Non-adjacent Form Research Articles

An Optimized Hardware Implementation of a Non-Adjacent Form Algorithm Using Radix-4 Multiplier for Binary Edwards Curves

Binary Edwards Curves (BEC) play a pivotal role in modern cryptographic processes and applications, offering a combination of robust security as well as computational efficiency. For robust security, this article harnesses the inherent strengths of BEC for the cryptographic point multiplication process by utilizing the Non-Adjacent Form (NAF) algorithm. For computational efficiency, a hardware architecture for the NAF algorithm is proposed. Central to this architecture is an Arithmetic Logic Unit (ALU) designed for streamlined execution of essential operations, including addition, squaring, and multiplication. One notable innovation in our ALU design is the integration of multiplexers, which maximize ALU efficiency with minimal additional hardware requirements. Complementing the optimized ALU, the proposed architecture incorporates a radix-4 multiplier, renowned for its efficiency in both multiplication and reduction. It eliminates resource-intensive divisions, resulting in a substantial boost to overall computational speed. The architecture is implemented on Xilinx Virtex series Field-Programmable Gate Arrays (FPGAs). It achieves throughput-to-area ratios of 14.819 (Virtex-4), 25.5 (Virtex-5), 34.58 (Virtex-6), and 37.07 (Virtex-7). These outcomes underscore the efficacy of our optimizations, emphasizing an equilibrium between computational performance and area utilization.

Blockchain Security Mechanism Design Based on Chinese Cryptosystem SM2 Algorithm

The cryptographic system is the foundation of blockchain security mechanisms. This paper constructs a blockchain security mechanism based on the Chinese national cryptographic SM2 algorithm, involving the generation of public–private key pairs and the signing of block information using key pairs. However, the original SM2 algorithm has a high computational complexity, which cannot meet the requirements of low on-chain delay in blockchain applications. To solve this problem, this paper optimizes the SM2 digital signature algorithm. The scalar multiplication operation in the digital signature algorithm has a large computational complexity, which affects the efficiency of a digital signature. To reduce its computational complexity, this paper improves the traditional window non-adjacent form (w-NAF) algorithm for the scalar multiplication operation and proposes a low computational complexity Fw-NAF algorithm based on the prime number precomputation window, which greatly reduces its computational complexity.

An SM9-Based Secondary Authentication Framework for 5G Technology

As the latest generation of cellular mobile communication technology, the fifth-generation (5G) mobile communication technology is proposed to build a network-based society. As a result, industry and academia are placing great emphasis on security research for 5G technology. 5G technology has higher security requirements than previous systems and the secondary authentication is an important technology to ensure the security of 5G. In this paper, we propose to incorporate the SM9 encryption algorithm with 5G secondary authentication to provide higher security assurance. Due to the high computational overhead of the SM9 algorithm, we further propose an acceleration strategy for the SM9 algorithm based on the non-adjacent form (NAF) representation method to improve the performance of the entire system.

ON SOME PATTERNS OF TNAF FOR SCALAR MULTIPLICATION OVER KOBLITZ CURVE

A τ-adic non-adjacent form (TNAF) of an element α of the ring Z(τ) is an expansion whereby the digits are generated by iteratively dividing α by τ, allowing the remainders of -1,0 or 1. The application of TNAF as a multiplier of scalar multiplication (SM) on the Koblitz curve plays a key role in Elliptical Curve Cryptography (ECC). There are several patterns of TNAF (α) expansion in the form of [c0,0,…,0,cl-1 ], [c0,0,…,c(l-1)/2,…,0,c(l-1)], 2+2k, 3+4k, 5+4k and 8k1+8k2 that have been produced in prior work in the literature. However, the construction of their properties based upon pyramid number formulas such as Nichomacus’s theorem and Faulhaber’s formula remains to be rather complex. In this work, we derive such types of TNAF in a more concise manner by applying the power of Frobenius map (τm) based on v-simplex and arithmetic sequences.

An alternative proof of the minimality of the Hamming weight of width-3 non-adjacent form

In elliptic curve cryptography (ECC for short), point multiplication (or scalar multiplication) is the dominant operation. It is a very important matter to improve the efficiency of point multiplication for practical use. In ECC, recoding methods of the scalars play an important role in the performance of the algorithm used. One such example is the width-ω non-adjacent form (ω-NAF). Bosma (2001) proved that the Hamming weight of the NAF (2-NAF) is minimal, and Muir and Stinson (2005) proved that the Hamming weight of the ω-NAF is minimal. As other examples, the generalized non-adjacent form (GNAF) and the τ-adic non-adjacent form (τ -NAF) are known. Clark and Liang (1973) proved that the Hamming weight of the GNAF is minimal by constructing an injective map. A similar strategy was adopted by Hakuta, Sato, and Takagi (2010) to prove the minimality of the Hamming weight of the τ -NAF. In this paper, we shall give an alternative proof of the minimality of the Hamming weight of the 3-NAF (ω-NAF with ω = 3). We also discuss that our alternative proof may not work in the case ω ≥ 4.

An efficient signed digit montgomery modular multiplication algorithm

In this paper, we present a novel radix-2 Montgomery modular multi+d its corresponding architecture for high speed and low ATP implementation. The proposed multiplier, which is based on the Signed Digit Adder(SDA), requires binary input and also generates a modular product in binary form. To speed up radix-2 Montgomery modular multiplication process, we first design a new Simplified SDA(SSDA), dedicated to multipliers based on Non-Adjacent-Form(NAF) methods, which can not only avoid the carry propagation at each addition operation of add-shift loop, but also reduce the addition rounds efficiently. Second, a detecting and skipping mechanism based on the proposed SSDA has been studied to bypass some unnecessary addition rounds which would further reduce the addition rounds. In addition, our proposed Montgomery modular multiplier uses only one-level SDA architecture, and this architecture is also used to both the operand pre-computation and the final format conversion, which lead to a low hardware cost and short critical path delay. At last, we compare the area, the critical path and the cycle number of the proposed multiplier with those of other multipliers based on CSAs and SDAs. Experimental results show that the proposed Montgomery modular multiplier can achieve higher speed(6.9%) and area–time product improvement(19.4%) when compared with previous designs.

On Multi-Scalar Multiplication Algorithms for Register-Constrained Environments

A basic but expensive operation in the implementations of several famous public-key cryptosystems is the computation of the multi-scalar multiplication in a certain finite additive group defined by an elliptic curve. We propose an adaptive window method for the multi-scalar multiplication, which aims to balance the computation cost and the memory cost under register-constrained environments. That is, our method can maximize the computation efficiency of multi-scalar multiplication according to any small, fixed number of registers provided by electronic devices. We further demonstrate that our method is efficient when five registers are available. Our method is further studied in detail in the case where it is combined with the non-adjacent form (NAF) representation and the joint sparse form (JSF) representation. One efficiency result is that our method with the proposed improved NAF n-bit representation on average requires 209n/432 point additions. To the best of our knowledge, this efficiency result is optimal compared with those of similar methods using five registers. Unlike the previous window methods, which store all possible values in the window, our method stores those with comparatively high probabilities to reduce the number of required registers.

On Some Specific Patterns of τ-Adic Non-Adjacent Form Expansion over Ring Z(τ)

On Some Specific Patterns of τ-Adic Non-Adjacent Form Expansion over Ring Z(τ)

A High-Performance Elliptic Curve Cryptographic Processor of SM2 over GF(p)

Elliptic curve cryptography (ECC) is widely used in practical applications because ECC has far fewer bits for operands at the same level of security than other public-key cryptosystems such as RSA. The performance of an ECC processor is usually determined by modular multiplication (MM) and point multiplication (PM) operations. For recommended prime field, MM operation can consist of multiplication and fast reduction operations. In this paper, a 256-bit multiplication operation is implemented by a 129-bit (half-word) multiplier using Karatsuba–Ofman multiplication algorithm. The fast reduction is a modulo operation, which gets 512-bit input data from multiplication and outputs a 256-bit result ( 0 ≤ Z < p ) . We propose a two-stage fast reduction algorithm (TSFR) over SCA-256 prime field, which can obtain an intermediate result of 0 ≤ Z < 2 p instead of 0 ≤ Z < 14 p in traditional algorithm, avoiding a lot of repetitive subtraction operations. The PM operation is implemented in width nonadjacent form (NAF) algorithm and its operational schedules are improved to increase the parallelism of multiplication and fast reduction operations. Synthesized with a 0.13 μ m complementary metal oxide semiconductor (CMOS) standard cell library, the proposed processor costs an area of 280 k gates and PM operation takes 0.057 ms at the frequency of 250 MHz. The design is also implemented on Xilinx Virtex-6 platform, which consumes 27.655 k LUTs and takes 0.37 ms to perform one 256-bit PM operation, attaining six times speed-up over the state-of-the-art. The processor makes a tradeoff between area and performance, thus it is better than other methods.

Improving the Performance of {0,1,3}-NAF Recoding Algorithm for Elliptic Curve Scalar Multiplication

Although scalar multiplication is highly fundamental to elliptic curve cryptography (ECC), it is the most time-consuming operation. The performance of such scalar multiplication depends on the performance of its scalar recoding which can be measured in terms of the time and memory consumed, as well as its level of security. This paper focuses on the conversion of binary scalar key representation into {0, 1, 3}-NAF non-adjacent form. Thus, we propose an improved {0, 1, 3}-NAF lookup table and mathematical formula algorithm which improves the performance of {0, 1, 3}-NAF algorithm. This is achieved by reducing the number of rows from 15 rows to 6 rows, and reading two (instead of three) digits to produce one. Furthermore, the improved lookup table reduces the recoding time of the algorithm by over 60% with a significant reduction in memory consumption even with an increase in key size. Specifically, the improved lookup table reduces the memory consumption by as much as 75% for the big key, which shows its higher level of resilience to side channel attacks.

Even And Odd Nature For Pseudo Τ-Adic Non-Adjacent Form

An algorithm was developed by previous researcher for elliptic scalar multiplication (SM) on Koblitz curve where the multiplier of SM is in the form of Pseudo -adic Non-Adjacent (pseudoTNAF). PseudoTNAF of an element of the ring Z ) where is an expansion where the digits are generated by successively dividing by , allowing remainders of , 0 or 1. Such a multiplier is in the form of . In this paper, we refine some properties of the multiplier from previous researchers focusing on even and odd situation for and . We also propose two properties of when is even and is odd. As a result, the nature of and are depends on the nature of and when is even. Whereas, the nature of and are not depends on the nature of and when is odd. Keywords: Pseudo -adic Non-Adjacent Form (pseudoTNAF); scalar multiplication (SM); Koblitz curve

Winternitz Signature Scheme Using Nonadjacent Forms

Hash-based signatures are gaining attention as one of the alternatives that can replace current digital signatures that are not secure against an attack by quantum computers along with lattice-based signatures, multivariate signatures, and code-based signatures. Up to now, all hash-based signatures have used binary representations to generate signatures. In this paper, we propose using the nonadjacent form (NAF) when generating signatures in hash-based signatures. Concretely, we propose a hash-based signature scheme, WSS-N, which is obtained by applying nonadjacent forms (NAF) to the Winternitz signature scheme. We prove that WSS-N is existentially unforgeable under chosen message attacks in the standard model. And we show that WSS-N needs less hash function calls compared to the Winternitz signature scheme using the binary representation, WSS-B. For a specific parameter with a 256-bit security, we can see that WSS-N generates signatures faster than WSS-B by 8%. Finally, we implement both WSS-N and WSS-B and show that WSS-N generates signatures faster than WSS-B on a desktop computer.

Fast elliptic curve point multiplication based on window Non-Adjacent Form method

This article presents window Non-Adjacent Form (wNAF) of scalar representation method for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. A new efficient wNAF-based algorithm has been presented. The algorithm was developed based on simple and composite operations with a point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. The theorem concerning its computational complexity is formulated and proved for this new algorithm. In the end of this article, an efficiency analysis of the proposed algorithm depending on various parameters is presented, namely, the characteristic of the field over which the curve is defined, the scalar length and the window width of the precomputations and the coordinates of the precomputed points. A comparative analysis of the new algorithm and previously known efficient algorithm is also presented.

Improved Algorithms and Implementations for Integer to $\tau $ NAF Conversion for Koblitz Curves

The conversion from an integer scalar to a short and sparse τ-adic nonadjacent form (τNAF) is crucial for efficient elliptic curve scalar multiplication over Koblitz curves. Currently the conversion is costly both in time and area, limiting the application of Koblitz curves. In this paper, we propose improved algorithms and implementations for both the single-digit and double-digit scalar conversions. Area reduction is achieved by removing the τ-and-add calculation of the remainder upon division by τ m for lazy reduction or the τ2-and-add one for the double lazy reduction. The τNAF and the double τNAF algorithms are modified accordingly to support a mixed-formreduced scalar from the new reduction algorithms. Furthermore, fair pipelining is explored to speed up conversion with only a slight increase in area. Implementation results on Altera Stratix II FPGA show that the proposed single-digit converters are both smaller and faster than existing works, and the 4-stage pipelined one achieves at least 42.3% area reduction and 78.9% better area-time product (ATP) performance. On Xilinx Virtex IV, our non-pipelined double-digit converters are at least 44.5% smaller but slightly slower, while the 4-stage pipelined one can run faster with averagely 46.6% better ATP than previous equivalent works.

Incidental acquisition of collocations in L2

AbstractAs collocations are very numerous, at least a part of them has to be learned incidentally when acquiring a second language. There is now an increasing body of evidence showing that incidental learning of collocations seems to be possible. The present study sets out to investigate whether collocations can be learned incidentally if encountered in their nonadjacent form (with words intervening between collocates). It also looks at the role of prior vocabulary knowledge in incidental acquisition of adjacent and nonadjacent collocations. The results show that nonadjacent collocations can be learned equally well as adjacent ones, at least to the recognition level of knowledge. Also, larger prior vocabulary knowledge seems to lead to better chances of learning both adjacent and nonadjacent collocations incidentally.

Fast elliptic curve point multiplication based on binary and binary non-adjacent scalar form methods

This article presents two methods for developing algorithms of computing scalar multiplication in groups of points on an elliptic curve over finite fields. Two new effective algorithms have been presented: one of them is based on a binary Non-Adjacent Form of scalar representation and another one on a binary of scalar representation method. All algorithms were developed based on simple and composite operations with point and also based on affine and Jacobi coordinates systems taking into account the latest achievements in computing cost reduction. Theorems concerning their computational complexity are formulated and proved for these new algorithms. In the end of this article comparative analysis of both new algorithms among themselves and previously known algorithms are represented.

An Improved Signed Digit Representation of Integers

This research modifies the GNAF (Generalized NAF), which is one of the signed-digit representations that have been used to speed up point multiplication in pairing-based cryptosystems. The hamming weight of MGNAF (Modified Generalized Non-Adjacent Form), which is proposed in this paper, is less than GNAF for radixes higher than 2 (in cost of a bigger digit set). Moreover, in radix 2 the hamming weight of MGNAF is as low as GNAF and the well-known NAF (Non-Adjacent Form) with the same digit set. Keywords: Cryptography, Digit Set, Generalized NAF, Hamming Weight, Non-Adjacent Form, Radix-r Representation

Some properties of $${\tau }$$ τ -adic expansions on hyperelliptic Koblitz curves

In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the \({\tau }\)-adic non-adjacent form (\(\tau \)-NAF). The \(\tau \)-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. On the other hand, Gunther et al. (Speeding up the arithmetic on koblitz curves of genus two. LNCS, vol. 2012, pp. 106–117. Springer, Heidelberg, 2001) have proposed two generalizations of \(\tau \)-NAF for a family of hyperelliptic curves (hyperelliptic Koblitz curves) which have been proposed by Koblitz (J Cryptol 1(3):139–150, 1989). We call these generalizations \({\tau }\)-adic sparse expansion, and \({\tau }\)-NAF, respectively. To our knowledge, it is not known whether the three properties are true or not, especially, the existence must be satisfied for concrete cryptographic implementations. We provide an answer to the question. Our investigation shows that the \({\tau }\)-adic sparse expansion has only the existence and the \({\tau }\)-NAF has the existence and uniqueness. Our results guarantee the concrete cryptographic implementations of these generalizations.

Attacking OpenSSL ECDSA with a small amount of side-channel information

In this work, we mount a lattice attack on the ECDSA signatures implemented by the latest version of OpenSSL which uses the windowed non-adjacent form method to implement the scalar multiplication. We first develop a new way of extracting information from the side-channel results of the ECDSA signatures. Just given a small fraction of the information about a side-channel result denoted as double-and-add chain, we take advantage of the length of the chain together with positions of two non-zero digits to recover information about the ephemeral key. Combining the information of both the most significant digits and the least significant bits, we are able to gain more information about the ephemeral key. The problem of recovering ECDSA secret key is then translated to the hidden number problem which can be solved by lattice reduction algorithms. Our attack is mounted to the series secp256k1 curve, and the result shows that 85 signatures would be enough to recover the secret key, which is better than the result that previous attack gained only utilizing the information extracted from the least significant bits, using about 200 signatures to recover the secret key.

Non-minimality of the width-$w$ non-adjacent form in conjunction with trace one $\tau $-adic digit expansions and Koblitz curves in characteristic two

Non-minimality of the width-$w$ non-adjacent form in conjunction with trace one $\tau $-adic digit expansions and Koblitz curves in characteristic two