New attacks on RSA with Moduli N = p^r q

International audienceWe present three attacks on the Prime Power RSA with mod-ulus N = p^r q. In the first attack, we consider a public exponent e satisfying an equation ex − φ(N)y = z where φ(N) = p^(r−1 )(p − 1)(q − 1). We show that one can factor N if the parameters |x| and |z| satisfy |xz| < N r(r−1) (r+1)/ 2 thereby extending the recent results of Sakar [16]. In the second attack, we consider two public exponents e1 and e2 and their corresponding private exponents d1 and d2. We show that one can factor N when d1 and d2 share a suitable amount of their most significant bits, that is |d1 − d2| < N r(r−1) (r+1) /2. The third attack enables us to factor two Prime Power RSA moduli N1 = p1^r q1 and N2 = p2^r q2 when p1 and p2 share a suitable amount of their most significant bits, namely, |p1 − p2| < p1/(2rq1 q2)

Cryptanalysis of NTRU with two public keys

NTRU is a fast public key cryptosystem presented in 1996 by Hoffstein, Pipher and Silverman. It operates in the ring of truncated polynomials. In NTRU, a public key is a polynomial defined by the combination of two private polynomials. In this paper, we consider NTRU with two different public keys defined by different private keys. We present a lattice-based attack to recover the private keys assuming that the public keys share polynomials with a suitable number of common coefficients

Implicit factorization of unbalanced RSA moduli

International audienceLet N1 = p1q1 and N2 = p2q2 be two RSA moduli, not necessarily of the same bit-size. In 2009, May and Ritzenhofen proposed a method to factor N1 and N2 given the implicit information that p1 and p2 share an amount of least significant bits. In this paper, we propose a generalization of their attack as follows: suppose that some unknown multiples a1p1 and a2p2 of the prime factors p1 and p2 share an amount of their Most Significant Bits (MSBs) or an amount of their Least Significant Bits (LSBs). Using a method based on the continued fraction algorithm, we propose a method that leads to the factorization of N1 and N2. Using simultaneous diophantine approximations and lattice reduction , we extend the method to factor k ≥ 3 RSA moduli Ni = piqi, i = 1,. .. , k given the implicit information that there exist unknown multiples a1p1,. .. , ak pk sharing an amount of their MSBs or their LSBs. Also, this paper extends many previous works where similar results were obtained when the pi's share their MSBs or their LSBs

Cryptanalysis of the Randomized Version of a Lattice-Based Signature Scheme from PKC'08

International audienceIn PKC'08, Plantard, Susilo and Win proposed a lattice-based signature scheme, whose security is based on the hardness of the closest vector problem with the infinity norm (CVP∞). This signature scheme was proposed as a countermeasure against the Nguyen-Regev attack, which improves the security and the efficiency of the Goldreich, Goldwasser and Halevi scheme (GGH). Furthermore, to resist potential side channel attacks, the authors suggested modifying the determinis-tic signing algorithm to be randomized. In this paper, we propose a chosen message attack against the randomized version. Note that the randomized signing algorithm will generate different signature vectors in a relatively small cube for the same message, so the difference of any two signature vectors will be relatively short lattice vector. Once collecting enough such short difference vectors, we can recover the whole or the partial secret key by lattice reduction algorithms, which implies that the randomized version is insecure under the chosen message attack

Bitcoin Security with a Twisted Edwards Curve

International audienceThe security of the Bitcoin cryptocurrency system depends on the Koblitz curve secp256k1 combined with the digital signature ECDSA and the hash function SHA-256. In this paper, we show that the security of Bitcoin with ECDSA and secp256k1 is not optimal and present a detailed study of the efficiency of Bitcoin with the digital signature algorithm Ed25519 combined with the twisted Edwards curve CurveEd25519 and the hash function SHA-512. We show that Bitcoin is more secure and more efficient with the digital signature algorithm Ed25519 and the twisted Edwards curve CurveEd25519. Subject Classifications: 94A6

A New RSA Variant Based on Elliptic Curves

We propose a new scheme based on ephemeral elliptic curves over the ring $\mathbb{Z}/n\mathbb{Z}$ where $n=pq$ is an RSA modulus with $p=u_p^2+v_p^2$, $q=u_q^2+v_q^2$, $u_p\equiv u_q\equiv 3\pmod 4$. The new scheme is a variant of both the RSA and the KMOV cryptosystems. The scheme can be used for both signature and encryption. We study the security of the new scheme and show that is immune against factorization attacks, discrete logarithm problem attacks, sum of two squares attacks, sum of four squares attacks, isomorphism attacks, and homomorphism attacks. Moreover, we show that the private exponents can be much smaller than the ordinary exponents for RSA and KMOV, which makes the decryption phase in the new scheme more efficient

A Unified Method for Private Exponent Attacks on RSA using Lattices

International audienceLet (n = pq, e = n^β) be an RSA public key with private exponent d = n^δ , where p and q are large primes of the same bit size. At Eurocrypt 96, Coppersmith presented a polynomial-time algorithm for finding small roots of univariate modular equations based on lattice reduction and then succussed to factorize the RSA modulus. Since then, a series of attacks on the key equation ed − kφ(n) = 1 of RSA have been presented. In this paper, we show that many of such attacks can be unified in a single attack using a new notion called Coppersmith's interval. We determine a Coppersmith's interval for a given RSA public key (n, e). The interval is valid for any variant of RSA, such as Multi-Prime RSA, that uses the key equation. Then we show that RSA is insecure if δ < β + 1/3 α − 1/3 √ (12αβ + 4α^2) provided that we have approximation p0 ≥ √ n of p with |p − p0| ≤ 1/2 n^α , α ≤ 1/2. The attack is an extension of Coppersmith's result

Lattice Attacks on the DGHV Homomorphic Encryption Scheme

In 2010, van Dijk, Gentry, Halevi, and Vaikuntanathan described the first fully homomorphic encryption over the integers, called DGHV. The scheme is based on a set of $m$ public integers $c_i=pq_i+r_i$, $i=1,\cdots,m$, where the integers $p$, $q_i$ and $r_i$ are secret. In this paper, we describe two lattice-based attacks on DGHV. The first attack is applicable when $r_1=0$ and the public integers $c_i$ satisfy a linear equation $a_2c_2+\ldots+a_mc_m=a_1q_1$ for suitably small integers $a_i$, $i=2,\ldots,m$. The second attack works when the positive integers $q_i$ satisfy a linear equation $a_1q_1+\ldots+a_mq_m=0$ for suitably small integers $a_i$, $i=1,\ldots,m$. We further apply our methods for the DGHV recommended parameters as specified in the original work of van Dijk, Gentry, Halevi, and Vaikuntanathan

Applications of Neural Network-Based AI in Cryptography

Artificial intelligence (AI) is a modern technology that allows plenty of advantages in daily life, such as predicting weather, finding directions, classifying images and videos, even automatically generating code, text, and videos. Other essential technologies such as blockchain and cybersecurity also benefit from AI. As a core component used in blockchain and cybersecurity, cryptography can benefit from AI in order to enhance the confidentiality and integrity of cyberspace. In this paper, we review the algorithms underlying four prominent cryptographic cryptosystems, namely the Advanced Encryption Standard, the Rivest--Shamir--Adleman, Learning With Errors, and the Ascon family of cryptographic algorithms for authenticated encryption. Where possible, we pinpoint areas where AI can be used to help improve their security

Function Call Graph Score for Malware Detection

Metamorphic malware changes its internal structure with each infection, while maintaining its core functionality. Detecting such malware is a challenging research problem. Function call graph analysis has previously shown promise in detecting such malware. In this research, we analyze the robustness of a function call graph score with respect to various code morphing strategies. We also consider modifications of the score that make it more robust in the face of such morphing