Cryptanalysis Attacks on Multi Prime Power Modulus Through Analyzing Prime Difference (original) (raw)
New Cryptanalytic Attack on RSA Modulus N=pq Using Small Prime Difference Method
Cryptography, 2018
This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus N = p q in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form | b 2 p - a 2 q | < N γ where the ratio of q p is close to b 2 a 2 , which yields a bound d < 3 2 N 3 4 - γ from the convergents of the continued fraction expansion of e N - ⌈ a 2 + b 2 a b N ⌉ + 1 . The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli N s = p s q s for s = 1 , 2 , … , t where we use N - ⌈ a 2 + b 2 a b N ⌉ + 1 as an approximation of ϕ ( N ) satisfying generalized key equations of the shape e s d - k s ϕ ( N s ) = 1 , e s d s - k ϕ ( N s ) = 1 , e s d - k s ϕ ( N s ) = z s , and e s d s - k ϕ ( N s ) = z s for unknown positive integers d , k s , d s , k s , and z s , where we establish that t RSA moduli can be simultaneously factored in polynomial time using...
Cryptanalysis on RSA Using Decryption Exponent
In this paper, we present two new decryption exponent cryptanalysis on RSA, which successfully lead to the factorization of RSA modulus = = 2 in polynomial time. We applied Wiener's technique of attack in RSA and developed the new attacks. In the first attack, we consider RSA with modulus = , < < 2 , with public encryption exponent e and private decryption exponent d. If in polynomial time.
2013
The RSA cryptosystem, named after Ron Rivest, Adi Shamir and Leonard Adleman, who first publicly described it in 1978, is a cryptographic public-key system based on the presumed difficulty of factoring integers. To receive an RSA-encrypted message a user selects two large prime numbers and publishes the product, along with an auxiliary value, as public key. The prime factors must be kept secret. Anyone can use this public key to encrypt a message. Someone knowing the prime factors can feasibly decode the message. But there exist several approaches to break the cryptographic system without this knowledge. In this project, we implement and study the efficiency and effectiveness of three RSA attacks - Integer Factorisation, Guessing plaintext, and Guessing φ(N) attack. In order to achieve this aim, we study the RSA algorithm and implement our version of the RSA algorithm. In our study of the RSA algorithm, we look at various algorithms and number theory relevant for the implementation of RSA.
On Some Attacks on Multi-prime RSA
Lecture Notes in Computer Science, 2003
Using more than two factors in the modulus of the RSA cryptosystem has the arithmetic advantage that the private key computations can be speeded up using Chinese remaindering. At the same time, with a proper choice of parameters, one does not have to work with a larger modulus to achieve the same level of security in terms of the difficulty of the integer factorization problem. However, numerous attacks on specific instances on the RSA cryptosystem are known that apply if, for example, the decryption or encryption exponent are chosen too small, or if partial knowledge of the private key is available. Little work is known on how such attacks perform in the multi-prime case. It turns out that for most of these attacks it is crucial that the modulus contains exactly two primes. They become much less effective, or fail, when the modulus factors into more than two distinct primes.
Public key exponent attacks on multi-prime power modulus using continued fraction expansion method
Caliphate Journal of Science and Technology, 2023
This paper proposes three public key exponent attacks of breaking the security of the prime power modulus = 2 2 where and are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus = 2 2 for q < < 2q using key equation − () = 1 where () = 2 2 (− 1)(− 1) can be broken by recovering the secret keys from the convergents of the continued fraction expansion of e −2 3 4 + 1 2. The paper also reports the second and third approaches of factoring multi-prime power moduli = 2 2 simultaneously through exploiting generalized system of equations − () = 1 and − () = 1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for = 1, 2, … , .
A Comparative Study of RSA based Cryptographic Algorithms
Iasse, 2004
In 1978 a powerful and practical public-key scheme Hadi Otrokwas produced by RSA; there work was applied using 2 large random odd primes p and q, each roughly of the same size. El-Kassar and Awad modi-…ed the RSA public-key encryption scheme from the domain of natural integers, Z, to two principal ideal domains, namely the domain of Gaussian integers, Z[i], and the domain of the rings of polynomials over …nite …elds, F [x], by extending the arithmetic needed for the modi…cations to these domains. In this work we implement the classical and modi…ed RSA cryptosystem to compare and to test their functionality, realiability and security. To test the security of the algorithms we implement an attack algorithm to solve the integer factorization problem. After factorization is found, the RSA problem could be solved by computing the order ©(n), and then …nding the private key using the extended Euclidean algorithm for integers.
Cryptanalysis of Multi Prime RSA with Secret Key Greater than Public Key
The efficiency of decryption process of Multi prime RSA, in which the modulus contains more than two primes, can be speeded up using Chinese remainder theorem (CRT). On the other hand, to achieve the same level of security in terms integer factorization problem the length of RSA modulus must be larger than the traditional RSA case. In [9], authors studied the RSA public key cryptosystem in a special case with the secret exponent d larger than the public exponent e. In this paper, we show that how such attack is performed in the multi-prime RSA case.
ijser.org
The RSA cryptosystem is most widely used cryptosystem it may be used to provide both secrecy and digital signatures and its security is based on the intractability of the integer factorization. The security of RSA algorithm depends on the ability of the hacker to factorize numbers. New, faster and better methods for factoring numbers are constantly being devised. The Trent best for long numbers is the Number Field Sieve. Although the past work has proven that none of the attacks on RSA cryptosystem were dangerous. Indeed most of the dangers were because of improper use of RSA. In this paper what I am trying to do is to analyze the different types of possible attacks on RSA Cryptosystem.
Forty years of attacks on the RSA cryptosystem: A brief survey
Journal of Discrete Mathematical Sciences and Cryptography, 2019
RSA public key cryptosystem is the de-facto standard use in worldwide technologies as a strong encryption/decryption and digital signature scheme. RSA successfully defended forty years of attack since invention. In this study we survey, its past, present advancements and upcoming challenges that needs concrete analysis and as a counter measure against possible threats according to underlying algebraic structure. Past studies shows us some attacks on RSA by inspecting flaws on relax model using weak public/private keys, integer factorization problem, and some specific low parameter selection attacks. Such flaws can not hamper the security of RSA cryptosystem by at large, but can explore possible vulnerabilities for more deep understanding about underlying mathematics and improper parameter selection. We describe a brief survey of past findings and detail description about specific attacks. A comprehensive survey of known attacks on RSA cryptosystem shows us that a well implemented algorithm is unbreakable and it survived against a number of cryptanalytic attacks since last forty years.
Factorization strategies of N = pq and N = pʳq and relation to its decryption exponent bound
2018
The major RSA underlying security problems rely on the difficulty of factoring a very large composite integer N into its two nontrivial prime factors of p and q in polynomial time, the ability to solve a given Diophantine equation ed = 1 + kφ (N) where only the public key e is known and the parameters d, k and φ (N) are un- known and finally the failure of an adversary to compute the decryption key d from the public key pair (e, N). This thesis develops three new strategies for the factorization of RSA modulus N = pq through analyzing small prime difference satisfying inequalities |b2 p − a2q| < Nγ , |bi p − a jq| < Nγ and |b j p − a jq| < for... This research work also focuses on successful factorization of t RSA moduli Ns = psqs. By using good approximation of φ (N) and generalized key equations of the form esd ksφ (Ns) = 1, esds kφ (Ns) = 1, esd kφ (Ns) = zs and esds kφ (Ns) = zs for s = 1, 2, . . . , t. This method leads to simultaneous factoring of t RSA moduli Ns = ps...