-based Arnold-Logistic security scheme from the security point of view.
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THE KEY EXCHANGE CRYPTOSYSTEM USED WITH HIGHER ORDER DIOPHANTINE EQUATIONS
1. International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.2, March 2011
DOI : 10.5121/ijnsa.2011.3204 43
THE KEY EXCHANGE CRYPTOSYSTEM USED WITH
HIGHER ORDER DIOPHANTINE EQUATIONS
Harry Yosh
HECO Ltd, Canberra, Australia
square17320508@yahoo.com
ABSTRACT
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.
KEYWORDS
public-key cryptography, one-way function, Diophantine equation, quotient ring, NP-complete
1. INTRODUCTION
Since the public-key cryptography introduced by Diffie and Hellman,[1] various public-
key cryptosystems based on integer factorization or discrete logarithm such as RSA, El Gamal,
or those using elliptic curve techniques have been developed.[2] They obtain the computational
security by using the sufficiently large key which imposes the attackers too heavy
computational cost to decipher through solving integer factorization or discrete logarithm
problems. Although no polynomial-time method for factoring large integers or solving the
discrete logarithm problem has been known, it has not been proven that there exist no
polynomial-time solutions for those problems.[3] Therefore the cryptosystems which are
proven that there is no polynomial-time solution for attackers are thought to be significantly
secure against their attacks.
Here a new cryptosystem is discussed. In this cryptosystem the public key for recipient is
expressed with Diophantine equation and his/her private key is expressed with a value on the
quotient ring defined with another Diophantine equation.
A Diophantine equation has plural unknown variables and defines an algebraic curve or
algebraic surface. It is know to find the lattice points as the solutions of Diophantine equation is
difficult in general when the order of that Diophantine equation is higher than one.[7] This
difficulty is derived from that the equation may have no nontrivial solution, a finite number or
an infinite number of solutions. Regarding its solvability, there is no general method to
determine by a finite number of operations whether the equation is solvable or not.[8] However
2. International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.2, March 2011
44
these properties of Diophantine equation are also useful for encrypting message to keep it secret
and have been applied to some public key cryptosystems.[9][10] For example, the cryptosystem
proposed by Lin, et al. is based on that the Diophantine equation dealt with in the system is
practically non-soluble.[11] Precisely saying, the practically non-soluble Diophantine equation
mentioned here indicates that there is no algorithm for solving it running in polynomial time.
However it has been revealed that some alternative methods such as using AI[12], A*
search[13] or genetic algorithm[14] are effective for solving numerically some specific types of
Diophantine equation. It implies the cryptosystem built on a certain type of Diophantine
equation might be broken within polynomial time by using those methods.
On the other hand, on the cryptosystem proposed here both sender and recipient can
choose the form of Diophantine equation arbitrary. Therefore they can avoid the use of the
specific types of Diophantine equation which are vulnerable to the methods mentioned above.
The Diophantine equation used for recipient's public key is given by sender who has the
trapdoor which is explained later and he/she can recovers the shared secret with that trapdoor,
while attackers who have obtained the Diophantine equations as public keys for sender and
recipient must solve them without the trapdoor. Thus this cryptosystem is expected to have the
computational security stemmed from the difficulty of Diophantine problem.
In the following sections we discuss the basic idea for that cryptosystem, practical scheme
and hardness of decryption by attackers. Also positive and negative aspects comparing with
other cryptosystems are discussed.
2. KEY EXCHANGE SCHEME
Firstly we discuss the basic idea on this key exchange scheme using examples. To simplify
the discussion we use simpler Diophantine equations and quotient ring in the following
examples than those used in practical cases.
This key exchange scheme begins from the recipient side. The recipient sets the integer values
of variables, say, x, y as x = 2, y = 3. Using those variables the recipient constructs a
Diophantine equation such as,
x3
– y2
+ 1 = 0
1)
The recipient sends the information of that Diophantine equation to sender with keeping the
values of x and y secret. The above Diophantine equation (1) is the public key and x, y are the
private keys for recipient.
Sender constructs the polynomial as Diophantine equation on the quotient ring Z[X,Y] /
(X3
– Y2
+ 1) through the following procedures; Firstly the sender defines the following
operator on that quotient ring.
T[a,b;c] : x → (x + a) c
+ b
2)
where a, b are integers and c is odd (c > 0). The inverse operator for it is given as,
3. International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.2, March 2011
45
T[a,b;c]
-1
: y → (y – b)1/c
– a
3)
Sender then sets an element on the quotient ring, say xy2
, and applies the operator (2)
repeatedly to that element as,
T[a1,b1;c1] (T[a2,b2;c2] (...(T[an,bn;cn] (xy2
))...) = T[a1,b1;c1] T[a2,b2;c2] ... T[an,bn;cn]
(xy2
)
= h(x,y)
4)
where h(x,y) is an element in Z[X,Y] / (X3
– Y2
+ 1). For example,
T[1,2;3] T[0,3;1] (xy2
) = T[1,2;3] (xy2
+ 3)
= (xy2
+ 4)3
+ 2 = x3
y6
+ 12x2
y4
+ 48xy2
+ 66
5)
Sender keeps the parameters in the operators (i.e. 1, 2, 3 and 0, 3, 1 in T[1,2;3] and T[0,3;1] ) secret.
Those parameters are the private keys for sender and actually form a "trapdoor". x3
y6
+ 12x2
y4
+
48xy2
+ 66 is expressed with various ways in Z[X,Y] / (X3
– Y2
+ 1). Sender chooses one of
those representations, say, (y2
– 1) y6
+ 12x2
y4
+ 48xy2
+ 66 = y8
– y6
+ 12x2
y4
+ 48xy2
+ 66 and
sends it with xy2
to the recipient. The form of polynomial y8
– y6
+ 12x2
y4
+ 48xy2
+ 66 is the
public key for sender.
Recipient calculates y8
– y6
+ 12x2
y4
+ 48xy2
+ 66 and xy2
using his/her private keys x = 2,
y = 3 as,
y8
– y6
+ 12x2
y4
+ 48xy2
+ 66 = 10650
6)
xy2
= 18
7)
and returns the value 10650 to sender with keeping xy2
= 18 secret. The Diophantine equation
(6) is another public key for recipient, and the value of xy2
is actually the shared secret between
sender and recipient as shown below; Sender calculates xy2
using the inverse operators as,
T[0,3:1]
-1
T[1,2:3]
-1
(10650) = T[0,3:1]
-1
((10650 – 2)1/3
– 1)
= T[0,3:1]
-1
(21) = (21 – 3) – 0 = 18
8)
and sender could recover the value of xy2
as 18.
Attackers who have had the public keys for both sender and recipient must solve the
following system of equations to obtain the value of xy2
.
4. International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.2, March 2011
46
x3
– y2
+ 1 = 0
9)
y8
– y6
+ 12x2
y4
+ 48xy2
+ 66 = 10650
10)
Since the above system is zero-dimensional, the attackers can solve it numerically or using
Groebner basis.[5] However when the number of variables are greater than two, or the system
is positive-dimensional, solving it is difficult in general.[6]
Based on the above discussion, we construct the general scheme for the key exchange
cryptosystem.
Step 1
Recipient sets the integer values of variables x1, x2, ... , xm as,
x1 = k1, x2 = k2, ... , xm = km (kj : integers)
11)
and constructs Diophantine equation as recipient's public key using those variables as,
f(x1, x2, ... , xm) = 0
12)
Recipient keeps k1, k2, ... , km secret.
Step 2
Recipient sends the above Diophantine equation (12) to sender.
Step 3
Sender sets an element g(x1, x2, ... , xm) in the quotient ring Z[X1, X2, ... , Xm] / (f(X1, X2, ... ,
Xm)) and defines the operators T[aj,bj:cj] (j = 1, ... , n) on that quotient ring as,
T[aj,bj:cj] : x → (x + aj) cj
+ bj
13)
where aj, bj are integers and cj is odd (cj > 0). Then the sender applies those operators to g(x1,
x2, ... , xm) as,
T[a1,b1:c1] (T[a2,b2:c2] ( ... (T[an,bn:cn] (g(x1, x2, ... , xm))) ... )
= T[a1,b1:c1] T[a2,b2:c2] ... T[an,bn:cn] (g(x1, x2, ... , xm))
= h(x1, x2, ... , xm)
14)
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47
where h(x1, x2, ... , xm) is an element in Z[X1, X2, ... , Xm] / (f(X1, X2, ... , Xm)) and has generally
various representations. Sender chooses one of them as his/her public key.
Step 4
Sender sends the public key h(x1, x2, ... , xm) chosen in the previous step with the form g(x1, x2,
... , xm) to recipient.
Step 5
Recipient calculates h(x1, x2, ... , xm) = p and g(x1, x2, ... , xm) = s using (11) where p and s are
some integers.
Step 6
Recipient sends back the value p to sender with keeping the value s secret.
Step 7
Sender recovers the value s of g(x1, x2, ... , xm) as,
s = T[an,bn:cn]
-1
T[an-1,bn-1:cn-1]
-1
... T[a1,b1:c1]
-1
(p)
15)
Thus sender and recipient could share the secret s.
The parameter c in the operator T[a,b:c] must be odd, otherwise the image of inverse
operator;
T[a,b;c]
-1
: y → (y – b)1/c
– a
16)
may not be determined uniquely and it causes the ambiguity on the obtained secret s.
It is possible to modify the above scheme to that on finite ring. In that case the
Diophantine equation (12) is modified to a congruence equation with modulus w and the
quotient ring is rewritten as,
Zw [X1, X2, ... , Xm] / (f(X1, X2, ... , Xm))
17)
where Zw is the quotient ring composed of 0, 1, ... , w-1. The operator defined in Step 3 is
modified as,
T[a,b:c] : x → (x + a) c
+ b mod w
18)
and its inverse operator is expressed as,
6. International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.2, March 2011
48
T[a,b;c]
-1
: y → (y – b) c'
– a mod w
19)
where cc' = 1 mod ϕ(w) (ϕ(w): Euler's totient function).
3. HARDNESS OF DECRYPTION FOR ATTACKERS
Attackers who have had the public information of sender and recipient are to solve the
following system of two Diophantine equations;
f(x1, x2, ... , xm) = 0 ... in Step 1
20)
h(x1, x2, ... , xm) = p ... in Step 5
21)
and estimate the secret s = g(x1, x2, ... , xm) with the obtained solution. As mentioned in earlier
section, it is difficult to solve (20) and (21) in integers when m is greater than two in general.
As another strategy, attackers may consider to decipher the sequence of operators appeared
in Step 3;
T[a1,b1:c1] T[a2,b2:c2] ... T[an,bn:cn] (g(x1, x2, ... , xm)) = h(x1, x2, ... , xm)
22)
by using (20) and (21). Once the above sequence has been deciphered, attackers can estimate
the secret s = g(x1, x2, ... , xm) applying the inverse sequence of the above sequence. However
the highest order among x1, x2, ... , xm in h(x1, x2, ... , xm) becomes greater in general along with
the increase of the iteration number n expressed in (22). Since the greater the highest order is,
the more equivalent expressions in the quotient ring Z[X1, X2, ... , Xm] / (f(X1, X2, ... , Xm)) are,
it is also virtually impossible to decipher the sequence of operators (22) unless attackers have
known the iteration number n previously or the Diophantine equation (12) is linear.
When the system of higher order congruence equations are used for the scheme instead of
Diophantine equations (20) and (21) as mentioned in earlier section, the size of secret key is
generally determined by that of modulus w. Although the solution space for the system of
congruence equations is finite, obtaining the secret s is still difficult for attackers because in
general the problem to solve such system of higher order congruence equations is NP-complete
when m > 2.[4]
4. PRO AND CON
Attackers must solve the system of Diophantine equations to estimate the shared secret
between sender and recipient. Generally it is hard to solve that system when it is positive-
dimensional. Attackers also subject the similar constraint on the corresponding higher order
congruence equation system. That feature brings the proven security which does not depend on
the performance of attackers' computational capacity. Namely sender and recipient can choose
the public keys which have the length shorter than that of public keys used for other
cryptosystems without harming the security, and the shorter length of public keys would
alleviate the computational load considerably.
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49
Comparing with other key exchange cryptosystem such as RSA or Diffie-Hellman scheme,
this cryptosystem needs additional key exchange step, that is, recipient must tell a Diophantine
equation from which the quotient ring is constructed to sender prior to key exchange between
them. It may increase the vulnerability regarding the security on this key exchange
cryptosystem.
Sender and recipient exchange Diophantine equations as their public keys. Although
sender and recipient can choose their Diophantine equations arbitrary, some specific
considerations are required to construct those Diophantine equations practically in the actual
key exchange scheme as discussed below.
Firstly in Step 1 described in the section: Key exchange scheme, the Diophantine equation
(12) must be constructed carefully to avoid the case that it has unique solution, otherwise as
soon as solving that equation, attackers get the secret keys expressed in (11). Next in Step 3, the
Diophantine equation (14) must not have unique solution as well as (12), also it must be
complicated enough to prevent from inferring the sequence of operators used for constructing
(14) by attackers. It is not practical that the above Diophantine equations are constructed
manually by sender and recipient in general cases, and it will be necessary to implement
appropriate programs into that cryptosystem to generate those equations automatically based on
the values of variables x1, x2, ... , xm set in Step 1, and aj, bj, cj and n set in Step 3. To prevent
from breaking the patterns of generating them by attackers, the forms of Diophantine equations
determined by those programs must not be static, but dynamically change according with the
input, or randomly. Thus the implementation of this cryptosystem will impose additional
calculation load for them with that for dealing with the Diophantine equations symbolically.
5. SUMMARY
In the proposed key exchange cryptosystem the system of two higher order Diophantine
equations is considered for encrypting shared secret between sender and recipient.
The first Diophantine equation is created by recipient and sender creates another
Diophantine equation on the quotient ring based on the first Diophantine equation. The shared
secret has the form of polynomial and both sender and recipient can obtain its value by a
trapdoor without solving the system of those Diophantine equations explicitly.
On the other hand, attackers must solve the following system of two Diophantine
equations mentioned above in order to obtain the secret.
f(x1, x2, ... , xm) = 0
23)
h(x1, x2, ... , xm) = p
24)
In general it is hard to solve the above system in integers for x1, x2, ... , xm when m is greater
than two.
Although this key exchange cryptosystem has the intrinsic security mentioned above, it
requires rather complicated implementation comparing with other key exchange cryptosystems.
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50
This complexity is due to generate Diophantine equations used as public keys with
unpredictable manner lest attackers detect the pattern of their generations. The specifically
designed program must be implemented in the cryptosystem for that purpose.
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[11] C. H. Lin, C. C. Chang, R. C. T. Lee, A New Public-Key Cipher System Based Upon the
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Author
Harry Yosh is a knowledge manager at HECO Ltd. He is also a data analyst at APA Group,
Canberra, having graduated from University of Canberra with a Master of IT. His hobbies are
traveling and swimming.
Paper: Harry Yosh, General Addition Formula for Meromorphic Functions Derived from
Residue Theorem. Journal of Mathematical Physics, Analysis, Geometry, v. 6, No 2, p.
183-191, 2010. (http://jmage.ilt.kharkov.ua/abstract.php?uid=jm06-0183e)