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

On Deterministic Polynomial-time Equivalence of Computing the CRT-RSA Secret Keys and Factoring

Let N = pq be the product of two large primes. Consider Chinese remainder theorem-Rivest, Shamir, Adleman (CRT-RSA) with the public encryption exponent e and private decryption exponents dp, dq. It is well known that given any one of dp or dq (or both) one can factorise N in probabilistic poly(log N) time with success probability almost equal to 1. Though this serves all the practical purposes, from theoretical point of view, this is not a deterministic polynomial time algorithm. In this paper, we present a lattice-based deterministic poly(log N) time algorithm that uses both dp, dq (in addition to the public information e, N) to factorise N for certain ranges of dp, dq. We like to stress that proving the equivalence for all the values of dp, dq may be a nontrivial task.Defence Science Journal, 2012, 62(2), pp.122-126, DOI:http://dx.doi.org/10.14429/dsj.62.171

A polynomial time attack on RSA with private CRT-exponents smaller than N0.073N^{0.073}

Wiener’s famous attack on RSA with d

A Secured Data Protocol for the Trusted Truck(R) System

Security has become one of the major concerns in the Intelligent Transportation Systems (ITS). The Trusted Truck(R) System, provides an efficient wireless communication mechanism for safe exchange of messages between the moving vehicles (trucks) and the roadside inspection stations. The vehicles and the station are equipped with processing units but with different computational capabilities. To make this Trusted Truck(R) system more secure, this thesis proposes a secured data protocol which ensures data integrity, message authentication and non-repudiation. The uniqueness of the protocol is: it is cost-effective, resource-efficient and embeds itself into the Trusted Truck (R) environment without demanding any additional infrastructure. The protocol also balances the computational load between the vehicle and station by incorporating an innovative key transport mechanism. Digital signatures and encryption techniques are used for authentication and data condentiality. Cryptography algorithms along with optimization methods are used for the digital signatures. The computational time for the algorithms are analyzed. Combining all these techniques, an efficient secured data protocol is developed and implemented successfully

Fast signing method in RSA with high speed verification

In this paper, we propose the method to speed up signature generation in RSA with small public exponent. We first divide the signing algorithm into two stages. One is message generating stage and the other is signing stage. Next, we modify the RSA signature so that the bulk of the calculation cost is allocated to message generating stage. This gives the possibility to propose the RSA signature schemes which have fast signature generation and very fast verification. Our schemes are suited for the applications in which a message is generated offline, but needs to be quickly signed and verified online

CCA secure ElGamal encryption over an integer group where ICDH assumption holds

In order to prove the ElGamal CCA (Chosen Ciphertext Attack) security in the random oracle model, it is necessary to use the group (i.e., ICDH group) where ICDH assumption holds. Until now, only bilinear group where ICDH assumption is equivalent to CDH assumption has been known as the ICDH group. In this paper, we introduce another ICDH group in which ICDH assumption holds under the RSA assumption. Based on this group, we propose the CCA secure ElGamal encryption. And we describe the possibility to speed up decryption by reducing CRT (Chinese Remainder Theorem) exponents in CCA secure ElGamal

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