Multipliers are used in most arithmetic computing systems such as 3D graphics, signal processing, and etc. It is inherently a slow operation as a large number of partial products are added to produce the result. There has been much work done on designing multipliers [1]-[6]. In first stage, Multiplication is implemented by accumulation of partial products, each of which is conceptually produced via multiplying the whole multi-digit multiplicand by a weighted digit of multiplier. To compute partial products, most of the approaches employ the Modified Booth Encoding (MBE) approach [3]-[5], [7], for the first step because of its ability to cut the number of partial products rows in half. In next step the partial products are reduced to a row of sums and a row of caries which is called reduction stage. / Multiplication is one of the major bottlenecks in most digital computing and signal processing systems, which depends on the word size to be executed. This paper presents three deferent designs for three-operand 4-bit multiplier for positive integer multiplication, and compares them in regard to timing, dynamic power, and area with classical method of multiplication performed on today architects. The three-operand 4-bit multipliers structure introduced, serves as a building block for three-operand multipliers in general

Large integers are used in several key areas such as RSA (Rivest-Shamir-Adleman) public-key cryptographic system and elliptic curve public-key cryptographic system. To achieve higher levels of security requires larger key size and this becomes a limiting factor in prime finite field GF(p) arithmetic using large integers because operations on large integers suffer from the long carry propagation problem. Large integer representation has direct impact on the efficiency of the calculations and the hardware and software implementations. Attempts to use different representations such as residue number systems suffer from their own problems. In this dissertation, we propose a novel and efficient attribute-based large integer representation scheme capable of efficiently representing the large integers that are commonly used in cryptography such as the five NIST primes and the Pierpont primes used in supersingular isogeny Diffie-Hellman (SIDH) used in post-quantum cryptography. Moreover, we propose algorithms for this new representation to perform arithmetic operations such as conversions from and to binary representation, two’s complement, left-shift, numbers comparison, addition/subtraction, modular addition/subtraction, modular reduction, multiplication, and modular multiplication. Extensive numerical simulations and software implementations are done to verify the performance of the new number representation. Results show that the attribute-based large integer arithmetic operations are done faster in our proposed representation when compared with binary and residue number representations. This makes the proposed representation suitable for cryptographic applications on embedded systems and IoT devices with limited resources for better security level. / Graduate / 2020-07-04