28 research outputs found
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)
Reducing the Complexity of Normal Basis Multiplication
In this paper we introduce a new transformation method and a multiplication
algorithm for multiplying the elements of the field GF
expressed in a normal basis. The number of XOR gates for the proposed
multiplication algorithm is
fewer than that of the optimal normal basis multiplication, not taking into
account the cost of forward and backward transformations. The algorithm is
more suitable for applications in which tens or hundreds of field multiplications
are performed before needing to transform the results back
Yet another Improvement of Plantard Arithmetic for Faster Kyber on Low-end 32-bit IoT Devices
This paper presents another improved version of Plantard arithmetic that
could speed up Kyber implementations on two low-end 32-bit IoT platforms (ARM
Cortex-M3 and RISC-V) without SIMD extensions. Specifically, we further enlarge
the input range of the Plantard arithmetic without modifying its computation
steps. After tailoring the Plantard arithmetic for Kyber's modulus, we show
that the input range of the Plantard multiplication by a constant is at least
2.45 times larger than the original design in TCHES2022. Then, two optimization
techniques for efficient Plantard arithmetic on Cortex-M3 and RISC-V are
presented. We show that the Plantard arithmetic supersedes both Montgomery and
Barrett arithmetic on low-end 32-bit platforms. With the enlarged input range
and the efficient implementation of the Plantard arithmetic on these platforms,
we propose various optimization strategies for NTT/INTT. We minimize or
entirely eliminate the modular reduction of coefficients in NTT/INTT by taking
advantage of the larger input range of the proposed Plantard arithmetic on
low-end 32-bit platforms. Furthermore, we propose two memory optimization
strategies that reduce 23.50% to 28.31% stack usage for the speed-version Kyber
implementation when compared to its counterpart on Cortex-M4. The proposed
optimizations make the speed-version implementation more feasible on low-end
IoT devices. Thanks to the aforementioned optimizations, our NTT/INTT
implementation shows considerable speedups compared to the state-of-the-art
work. Overall, we demonstrate the applicability of the speed-version Kyber
implementation on memory-constrained IoT platforms and set new speed records
for Kyber on these platforms
Parallel Prefix Computation with Few Processors
We present a parallel prefix algorithm which uses..
Finite field arithmetic for cryptography
Cryptography is one of the most prominent application areas of the finite field arithmetic. Almost all public-key cryptographic algorithms including the recent algorithms such as elliptic curve and pairing-based cryptography rely heavily on finite field arithmetic, which needs to be performed efficiently to meet the execution speed and design space constraints. These objectives constitute massive challenges that necessitate interdisciplinary research efforts that will render the best algorithms, architectures, implementations, and design practices. This paper aims to provide a concise perspective on designing architectures for efficient finite field arithmetic for usage in cryptography. We present different architectures, methods and techniques for fast execution of cryptographic operations as well as high utilization of resources in the realization of cryptographic algorithms. While it is difficult to have a complete coverage of all related work, this paper aims to reflect the current trends and important implementation issues of finite field arithmetic in the context of cryptography