An efficient and secure RSA--like cryptosystem exploiting R\'edei rational functions over conics

We define an isomorphism between the group of points of a conic and the set of integers modulo a prime equipped with a non-standard product. This product can be efficiently evaluated through the use of R\'edei rational functions. We then exploit the isomorphism to construct a novel RSA-like scheme. We compare our scheme with classic RSA and with RSA-like schemes based on the cubic or conic equation. The decryption operation of the proposed scheme turns to be two times faster than RSA, and involves the lowest number of modular inversions with respect to other RSA-like schemes based on curves. Our solution offers the same security as RSA in a one-to-one communication and more security in broadcast applications.Comment: 18 pages, 1 figur

Hard isogeny problems over RSA moduli and groups with infeasible inversion

We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders. Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.Comment: Significant revision of the article previously titled "A Candidate Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the constructions by giving toy examples, added "The Parallelogram Attack" (Sec 5.3.2). 54 pages, 8 figure

An Authentication Protocol for Future Sensor Networks

Authentication is one of the essential security services in Wireless Sensor Networks (WSNs) for ensuring secure data sessions. Sensor node authentication ensures the confidentiality and validity of data collected by the sensor node, whereas user authentication guarantees that only legitimate users can access the sensor data. In a mobile WSN, sensor and user nodes move across the network and exchange data with multiple nodes, thus experiencing the authentication process multiple times. The integration of WSNs with Internet of Things (IoT) brings forth a new kind of WSN architecture along with stricter security requirements; for instance, a sensor node or a user node may need to establish multiple concurrent secure data sessions. With concurrent data sessions, the frequency of the re-authentication process increases in proportion to the number of concurrent connections, which makes the security issue even more challenging. The currently available authentication protocols were designed for the autonomous WSN and do not account for the above requirements. In this paper, we present a novel, lightweight and efficient key exchange and authentication protocol suite called the Secure Mobile Sensor Network (SMSN) Authentication Protocol. In the SMSN a mobile node goes through an initial authentication procedure and receives a re-authentication ticket from the base station. Later a mobile node can use this re-authentication ticket when establishing multiple data exchange sessions and/or when moving across the network. This scheme reduces the communication and computational complexity of the authentication process. We proved the strength of our protocol with rigorous security analysis and simulated the SMSN and previously proposed schemes in an automated protocol verifier tool. Finally, we compared the computational complexity and communication cost against well-known authentication protocols.Comment: This article is accepted for the publication in "Sensors" journal. 29 pages, 15 figure

A New Attack on Three Variants of the RSA Cryptosystem

International audienceIn 1995, Kuwakado, Koyama and Tsuruoka presented a new RSA-type scheme based on singular cubic curves y^2 ≡ x^3 + bx^2 (mod N) where N = pq is an RSA modulus. Then, in 2002, Elkamchouchi, Elshenawy and Shaban introduced an extension of the RSA scheme to the field of Gaussian integers using a modulus N = P Q where P and Q are Gaussian primes such that p = |P | and q = |Q| are ordinary primes. Later, in 2007, Castagnos's proposed a scheme over quadratic fields quotients with an RSA modulus N = pq. In the three schemes, the public exponent e is an integer satisfying the key equation ed − k^(p^2 − 1) (q^2 − 1) = 1. In this paper, we apply the continued fraction method to launch an attack on the three schemes when the private exponent d is sufficiently small. Our attack can be considered as an extension of the famous Wiener attack on RSA

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