Fast Large Integer Modular Addition in GF(p) Using Novel Attribute-Based Representation

Abstract

Addition is an essential operation in all cryptographic algorithms. Higher levels of security require larger key sizes and this becomes a limiting factor in GF(p) using large integers because of the carry propagation problem. We propose a novel and efficient attribute-based large integer representation scheme suitable for large integers commonly used in cryptography such as the five NIST primes and the Pierpont primes used in supersingular isogeny Diffie-Hellman (SIDH) for post-quantum cryptography. Algorithms are proposed for this new representation to implement arithmetic operations such as two's complement, addition/subtraction, comparison, sign detection, and modular reduction. Algorithms are also developed for converting binary numbers to attribute representation and vice versa. The extensive numerical simulations were done to verify the performance of the new number representation. Results show that addition is done faster in our proposed representation when compared with binary and residue number system (RNS)-based additions. Attribute addition outperformed RNS addition for all values of m where 128 ≤ m ≤ 32 768 bits for all machine word sizes w where 4 ≤ w ≤ 128 bits. Attribute-based addition outperforms Kogge-Stone binary adders for a wide range of m when w is small. For increasing values of w, the speed advantages are evident only for large values of m. This makes the proposed number representation suitable for implementing cryptographic applications in embedded processors for IoT and consumer electronic devices where w is small. INDEX TERMS Prime fields GF(p), large integer arithmetic, modular arithmetic, Kogge-Stone adder, number representation, post-quantum cryptography, SIDH, cryptographic processor, embedded systems, NIST primes, generalized pierpont prime, parallel algorithms.BADER ALHAZMI received the B.Sc. degree in electrical and computer engineering from King Abdulaziz University, Jeddah, Saudi Arabia, and the master's degree in information systems security from Concordia University, Montreal, QC, Canada. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, University of Victoria, Canada. His research interests include cryptosystems, hardware security, computer arithmetic, and parallel algorithms.FAYEZ GEBALI received the B.Sc. degree (Hons.) in electrical engineering from Cairo University, the B.Sc. degree (Hons.) in mathematics from Ain Shams University, and the Ph.D. degree in electrical engineering from the University of British Columbia. He is currently a Professor of electrical and computer engineering with the University of Victoria. His research interests include parallel algorithms, dataflow computing, finite-field arithmetic, and radar signal processing.

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“…Over , carry skip (CSK), carry select (CSL), carry-save (CSA), carry-lookahead (CLA), and parallel prefix (PPF) adders are some of the fast binary adders proposed in the literature [ 23 ]. The Kogge–Stone adder (KSA) is the fastest adder in the literature because it has a lower fan-out at each stage, which then increases its performance, and is thus widely considered as a standard adder in the industry for high-performance arithmetic circuits [ 24 ].…”

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

“…As the newly designed addition circuit is the most fundamental operation, developing an efficient addition circuit leads to the practical design of other essential operations such as multiplication, division, and modular addition, which are primitives for solving various problems [10][11][12]. Thus far, various quantum modular adders for specific fields have been proposed based on existing classical addition algorithms [13], utilizing quantum adders, such as Quantum Ripple Carry Adder (QRCA), Quantum Carry Save Adder (QCSA), and Quantum Carry Lookahead Adder (QCLA), classified by how to handle the carry propagation [14]. A more special addition circuit, such as Lu's quantum adder for superposition states, as one of the quantum principles has also been proposed [12].…”