A new algorithm for multiplication in finite fields

Cover title.Includes bibliographical references.by Antonio Pincin

The construction of self-dual normal polynomials over GF(2) and their applications to the Massey-Omura algorithm

Gaussian periods are used to locate a normal element of the finite field GF(2e) of odd degree e and an algorithm is presented for the construction of self-dual normal polynomials over GF(2) for any odd degree. This gives a new constructive proof of the existence of a self-dual basis for odd degree. The use of such polynomials in the Massey-Omura multiplier improves the efficiency and decreases the complexity of the multiplie

Stream ciphers: A Practical Solution for Efficient Homomorphic-Ciphertext Compression

In typical applications of homomorphic encryption, the first step consists for Alice to encrypt some plaintext m under Bob’s public key pk and to send the ciphertext c = HE_pk(m) to some third-party evaluator Charlie. This paper specifically considers that first step, i.e. the problem of transmitting c as efficiently as possible from Alice to Charlie. As previously noted, a form of compression is achieved using hybrid encryption. Given a symmetric encryption scheme E, Alice picks a random key k and sends a much smaller ciphertext c′ = (HE_pk(k), E_k(m)) that Charlie decompresses homomorphically into the original c using a decryption circuit C_E^{−1}. In this paper, we revisit that paradigm in light of its concrete implementation constraints; in particular E is chosen to be an additive IV-based stream cipher. We investigate the performances offered in this context by Trivium, which belongs to the eSTREAM portfolio, and we also propose a variant with 128-bit security: Kreyvium. We show that Trivium, whose security has been firmly established for over a decade, and the new variant Kreyvium have an excellent performance

High-speed algorithms & architectures for number-theoretic cryptosystems

Computer and network security systems rely on the privacy and authenticity of information, which requires implementation of cryptographic functions. Software implementations of these functions are often desired because of their flexibility and cost effectiveness. In this study, we concentrate on developing high-speed and area-efficient modular multiplication and exponentiation algorithms for number-theoretic cryptosystems. The RSA algorithm, the Diffie-Hellman key exchange scheme and Digital Signature Standard require the computation of modular exponentiation, which is broken into a series of modular multiplications. One of the most interesting advances in modular exponentiation has been the introduction of Montgomery multiplication. We are interested in two aspects of modular multiplication algorithms: development of fast and convenient methods on a given hardware platform, and hardware requirements to achieve high-performance algorithms. Arithmetic operations in the Galois field GF(2[superscript]k) have several applications in coding theory, computer algebra, and cryptography. We are especially interested in cryptographic applications where k is large, such as elliptic curve cryptosystems