Improved Factoring Attacks on Multi-Prime RSA with Small Prime Difference

In this paper, we study the security of multi-prime RSA with small prime difference and propose two improved factoring attacks. The modulus involved in this variant is the product of r distinct prime factors of the same bit-size. Zhang and Takagi (ACISP 2013) showed a Fermat-like factoring attack on multi-prime RSA. In order to improve the previous result, we gather more information about the prime factors to derive r simultaneous modular equations. The first attack is to combine all the equations and solve one multivariate equation by generic lattice approaches. Since the equation form is similar to multi-prime Phi-hiding problem, we propose the second attack by applying the optimal linearization technique. We also show that our attacks can achieve better bounds in the experiments

Cryptanalysis of Server-Aided RSA Protocols with Private-Key Splitting

International audienceWe analyze the security and the efficiency of interactive protocols where a client wants to delegate the computation of an RSA signature given a public key, a public message and the secret signing exponent. We consider several protocols where the secret exponent is splitted using some algebraic decomposition. We first provide an exhaustive analysis of the delegation protocols in which the client outsources a single RSA exponentiation to the server. We then revisit the security of the protocols RSA-S1 and RSA-S2 that were proposed by Matsumoto, Kato and Imai in 1988. We present an improved lattice-based attack on RSA-S1 and we propose a simple variant of this protocol that provides better efficiency for the same security level. Eventually, we present the first attacks on the protocol RSA-S2 that employs the Chinese Remainder Theorem to speed up the client's computation. The efficiency of our (heuristic) attacks has been validated experimentally

Solving Linear Equations Modulo Unknown Divisors: Revisited

We revisit the problem of finding small solutions to a collection of linear equations modulo an unknown divisor $p$ for a known composite integer $N$. In CaLC 2001, Howgrave-Graham introduced an efficient algorithm for solving univariate linear equations; since then, two forms of multivariate generalizations have been considered in the context of cryptanalysis: modular multivariate linear equations by Herrmann and May (Asiacrypt\u2708) and simultaneous modular univariate linear equations by Cohn and Heninger (ANTS\u2712). Their algorithms have many important applications in cryptanalysis, such as factoring with known bits problem, fault attacks on RSA signatures, analysis of approximate GCD problem, etc. In this paper, by introducing multiple parameters, we propose several generalizations of the above equations. The motivation behind these extensions is that some attacks on RSA variants can be reduced to solving these generalized equations, and previous algorithms do not apply. We present new approaches to solve them, and compared with previous methods, our new algorithms are more flexible and especially suitable for some cases. Applying our algorithms, we obtain the best analytical/experimental results for some attacks on RSA and its variants, specifically, \begin{itemize} \item We improve May\u27s results (PKC\u2704) on small secret exponent attack on RSA variant with moduli $N = p^rq$ ($r\geq 2$). \item We experimentally improve Boneh et al.\u27s algorithm (Crypto\u2798) on factoring $N=p^rq$ ($r\geq 2$) with known bits problem. \item We significantly improve Jochemsz-May\u27 attack (Asiacrypt\u2706) on Common Prime RSA. \item We extend Nitaj\u27s result (Africacrypt\u2712) on weak encryption exponents of RSA and CRT-RSA. \end{itemize

Notes on Small Private Key Attacks on Common Prime RSA

We point out critical deficiencies in lattice-based cryptanalysis of common prime RSA presented in ``Remarks on the cryptanalysis of common prime RSA for IoT constrained low power devices'' [Information Sciences, 538 (2020) 54--68]. To rectify these flaws, we carefully scrutinize the relevant parameters involved in the analysis during solving a specific trivariate integer polynomial equation. Additionally, we offer a synthesized attack illustration of small private key attacks on common prime RSA.Comment: 15 pages, 1 figur

Finite Fields: Theory and Applications

Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation

Partial Key Exposure Attack on Common Prime RSA

In this paper, we focus on the common prime RSA variant and introduces a novel investigation into the partial key exposure attack targeting it. We explore the vulnerability of this RSA variant, which employs two common primes $p$ and $q$ defined as $p=2ga+1$ and $q=2gb+1$ for a large prime $g$. Previous cryptanalysis of common prime RSA has primarily focused on the small private key attack. In our work, we delve deeper into the realm of partial key exposure attacks by categorizing them into three distinct cases. We are able to identify weak private keys that are susceptible to partial key exposure by using the lattice-based method for solving simultaneous modular univariate linear equations. To validate the effectiveness and soundness of our proposed attacks, we conduct experimental evaluations. Through these examinations, we demonstrate the validity and practicality of the proposed partial key exposure attacks on common prime RSA

Notes on Small Private Key Attacks on Common Prime RSA

We point out critical deficiencies in lattice-based cryptanalysis of common prime RSA presented in ``Remarks on the cryptanalysis of common prime RSA for IoT constrained low power devices\u27\u27 [Information Sciences, 538 (2020) 54--68]. To rectify these flaws, we carefully scrutinize the relevant parameters involved in the analysis during solving a specific trivariate integer polynomial equation. Additionally, we offer a synthesized attack illustration of small private key attacks on common prime RSA

Revisiting Orthogonal Lattice Attacks on Approximate Common Divisor Problems and their Applications

In this paper, we revisit three existing types of orthogonal lattice (OL) attacks and propose optimized cases to solve approximate common divisor (ACD) problems. In order to reduce both space and time costs, we also make an improved lattice using the rounding technique. Further, we present asymptotic formulas of the time complexities on our optimizations as well as three known OL attacks. Besides, we give specific conditions that the optimized OL attacks can work and show how the attack ability depends on the blocksize $\beta$ in the BKZ-$\beta$ algorithm. Therefore, we put forward a method to estimate the concrete cost of solving the random ACD instances. It can be used in the choice of practical parameters in ACD problems. Finally, we give the security estimates of some ACD-based FHE constructions from the literature and also analyze the implicit factorization problem with sufficient number of samples. In the above situations, our optimized OL attack using the rounding technique performs fastest in practice

Recovering cryptographic keys from partial information, by example

Side-channel attacks targeting cryptography may leak only partial or indirect information about the secret keys. There are a variety of techniques in the literature for recovering secret keys from partial information. In this tutorial, we survey several of the main families of partial key recovery algorithms for RSA, (EC)DSA, and (elliptic curve) Diffie-Hellman, the public-key cryptosystems in common use today. We categorize the known techniques by the structure of the information that is learned by the attacker, and give simplified examples for each technique to illustrate the underlying ideas

Small CRT-Exponent RSA Revisited

Since May (Crypto\u2702) revealed the vulnerability of the small CRT-exponent RSA using Coppersmith\u27s lattice-based method, several papers have studied the problem and two major improvements have been made. (1) Bleichenbacher and May (PKC\u2706) proposed an attack for small $d_q$ when the prime factor $p$ is significantly smaller than the other prime factor $q$; the attack works for $p<N^{0.468}$. (2) Jochemsz and May (Crypto\u2707) proposed an attack for small $d_p$ and $d_q$ when the prime factors $p$ and $q$ are balanced; the attack works for $d_p, d_q<N^{0.073}$. Even a decade has passed since their proposals, the above two attacks are still considered as the state-of-the-art, and no improvements have been made thus far. A novel technique seems to be required for further improvements since it seems that the attacks have been studied with all the applicable techniques for Coppersmith\u27s methods proposed by Durfee-Nguyen (Asiacrypt\u2700), Jochemsz-May (Asiacrypt\u2706), and Herrmann-May (Asiacrypt\u2709, PKC\u2710). In this paper, we propose two improved attacks on the small CRT-exponent RSA: a small $d_q$ attack for $p<N^{0.5}$ (an improvement of Bleichenbacher-May\u27s) and a small $d_p$ and $d_q$ attack for $d_p, d_q < N^{0.122}$ (an improvement of Jochemsz-May\u27s). The latter result is also an improvement of our result in the proceeding version (Eurocrypt \u2717); $d_p, d_q < N^{0.091}$. We use Coppersmith\u27s lattice-based method to solve modular equations and obtain the improvements from a novel lattice construction by exploiting useful algebraic structures of the CRT-RSA key generation equation. We explicitly show proofs of our attacks and verify the validities by computer experiments. In addition to the two main attacks, we also propose small $d_q$ attacks on several variants of RSA