In this paper, we present a new signature scheme based on factoring and discrete logarithm problems.Derived from a variant of ElGamal signature protocol and the RSA algorithm, this method can be seen asan alternative protocol if known systems are broken.
A New Signature Protocol Based on RSA and Elgamal SchemeZac Darcy
In this paper, we present a new signature scheme based on factoring and discrete logarithm problems.
Derived from a variant of ElGamal signature protocol and the RSA alg
New Digital Signature Protocol Based on Elliptic Curvesijcisjournal
In this work, a new digital signature based on elliptic curves is presented. We established its efficiency and
security. The method, derived from a variant of ElGamal signature scheme, can be seen as a secure alternative protocol if known systems are completely broken.
A SIGNATURE ALGORITHM BASED ON DLP AND COMPUTING SQUARE ROOTSZac Darcy
n this work, we present a new digital signature protocol based on the discrete logarithm problem and
computing square roots modulo a large composite number. This method can be used as an alternative if
known systems are broken.
A Signature Algorithm Based on DLP and Computing Square RootsZac Darcy
In this work, we present a new digital signature protocol based on the discrete logarithm problem and
computing square roots modulo a large composite number. This method can be used as an alternative if
known systems are broken.
The document discusses the RSA cryptosystem. It begins by explaining that RSA is an important public-key cryptosystem based on the difficulty of factoring large integers. It then provides examples of how RSA works, including choosing prime numbers p and q to generate the public and private keys, and using modular exponentiation to encrypt and decrypt messages. The document also discusses the importance of integer factorization for the security of RSA, and considerations for designing a secure RSA system, such as choosing sufficiently large prime numbers.
This document discusses implementing the RSA encryption/decryption algorithm on an FPGA. It provides an overview of the RSA algorithm, including key generation, encryption, and decryption steps. It describes how modular arithmetic operations like addition, multiplication, and exponentiation are used in RSA. The document presents the VHDL code developed to implement an RSA decryption engine on an FPGA. Synthesis results show the decryption engine utilizes under 15% of device resources and achieves a maximum clock frequency of 68.57MHz. The document concludes that an RSA decryption engine can be efficiently implemented on an FPGA.
This document summarizes an article about implementing the RSA encryption/decryption algorithm on an FPGA. It begins with an overview of cryptography and the RSA algorithm. It then describes the key steps in RSA - key generation, encryption, and decryption. The main mathematical operations required for RSA are also summarized - modular addition, multiplication, and exponentiation. The document then presents the design of a 32-bit RSA decryption engine in VHDL, along with synthesis results showing its resource usage and maximum clock frequency on an FPGA. It concludes that an RSA decryption engine can be efficiently implemented on an FPGA using limited resources.
RSA ALGORITHM WITH A NEW APPROACH ENCRYPTION AND DECRYPTION MESSAGE TEXT BY A...ijcisjournal
In many research works, there has been an orientation to studying and developing many of the applications of public-key cryptography to secure the data while transmitting in the systems, In this paper we present an approach to encrypt and decrypt the message text according to the ASCII(American Standard Code for Information Interchange) and RSA algorithm by converting the message text into binary representation and dividing this representation to bytes(8s of 0s and 1s) and applying a bijective function between the group of those bytes and the group of characters of ASCII and then using this mechanism to be compatible with using RSA algorithm, finally, Java application was built to apply this approach directly.
A New Signature Protocol Based on RSA and Elgamal SchemeZac Darcy
In this paper, we present a new signature scheme based on factoring and discrete logarithm problems.
Derived from a variant of ElGamal signature protocol and the RSA alg
New Digital Signature Protocol Based on Elliptic Curvesijcisjournal
In this work, a new digital signature based on elliptic curves is presented. We established its efficiency and
security. The method, derived from a variant of ElGamal signature scheme, can be seen as a secure alternative protocol if known systems are completely broken.
A SIGNATURE ALGORITHM BASED ON DLP AND COMPUTING SQUARE ROOTSZac Darcy
n this work, we present a new digital signature protocol based on the discrete logarithm problem and
computing square roots modulo a large composite number. This method can be used as an alternative if
known systems are broken.
A Signature Algorithm Based on DLP and Computing Square RootsZac Darcy
In this work, we present a new digital signature protocol based on the discrete logarithm problem and
computing square roots modulo a large composite number. This method can be used as an alternative if
known systems are broken.
The document discusses the RSA cryptosystem. It begins by explaining that RSA is an important public-key cryptosystem based on the difficulty of factoring large integers. It then provides examples of how RSA works, including choosing prime numbers p and q to generate the public and private keys, and using modular exponentiation to encrypt and decrypt messages. The document also discusses the importance of integer factorization for the security of RSA, and considerations for designing a secure RSA system, such as choosing sufficiently large prime numbers.
This document discusses implementing the RSA encryption/decryption algorithm on an FPGA. It provides an overview of the RSA algorithm, including key generation, encryption, and decryption steps. It describes how modular arithmetic operations like addition, multiplication, and exponentiation are used in RSA. The document presents the VHDL code developed to implement an RSA decryption engine on an FPGA. Synthesis results show the decryption engine utilizes under 15% of device resources and achieves a maximum clock frequency of 68.57MHz. The document concludes that an RSA decryption engine can be efficiently implemented on an FPGA.
This document summarizes an article about implementing the RSA encryption/decryption algorithm on an FPGA. It begins with an overview of cryptography and the RSA algorithm. It then describes the key steps in RSA - key generation, encryption, and decryption. The main mathematical operations required for RSA are also summarized - modular addition, multiplication, and exponentiation. The document then presents the design of a 32-bit RSA decryption engine in VHDL, along with synthesis results showing its resource usage and maximum clock frequency on an FPGA. It concludes that an RSA decryption engine can be efficiently implemented on an FPGA using limited resources.
RSA ALGORITHM WITH A NEW APPROACH ENCRYPTION AND DECRYPTION MESSAGE TEXT BY A...ijcisjournal
In many research works, there has been an orientation to studying and developing many of the applications of public-key cryptography to secure the data while transmitting in the systems, In this paper we present an approach to encrypt and decrypt the message text according to the ASCII(American Standard Code for Information Interchange) and RSA algorithm by converting the message text into binary representation and dividing this representation to bytes(8s of 0s and 1s) and applying a bijective function between the group of those bytes and the group of characters of ASCII and then using this mechanism to be compatible with using RSA algorithm, finally, Java application was built to apply this approach directly.
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
This paper provides an overview of the RSA algorithm, exploring the foundations and mathematics in detail. It then uses this base information to explore issues with the RSA algorithm that detract from the algorithms security.
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.
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.
This document discusses several topics in number theory including prime numbers, relatively prime numbers, modular arithmetic, Fermat's theorem, Euler's theorem, and the Chinese Remainder theorem. It provides examples and explanations of these concepts. It also discusses how some of these number theory concepts like modular arithmetic and the difficulty of factoring large numbers into primes are applied in public key cryptography algorithms like RSA.
Presentation on Cryptography_Based on IEEE_PaperNithin Cv
The document summarizes a seminar report on a hybrid cryptography architecture that uses multiple cryptographic algorithms. It discusses Elliptic Curve Cryptography (ECC), Elliptic Curve Diffie-Hellman (ECDH), Elliptic Curve Digital Signature Algorithm (ECDSA), and Dual-RSA. ECC is used to encrypt data onto an elliptic curve. ECDH generates shared secret keys between two parties for use in symmetric encryption. ECDSA allows digital signatures using elliptic curve parameters. Dual-RSA improves decryption efficiency using the Chinese Remainder Theorem to split computations between prime factors p and q. The hybrid architecture combines the strengths of these algorithms to provide secure encryption, authentication, and key exchange.
Security_Attacks_On_RSA~ A Computational Number Theoretic Approach.pptxshahiduljahid71
The document discusses various attacks on the RSA cryptosystem. It introduces factoring attacks which aim to factor the modulus N, as knowing the prime factors p and q would allow an attacker to calculate the private key. It notes that while factoring large integers remains difficult, other attacks exist that could decrypt messages without directly factoring N. The document then examines elementary attacks like common modulus and blinding attacks that could potentially extract information about encrypted messages without inverting the RSA function. Finally, it poses an open problem around whether an algorithm could efficiently factor N given access to the RSA encryption function.
The discrete logarithm problem (DLP) is the basis for elliptic curve cryptography (ECC) and differs from the integer factorization problem in RSA. In ECC over a finite field, the DLP is to find the exponent that computes one point on the elliptic curve as a multiple of another point, given the curve equation and two points. In RSA, the problem is to find the prime factors of a composite integer. While general algorithms exist to solve both, the DLP in ECC providing equivalent security to RSA requires smaller key sizes, making ECC more efficient.
A NEW ALGORITHM TO CONSTRUCT S-BOXES WITH HIGH DIFFUSIONijscmcj
In this paper is proposed a new algorithm to construct S-Boxes over GF(28) with Branch Number value at least 3. This is an important property that guarantees a high diusion in the S-Box [12]. Also are introduced some defnition and properties that show the be- havior of S-Boxes under the composition with affine functions. Finally is presented a comparision between this algorithm and the method pro-posed by Tavares [12].
A NEW ALGORITHM TO CONSTRUCT S-BOXES WITH HIGH DIFFUSIONijscmcj
In this paper is proposed a new algorithm to construct S-Boxes over GF(28) with Branch Number value at
least 3. This is an important property that guarantees a high diusion in the S-Box [12]. Also are introduced
some defnition and properties that show the be- havior of S-Boxes under the composition with affine
functions. Finally is presented a comparision between this algorithm and the method pro-posed by Tavares
[12].
AUTHENTICATED PUBLIC KEY ENCRYPTION SCHEME USING ELLIPTIC CURVE CRYPTOGRAPHYijujournal
Secure transformation of data is of prime importance in today’s world. In the present paper, we propose a
double fold authenticated public key encryption scheme which helps us in securely sending the confidential
data between sender and receiver. This scheme makes the encrypted data more secure against various
cryptographic attacks.
AUTHENTICATED PUBLIC KEY ENCRYPTION SCHEME USING ELLIPTIC CURVE CRYPTOGRAPHYijujournal
The document presents an authenticated public key encryption scheme using elliptic curve cryptography. It proposes a double encryption method to securely transmit confidential data between a sender and receiver. In the scheme, the sender and receiver first agree on an elliptic curve and generator point over a finite field. They generate private/public key pairs and specific public keys for each other. The sender encrypts the message points in two stages - first generating cipher points using a random integer, and then performing XOR operations on the point coordinates with other values. The receiver decrypts the cipher text in two stages to recover the original message points and plaintext. An example is provided to illustrate the encryption and decryption process.
Large Semi Primes Factorization with Its Implications to RSA.pdfDr. Richard Otieno
The document presents new techniques for factorizing large semi-primes, numbers with two prime factors, that could undermine the security of the RSA cryptosystem. It analyzes Fermat's Last Theorem and Arnold's Theorem to derive equations relating the prime factors to the semi-prime. A "top-to-bottom, bottom-to-top" search approach and Arnold's Digitized Summation Technique aim to drastically reduce the time needed to find the prime factors of large RSA numbers. The document shows how these new methods allow factorization of RSA semi-primes in polynomial time on classical computers by establishing limits on the possible difference between the prime factors.
Design Of Elliptic Curve Crypto Processor with Modified Karatsuba Multiplier ...ijdpsjournal
ECDSA stands for “Elliptic Curve Digital Signature Algorithm”, it’s used to create a digital signature of
data (a file for example) in order to allow you to verify its authenticity without compromising its security.
This paper presents the architecture of finite field multiplication. The proposed multiplier is hybrid
Karatsuba multiplier used in this processor. For multiplicative inverse we choose the Itoh-Tsujii
Algorithm(ITA). This work presents the design of high performance elliptic curve crypto processor(ECCP)
for an elliptic curve over the finite field GF(2^233). The curve which we choose is the standard curve for
the digital signature. The processor is synthesized for Xilinx FPGA.
The document proposes a new encryption scheme that combines Enhanced RSA and Elgamal encryption algorithms. Enhanced RSA is based on the Integer Factorization Problem and uses three prime numbers for key generation, providing faster encryption and decryption compared to standard RSA. Elgamal encryption is based on the Discrete Logarithm Problem. The proposed scheme generates public/private keys using Enhanced RSA and then encrypts messages using Elgamal encryption. Experimental results show the proposed scheme has lower encryption times and higher throughput than Elgamal encryption or a hybrid of standard RSA and Elgamal encryption.
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.
1) The study develops a symmetric encryption algorithm based on finding a third point from two points on a line. This rule, typically used for elliptic curves, is also found to apply to lines intersecting both axes.
2) When applying the rule sequentially, the line is found to wrap around at three points - the third point from points A and B is C, and the third point from B and C is A. This feature enables key generation.
3) ASCII character codes are represented as the sum of 10 random numbers for encryption. Keys are also 10 random numbers selected such that no numbers are the same as the character code. Encryption finds the third point of the key numbers relative to the character code numbers.
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.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
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Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
This paper provides an overview of the RSA algorithm, exploring the foundations and mathematics in detail. It then uses this base information to explore issues with the RSA algorithm that detract from the algorithms security.
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.
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.
This document discusses several topics in number theory including prime numbers, relatively prime numbers, modular arithmetic, Fermat's theorem, Euler's theorem, and the Chinese Remainder theorem. It provides examples and explanations of these concepts. It also discusses how some of these number theory concepts like modular arithmetic and the difficulty of factoring large numbers into primes are applied in public key cryptography algorithms like RSA.
Presentation on Cryptography_Based on IEEE_PaperNithin Cv
The document summarizes a seminar report on a hybrid cryptography architecture that uses multiple cryptographic algorithms. It discusses Elliptic Curve Cryptography (ECC), Elliptic Curve Diffie-Hellman (ECDH), Elliptic Curve Digital Signature Algorithm (ECDSA), and Dual-RSA. ECC is used to encrypt data onto an elliptic curve. ECDH generates shared secret keys between two parties for use in symmetric encryption. ECDSA allows digital signatures using elliptic curve parameters. Dual-RSA improves decryption efficiency using the Chinese Remainder Theorem to split computations between prime factors p and q. The hybrid architecture combines the strengths of these algorithms to provide secure encryption, authentication, and key exchange.
Security_Attacks_On_RSA~ A Computational Number Theoretic Approach.pptxshahiduljahid71
The document discusses various attacks on the RSA cryptosystem. It introduces factoring attacks which aim to factor the modulus N, as knowing the prime factors p and q would allow an attacker to calculate the private key. It notes that while factoring large integers remains difficult, other attacks exist that could decrypt messages without directly factoring N. The document then examines elementary attacks like common modulus and blinding attacks that could potentially extract information about encrypted messages without inverting the RSA function. Finally, it poses an open problem around whether an algorithm could efficiently factor N given access to the RSA encryption function.
The discrete logarithm problem (DLP) is the basis for elliptic curve cryptography (ECC) and differs from the integer factorization problem in RSA. In ECC over a finite field, the DLP is to find the exponent that computes one point on the elliptic curve as a multiple of another point, given the curve equation and two points. In RSA, the problem is to find the prime factors of a composite integer. While general algorithms exist to solve both, the DLP in ECC providing equivalent security to RSA requires smaller key sizes, making ECC more efficient.
A NEW ALGORITHM TO CONSTRUCT S-BOXES WITH HIGH DIFFUSIONijscmcj
In this paper is proposed a new algorithm to construct S-Boxes over GF(28) with Branch Number value at least 3. This is an important property that guarantees a high diusion in the S-Box [12]. Also are introduced some defnition and properties that show the be- havior of S-Boxes under the composition with affine functions. Finally is presented a comparision between this algorithm and the method pro-posed by Tavares [12].
A NEW ALGORITHM TO CONSTRUCT S-BOXES WITH HIGH DIFFUSIONijscmcj
In this paper is proposed a new algorithm to construct S-Boxes over GF(28) with Branch Number value at
least 3. This is an important property that guarantees a high diusion in the S-Box [12]. Also are introduced
some defnition and properties that show the be- havior of S-Boxes under the composition with affine
functions. Finally is presented a comparision between this algorithm and the method pro-posed by Tavares
[12].
AUTHENTICATED PUBLIC KEY ENCRYPTION SCHEME USING ELLIPTIC CURVE CRYPTOGRAPHYijujournal
Secure transformation of data is of prime importance in today’s world. In the present paper, we propose a
double fold authenticated public key encryption scheme which helps us in securely sending the confidential
data between sender and receiver. This scheme makes the encrypted data more secure against various
cryptographic attacks.
AUTHENTICATED PUBLIC KEY ENCRYPTION SCHEME USING ELLIPTIC CURVE CRYPTOGRAPHYijujournal
The document presents an authenticated public key encryption scheme using elliptic curve cryptography. It proposes a double encryption method to securely transmit confidential data between a sender and receiver. In the scheme, the sender and receiver first agree on an elliptic curve and generator point over a finite field. They generate private/public key pairs and specific public keys for each other. The sender encrypts the message points in two stages - first generating cipher points using a random integer, and then performing XOR operations on the point coordinates with other values. The receiver decrypts the cipher text in two stages to recover the original message points and plaintext. An example is provided to illustrate the encryption and decryption process.
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The document presents new techniques for factorizing large semi-primes, numbers with two prime factors, that could undermine the security of the RSA cryptosystem. It analyzes Fermat's Last Theorem and Arnold's Theorem to derive equations relating the prime factors to the semi-prime. A "top-to-bottom, bottom-to-top" search approach and Arnold's Digitized Summation Technique aim to drastically reduce the time needed to find the prime factors of large RSA numbers. The document shows how these new methods allow factorization of RSA semi-primes in polynomial time on classical computers by establishing limits on the possible difference between the prime factors.
Design Of Elliptic Curve Crypto Processor with Modified Karatsuba Multiplier ...ijdpsjournal
ECDSA stands for “Elliptic Curve Digital Signature Algorithm”, it’s used to create a digital signature of
data (a file for example) in order to allow you to verify its authenticity without compromising its security.
This paper presents the architecture of finite field multiplication. The proposed multiplier is hybrid
Karatsuba multiplier used in this processor. For multiplicative inverse we choose the Itoh-Tsujii
Algorithm(ITA). This work presents the design of high performance elliptic curve crypto processor(ECCP)
for an elliptic curve over the finite field GF(2^233). The curve which we choose is the standard curve for
the digital signature. The processor is synthesized for Xilinx FPGA.
The document proposes a new encryption scheme that combines Enhanced RSA and Elgamal encryption algorithms. Enhanced RSA is based on the Integer Factorization Problem and uses three prime numbers for key generation, providing faster encryption and decryption compared to standard RSA. Elgamal encryption is based on the Discrete Logarithm Problem. The proposed scheme generates public/private keys using Enhanced RSA and then encrypts messages using Elgamal encryption. Experimental results show the proposed scheme has lower encryption times and higher throughput than Elgamal encryption or a hybrid of standard RSA and Elgamal encryption.
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.
1) The study develops a symmetric encryption algorithm based on finding a third point from two points on a line. This rule, typically used for elliptic curves, is also found to apply to lines intersecting both axes.
2) When applying the rule sequentially, the line is found to wrap around at three points - the third point from points A and B is C, and the third point from B and C is A. This feature enables key generation.
3) ASCII character codes are represented as the sum of 10 random numbers for encryption. Keys are also 10 random numbers selected such that no numbers are the same as the character code. Encryption finds the third point of the key numbers relative to the character code numbers.
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.
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https://www.leewayhertz.com/generative-ai-use-cases-and-applications/
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#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
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- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
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Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
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A New Signature Protocol Based on RSA and Elgamal Scheme
1. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
DOI: 10.5121/ijitmc.2016.4302 11
A NEW SIGNATURE PROTOCOL BASED ON RSA
AND ELGAMAL SCHEME
J. Ettanfouhi and O. Khadir
Laboratory of Mathematics, Cryptography and Mechanics, Fstm,
University Hassan II of Casablanca, Morocco
ABSTRACT
In this paper, we present a new signature scheme based on factoring and discrete logarithm problems.
Derived from a variant of ElGamal signature protocol and the RSA algorithm, this method can be seen as
an alternative protocol if known systems are broken.
KEYWORDS
Factoring, DLP, PKC, ElGamal signature scheme, RSA.
MSC: 60
94A
1. INTRODUCTION
In 1977, Rivest, Shamir, and Adleman[6 ] described the famous RSA algorithm which is based
on the presumed difficulty of factoring large integers. In 1985, ElGamal [2] proposed a signature
digital protocol that uses the hardness of the discrete logarithm problem[5 p.116, 7 p. 213, 8
p. 228 ]. Since then, many similar schemes were elaborated and published[1,3].
Among them, a new variant was conceived in 2010 by the second author[ 4 ].In this work, we
apply a combination of the new variant of Elgamal and RSA algorithm to build a secure digital
signature. The efficiency of the method is discussed and its security analyzed.
The paper is organised as follows: In section 2, we describe the basic ElGamal digital signature
algorithm and its variant. Section 3 is devoted to our new digital signature method. We end with
the conclusion in section 4.
In the paper, we will respect ElGamal work notations [3]. N, Z are respectively the sets of
integers and non-negative integers. For every positive integer n , we denote by Z
Z n
/ the finite
ring of modular integers and by *
)
/
( Z
Z n the multiplicative group of its invertible elements. Let
a ,b , c be three integers. The GCD of a and b is written as )
,
( b
a
gcd . We write
]
[
0.1 c
cm
b
a ≡ if c divides b
a − , and c
b
a mod
= if a is the rest in the division of b by
c . The bit length of n is the number of bits in its binary model, with n an integer .We start by
presenting the basic ElGamal digital signature algorithm and its variant:
2. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
12
2. ELGAMAL SIGNATURE SCHEME
In this section we recall ElGamal signature scheme[2] and its variant[4].
1. Alice chooses three numbers:
- p , a large prime integer.
- α , a primitive root of the finite multiplicative group *
)
/
( Z
Z p
- x , a random element of {1,2,..., 1
−
p }
2. She computes x
y α
= mod p . Alice’s public key is )
,
,
( y
p α , and x is her private key.
3. To sign the document m , Alice must solve the problem:
(1)
where s
r, are the unknown variables.
Alice fixes arbitrary r to be k
r α
= mod p , where k is chosen randomly and invertible
modulo 1
−
p . Equation (1) is then equivalent to:
(2)
Since Alice has the secret key x , and as the number k is invertible modulo 1
−
p , she calculates
the other unknown variable s by
(3)
4. Bob can verify the signature by checking if congruence (1) is valid for the variables r and s
given by Alice.
3. VARIANT OF ELGAMAL SIGNATURE SCHEME
We present a variant of ElGamal digital signature system.
This variant Error! Reference source not found. is based on the equation:
(4)
t
s
r ,
, are the unknown parameters, and )
,
,
( y
p α are Alice public keys. p is an integer (a large
prime). α is a primitive root of *
p
Z . y is calculated by x
y α
= mod p . x is a random
element of {1,2,...,p-1}.
3. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
13
Let )
(
= M
h
m , where h is a hash function, and M the message to be signed by Alice.
To give the solution (3), she fixes randomly r as p
mod
r k
α
≡ , and s to be p
mod
s l
α
≡ ,
where l
k, are selected arbitrary in {1,2,...,p-1}.
Equation (3) is then equivalent to:
(5)
as Alice recognize the values of x
m
l
k
s
r ,
,
,
,
, , she is able to calculate the last unknown variable
t .
Bob verify the signature by verifying the congruence (4).
this system does not use the extended Euclidean algorithm for calculating 1)
(
1
−
−
p
mod
k .
We clarify the scheme by the example given by the creator of this alternative[4].
3.1 EXAMPLE
Let )
,
,
( y
p α be Alice public keys where: 509
=
p , 2
=
α and 482
=
y . We assert that we are
not confident if using a small value of α does not abate the protocol. The private key is
281
=
x . Suppose that Alice wants to generate a signature for the document M for which
432[508]
)
( ≡
≡ M
h
m with the exponents 208
=
k and 386
=
l are randomly taken. She
computes ]
332[
2208
p
r k
≡
≡
≡ α , ]
39[
2386
p
s l
≡
≡
≡ α and 1]
440[ −
≡
+
+
≡ p
lm
ks
rx
t .
Bob or anyone can verify the relation ]
[p
s
r
y m
s
r
t
≡
α . Indeed, we find that ]
436[p
≡
α and
]
436[p
s
r
y m
s
r
≡ .
4. OUR PROTOCOL
4.1 DESCRIPTION
In this section, we describe our new digital signature. The protocol is based simultaneously on
two hard problems.
We assume first that h is a public secure hash function like SHA1[5 p.348, 7 p. 242 , 8
p.133].
We suppose that Alice public keys are )
,
,
,
( e
y
P α where:
- 1
2
= +
pq
P , p , q are three primes.
- α , a primitive root of the multiplicative group *
)
/
( Z
Z p .
- x
y α
= mod P , where x is the private key of Alice, which is randomly taken in
1}
{1,2,..., −
P .
4. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
14
- Element e is the public exponent in the RSA cryptosystem.
We propose the following protocol:
If Alice wants to sign the message M , she must give a solution for the modular equation:
(6)
where )
(
= M
h
m mod p , and r , s , t are unknown.
To solve equation (5) , Alice starts by putting:
(7)
(8)
Equation (5) becomes:
(9)
Alice uses the new variant of Elgamal algorithm[4] to solve equation (9) and to get the values of
r′ , s′ and t .
Then with her RSA private key she solves equations (7) and (8) . The cupel r and s is her
signature for the message M .
Bob or anybody can check that the signature is valid by replacing r , s and t in relation (5) .
4.2 EXAMPLE
Let us illustrate the method by the following example.
Suppose that Alice’s public key is: 104543
=
1
313
*
167
*
2
= +
P , 5
=
α , 23292
=
y , 7
=
e .
The private keys for RSA and ElGamal systems are respectively: 9502
=
x , 7399
=
d .
Assume that 12345
=
)
(
= M
h
m is the hashed message that she likes to sign.
If she takes Randomly 845
=
k and 2561
=
l .
She will find from equation 8 that 17744
=
r′ , 31839
=
s′ .
Relation 4 implies 57764
=
t .
Alice uses (6) and (7) to obtain:
5. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
15
r ≡ 1]
[ −
′ P
r d
≡ 75282
s ≡ 1]
[ −
′ P
s d
≡ 19005.
To verify Bob puts 62833
=
mod
= P
A t
α , 79849
=
mod
= 1
mod
P
y
B P
e
r −
,
83421
=
mod
1)
mod
(
= 1)
mod
(
P
P
r
C P
e
s
e −
− and 212997
=
mod
1)
mod
(
= P
P
s
D m
e
− , and
checks if P
D
C
B
A mod
*
*
= .
4.3 SECURITY ANALYSIS
Now that we have presented the protocol, we will discuss some possible attacks. Assume that
Oscar is Alice’s opponent.
ATTACK 1: If the attacker try to imitate the computation made by Alice, he can find r and s ,
but to find t he needs the value of the private key x to solve equation 4 .
ATTACK 2: Suppose Oscar is capable to solve the discrete logarithm problem [2]. He cannot
calculate r and s from equation (7) and (8) he will be confronted to the factorisation of a
large composite modulus [5,8].
ATTACK 3: Suppose Oscar is capable to solve RSA equations (7) and (8) . Oscar cannot get t
from equation (9) since x is Alice’s secret key. If he tries to get t from equation (4) , he will be
stopped by the discrete logarithm problem.
4.4 COMPLEXITY OF OUR ALGORITHM
As in [1], let exp
T , mult
T and h
T be appropriately the time to calculate an exponentiation, a
multiplication and hash function of a document M . We neglect the time needed for modular
substraction, additions, comparisons and apply the conversion mult
exp T
T 240
= .
4.4.1 SIGNATURE COMPLEXITY
To sign the message M , Alice must compute the six parameters:
)
(
= M
h
m mod P , k
r α
≡
′ ]
[P , l
s α
≡
′ ]
[P , d
r
r ′
≡ 1]
[ −
P , d
s
s ′
≡ 1]
[ −
P ,
lm
s
k
r
x
t +
′
+
′
≡ 1]
[ −
P .
Alice needs to perform four modular exponentiations, three modular multiplications and one hash
function computation. So the global required time is :
h
mult
h
mult
exp T
T
T
T
T
T +
+
+ 963
=
3
4
=
1
4.4.2 VERIFICATION COMPLEXITY
Bob should calculate 4 exponentiations, 2 multiplications and one hash function. So the global
required time is :
h
mult
h
mult
exp T
T
T
T
T
T +
+
+ 962
=
2
4
=
2
6. International Journal of Information Technology, Modeling and Computing (IJITMC) Vol. 4, No.3, August 2016
16
5. CONCLUSION
In this work, we proposed a new signature protocol that can be an alternative if old systems are
broken. Our method is based simultanyously on RSA cryptosystem and DLP.
ACKNOWLEDGEMENTS
This work is supported by the MMS e-orientation project.
REFERENCES
[1] R. R. Ahmad, E. S. Ismail,and N. M. F. Tahat , A new digital signature scheme based on factoring
and discrete logarithms , J. of Mathematics and Statistics (4): (2008), pp. .
[2] T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithm problem,
IEEE Trans. Info. Theory, IT-31, (1985), pp. .
[3] L.C. Guillou, J.J. Quisquater, A Paradoxial Identity-based SIgnature Scheme Resulting from Zero-
Knowledge, Advances in cryptography, LNCS 403, (1990) pp. .
[4] O. Khadir, New variant of ElGamal signature scheme, Int. J. Contemp. Math. Sciences, Vol. 5, no.
34, (2010), pp. .
[5] A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of applied cryptography, CRC
Press, Boca Raton, Florida, 1997.
[6] R. Rivest, A. Shamir and L. Adeleman, A method for obtaining digital signatures and public key
cryptosystems, Communication of the ACM, Vol. no 21, (1978), pp. .
[7] J. Buchmann, Introduction to Cryptography,(Second Edition), Springer 2000.
[8] D. R. Stinson, Cryptography, theory and practice, second Edition, Chapman & Hall/CRC, 2006.
Authors
Jaouad Ettanfouhi holds an engineer degree in Computer Science from the University of
Hassan II of Casablanca (2011). Member of the laboratory of Mathematics, Cryptography
and Mechanics, he is preparing a thesis in public key cryptography.
Dr Omar Khadir received his Ph.D. degree in Computer Science from the University
of Rouen, France (1994). Co-founder of the Laboratory of Mathematics, Cryptography
and Mechanics at the University of Hassan II Casablanca, Morocco, where he is a
professor in the Department of Mathematics. He teaches cryptography for graduate
students preparing a degree in computer science. His current research interests include
public key cryptography, digital signature, primality, factorisation of large integers and
more generally, all subjects connected to the information technology.