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

Contributions to Lattice–based Cryptography

Post–quantum cryptography (PQC) is a new and fast–growing part of Cryptography. It focuses on developing cryptographic algorithms and protocols that resist quantum adversaries (i.e., the adversaries who have access to quantum computers). To construct a new PQC primitive, a designer must use a mathematical problem intractable for the quantum adversary. Many intractability assumptions are being used in PQC. There seems to be a consensus in the research community that the most promising are intractable/hard problems in lattices. However, lattice–based cryptography still needs more research to make it more efficient and practical. The thesis contributes toward achieving either the novelty or the practicality of lattice– based cryptographic systems

Security of signed ELGamal encryption

Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel one-more-decyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts

Implementation and evaluation of improved Gaussian sampling for lattice trapdoors

We report on our implementation of a new Gaussian sampling algorithm for lattice trapdoors. Lattice trapdoors are used in a wide array of lattice-based cryptographic schemes including digital signatures, attributed-based encryption, program obfuscation and others. Our implementation provides Gaussian sampling for trapdoor lattices with prime moduli, and supports both single- and multi-threaded execution. We experimentally evaluate our implementation through its use in the GPV hash-and-sign digital signature scheme as a benchmark. We compare our design and implementation with prior work reported in the literature. The evaluation shows that our implementation 1) has smaller space requirements and faster runtime, 2) does not require multi-precision floating-point arithmetic, and 3) can be used for a broader range of cryptographic primitives than previous implementations

The Rabin cryptosystem revisited

The Rabin public-key cryptosystem is revisited with a focus on the problem of identifying the encrypted message unambiguously for any pair of primes. In particular, a deterministic scheme using quartic reciprocity is described that works for primes congruent 5 modulo 8, a case that was still open. Both theoretical and practical solutions are presented. The Rabin signature is also reconsidered and a deterministic padding mechanism is proposed.Comment: minor review + introduction of a deterministic scheme using quartic reciprocity that works for primes congruent 5 modulo