RSA Weaknesses

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.

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.

The RSA is based on a trapdoor one-way function which is easy to compute but is most hard to revert without knowing the trapdoor. A cryptanalysis, presented in this paper, consists in finding a new decrypt key which plays the same role of the original trapdoor. To find this new decrypt key we must seek the maximum degree of ciphering function composition in a given modulus N. The maximum degree (d_max) is obtained by applying the ciphering function to a restricted set of residues in the modulus N. We then define the new decrypt key by (e ^d_max). Thanks to this new key, we can decrypt any ciphertext for a given modulus. The interest of this cryptanalysis, contrary to factorization, is that the search of the decrypt key is independent from the modulus size.

2010

The basic situation is that one party, A, say Albert, wants to send a message to another party, J, say Julia. However, there is a danger that some ill-intentioned third party, Machiavelli, may intercept the message and learn things that he is not supposed to know about and as a result, do evil things. The original message, understandable to all parties, is known as the plain text. To protect the content of the message, Albert encrypts his message. When Julia receives the encrypted message, she must decrypt it in order to be able to read it.

2013

The protection on many public key encoding schemes depended on the intractability of detecting the integer factoring problem such as RSA scheme. However, there are great deals of researches regarding the RSA factoring modulus compared with the other type of attack the RSA scheme. So the need for more methods of attacks other than RSA factoring modulus to find an effective and quicker algorithm to solve this problem is still crucial. This paper introduces a new algorithmic program which approaches the RSA scheme. The suggested algorithm aims to find the private key of the RSA scheme and then factoring the modulus based on the public key of the RSA scheme. The new idea exacted to be more efficient than the already existed algorithms particularly when the public key is small, since most of public key encryption schemes select a small public encryption key e in order to improve the efficiency of encryption. Also, the suggested algorithmic program is more effective since it is faster and...

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...

Progress in Cryptology – AFRICACRYPT 2014, 2014

This paper presents three new attacks on the RSA cryptosystem. The first two attacks work when k RSA public keys (Ni, ei) are such that there exist k relations of the shape eix − yiφ(Ni) = zi or of the shape eixi − yφ(Ni) = zi where Ni = piqi, φ(Ni) = (pi − 1)(qi − 1) and the parameters x, xi, y, yi, zi are suitably small in terms of the prime factors of the moduli. We show that our attacks enable us to simultaneously factor the k RSA moduli Ni. The third attack works when the prime factors p and q of the modulus N = pq share an amount of their least significant bits (LSBs) in the presence of two decryption exponents d1 and d2 sharing an amount of their most significant bits (MSBs). The three attacks improve the bounds of some former attacks that make RSA insecure.

2016

RSA always uses two big prime numbers to deal with the encryption process. The public key is obtained from the multiplication of both figures. However, we can break it by doing factorization to split the public key into two individual numbers. Cryptanalysis can perform the public key crack by knowing its value. The private key will be soon constructed after the two numbers retrieved. The public key is noted as " N " , while "N = P. Q". This technique is unclassified anymore to solve the RSA public and private key. If it is successfully factored into p and q then ɸ (N) = (P-1). (Q-1) can be further calculated. By having the public key e, the private key d will be solved. Factorization method is the best way to do the demolition. This study concerns to numbers factorization. GCD calculation will produce the encryption "E" and decryption "D" keys, but it depends on the computer speed.

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.

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.