Parallel-prefix structures for binary and modulo {2n - 1, 2n, 2n + 1} adders

Adders are the among the most essential arithmetic units within digital systems. Parallel-prefix structures are efficient for adders because of their regular topology and logarithmic delay. However, building parallel-prefix adders are barely discussed in literature. This work puts emphasis on how to build prefix trees and simple algorithms for building these architectures. One particular modification of adders is for use with modulo arithmetic. The most common type of modulo adders are modulo 2n -1 and modulo 2n + 1 adders because they have a common base that is a power of 2. In order to improve their speed, parallel-prefix structures can also be employed for modulo 2n +- 1 adders. This dissertation presents the formation of several binary and modulo prefix architectures and their modifications using Ling's algorithm. For all binary and modulo adders, both algorithmic and quantitative analysis are provided to compare the performance of different architectures. Furthermore, to see how process impact the design, three technologies, from deep submicron to nanometer range, are utilized to collect the quantitative data

Efficient Unified Arithmetic for Hardware Cryptography

The basic arithmetic operations (i.e. addition, multiplication, and inversion) in finite fields, GF(q), where q = pk and p is a prime integer, have several applications in cryptography, such as RSA algorithm, Diffie-Hellman key exchange algorithm [1], the US federal Digital Signature Standard [2], elliptic curve cryptography [3, 4], and also recently identity based cryptography [5, 6]. Most popular finite fields that are heavily used in cryptographic applications due to elliptic curve based schemes are prime fields GF(p) and binary extension fields GF(2n). Recently, identity based cryptography based on pairing operations defined over elliptic curve points has stimulated a significant level of interest in the arithmetic of ternary extension fields, GF(3^n)

On singular moduli that are S-units

Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.Comment: Version changed according to the referee's comments. The final version appears in Manuscripta Mathematica, https://doi.org/10.1007/s00229-020-01230-

Design and Implementation of an RNS-based 2D DWT Processor

No abstract availabl

Secure and Efficient RNS Approach for Elliptic Curve Cryptography

Scalar multiplication, the main operation in elliptic curve cryptographic protocols, is vulnerable to side-channel (SCA) and fault injection (FA) attacks. An efficient countermeasure for scalar multiplication can be provided by using alternative number systems like the Residue Number System (RNS). In RNS, a number is represented as a set of smaller numbers, where each one is the result of the modular reduction with a given moduli basis. Under certain requirements, a number can be uniquely transformed from the integers to the RNS domain (and vice versa) and all arithmetic operations can be performed in RNS. This representation provides an inherent SCA and FA resistance to many attacks and can be further enhanced by RNS arithmetic manipulation or more traditional algorithmic countermeasures. In this paper, extending our previous work, we explore the potentials of RNS as an SCA and FA countermeasure and provide an description of RNS based SCA and FA resistance means. We propose a secure and efficient Montgomery Power Ladder based scalar multiplication algorithm on RNS and discuss its SCAFA resistance. The proposed algorithm is implemented on an ARM Cortex A7 processor and its SCA-FA resistance is evaluated by collecting preliminary leakage trace results that validate our initial assumptions