Scalar Multiplication Algorithm Research Articles

Pyecsca: Reverse engineering black-box elliptic curve cryptography via side-channel analysis

Side-channel attacks on elliptic curve cryptography (ECC) often assume a white-box attacker who has detailed knowledge of the implementation choices taken by the target implementation. Due to the complex and layered nature of ECC, there are many choices that a developer makes to obtain a functional and interoperable implementation. These include the curve model, coordinate system, addition formulas, and the scalar multiplier, or lower-level details such as the finite-field multiplication algorithm. This creates a gap between the attack requirements and a real-world attacker that often only has black-box access to the target – i.e., has no access to the source code nor knowledge of specific implementation choices made. Yet, when the gap is closed, even real-world implementations of ECC succumb to side-channel attacks, as evidenced by attacks such as TPM-Fail, Minerva, the Side Journey to Titan, or TPMScan [MSE+20; JSS+20; RLM+21; SDB+24].We study this gap by first analyzing open-source ECC libraries for insight into realworld implementation choices. We then examine the space of all ECC implementations combinatorially. Finally, we present a set of novel methods for automated reverse engineering of black-box ECC implementations and release a documented and usable open-source toolkit for side-channel analysis of ECC called pyecsca.Our methods turn attacks around: instead of attempting to recover the private key, they attempt to recover the implementation configuration given control over the private and public inputs. We evaluate them on two simulation levels and study the effect of noise on their performance. Our methods are able to 1) reverse-engineer the scalar multiplication algorithm completely and 2) infer significant information about the coordinate system and addition formulas used in a target implementation. Furthermore, they can bypass coordinate and curve randomization countermeasures.

Analysis of Isomorphism Classes of a Family of Elliptic Curves Over Finite Fields

Doche et al. constructed a family of elliptic curves (DIK elliptic curves) and proposed more efficient tripling formulas leading to a fast scalar multiplication algorithm. In this paper we present a direct method to compute the number of \(\bar{F}\)q-isomorphism classes (isomorphism over \(\bar{F}\)q ) and \(\bar{F}\)q isomorphism classes of DIK family of elliptic curves defined over a finite field \(\bar{F}\)q. We give the explicit formulae for the number of \(\bar{F}\)q-isomorphism and an estimate formulae for the number of isomorphism classes. These result can be used in the elliptic curve cryptosystems.

An Optimized Point Multiplication Strategy in Elliptic Curve Cryptography for Resource-Constrained Devices

Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point.

Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.

A special scalar multiplication algorithm on Jacobi quartic curves

A special scalar multiplication algorithm on Jacobi quartic curves

A Hybrid Approach for Efficient and Secure Point Multiplication on Binary Edwards Curves

The focus of this article is to present a novel crypto-accelerator architecture for a resource-constrained embedded system that utilizes elliptic curve cryptography (ECC). The architecture is built around Binary Edwards curves (BEC) to provide resistance against simple power analysis (SPA) attacks. Furthermore, the proposed architecture incorporates several optimizations to achieve efficient hardware resource utilization for the point multiplication process over GF(2m). This includes the use of a Montgomery radix-2 multiplier and the projective coordinate hybrid algorithm (combination of Montgomery ladder and double and add algorithm) for scalar multiplication. A two-stage pipelined architecture is employed to enhance throughput. The design is modeled in Verilog HDL and verified using Vivado and ISE design suites from Xilinx. The obtained results demonstrate that the proposed BEC accelerator offers significant performance improvements compared to existing solutions. The obtained throughput over area ratio for GF(2233) on Virtex-4, Virtex-5, Virtex-6, and Virtex-7 Xilinx FPGAs are 9.43, 14.39, 26.14, and 28.79, respectively. The computation time required for a single point multiplication operation on the Virtex-7 device is 19.61 µs. These findings indicate that the proposed architecture has the potential to address the challenges posed by resource-constrained embedded systems that require high throughput and efficient use of available resources.

Differential addition on binary elliptic curves

This paper presents new and fast differential addition (i.e., the addition of two points with the known difference) and doubling formulas, as the core step in Montgomery scalar multiplication, for various forms of elliptic curves over binary fields. The formulas are provided for binary Edwards, binary Hessian and binary Huff elliptic curves with cost of 5M+4S+1D when the given difference point is in affine form. Here, M,S,D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively. In this paper, we show that every ordinary elliptic curve E over binary field F2n has Lopez and Dahab differential addition and doubling formulas with suitable function. This paper also presents, new complete differential addition formulas for binary Edwards curves with cost of 5M+4S+2D. Also, new complete Montgomery ladder and scalar multiplication algorithms are presented using the complete differential addition formulas.

Secure and Compact Elliptic Curve Scalar Multiplication with Optimized Inversion

Abstract Elliptic curve cryptography (ECC) is a typical public key cryptography technique that can ensure equivalent security with considerably smaller key sizes than Rivest-Shamir-Adleman (RSA). Hence, various implementations based on ECC are recommended for block chain and Internet of Things (IoT) devices. Because elliptic curve scalar multiplication (ECSM) is a fundamental computation in ECC, enhancing the security and efficiency of ECSM is important. ECSM specifies a scalar multiplication algorithm and elliptic curve addition formulae. Elliptic curve addition formulae on affine coordinates are compact from a memory cost perspective but weak against side channel attacks. Elliptic curve complete addition (CA) formulae can achieve secure ECSMs but are inefficient. A newly proposed secure ECSM, which uses a right-to-left (RL) scalar multiplication algorithm and (extended) affine coordinates, takes advantage of elliptic curve addition formulae on affine coordinates. However, it can only scan the input scalars from right to left. We propose new ECSMs, which can scan the input scalars from left to right (LR) based on (extended) affine coordinates. We also prove that our LR ECSMs satisfy secure generality without requiring exceptional computations. We enhance the efficiency of both LR and RL ECSMs with optimized inversion. Our LR ECSM with a memory of 12 field elements reduces that of the Montgomery ladder and Joye’s LR with CA formulae by 36.84% and that of 2-ary RL with (extended) affine coordinates by 14.29%, respectively. Our compact ECSMs are fit for applications on IoT devices and block chain, with a critical memory requirement.

High Order Side-Channel Security for Elliptic-Curve Implementations

Elliptic-curve implementations protected with state-of-the-art countermeasures against side-channel attacks might still be vulnerable to advanced attacks that recover secret information from a single leakage trace. The effectiveness of these attacks is boosted by the emergence of deep learning techniques for side-channel analysis which relax the control or knowledge an adversary must have on the target implementation. In this paper, we provide generic countermeasures to withstand these attacks for a wide range of regular elliptic-curve implementations. We first introduce a framework to formally model a regular algebraic program which consists of a sequence of algebraic operations indexed by key-dependent values. We then introduce a generic countermeasure to protect these types of programs against advanced single-trace side-channel attacks. Our scheme achieves provable security in the noisy leakage model under a formal assumption on the leakage of randomized variables. To demonstrate the applicability of our solution, we provide concrete examples on several widely deployed scalar multiplication algorithms and report some benchmarks for a protected implementation on a smart card.

Adversarial Attack Protection Scalar Multiplication for WSNs Resistance Machine-Learning Side-channel Attack

Wireless sensor networks (WSNs) have limited storage and low processing capabilities. However, these devices may be captured by opponents, so the security requirements are particularly strict. With the development of side-channel attacks based on the machine-learning, designing a safe elliptic curve scalar multiplication for computing-limited devices like WSNs has become a major challenge. Based on the adversarial attack technology, a secure scalar multiplication is proposed in this article. The main contributions are: (1) We propose an efficient non-zero form (NZF) encoding algorithm that can be applied to various types of elliptic curves; (2) we have designed a secure scalar multiplication algorithm that can resist against conventional side-channel attacks such as SPA, DA, DPA, RPA, and ZPA; and (3) we propose an adversarial protection mechanism based on blind point technology and NZF coding, which can prevent side-channel attacks based on machine learning. The algorithm has no precomputation and is suitable for low communication frequency, low calculation amount, and high security requirements. Especially, it can be applied to lightweight equipment such as WSN and IoT.

A high speed processor for elliptic curve cryptography over NIST prime field

Elliptic curve cryptography (ECC), as one of the public key cryptography systems, has been widely applied to many security applications. It is challenging to implement a scalar multiplication (SM) operation which has the highest computational complexity in ECC. In this study, we propose a hardware processor which achieves high speed and high security for ECC. We first present a three-clock cycle, divide-and-conquer multiplication algorithm which greatly reduces the number of execution cycles of multiplication. We then propose a dedicated multiplication hardware structure which reuses the multiplier and optimizes data path delay. To keep multiplication running in non-idle status and executing in parallel with other modular operations, the operation scheduling of point addition and point doubling has been re-designed and optimized based on an effective segmentation and pipeline strategy. Finally, under the premise of similar computing and hardware overhead, we propose an improved high-security SM algorithm which involves random points to resist side-channel attacks. On a 55 nm complementary metal oxide semiconductor application specific integrated circuit platform, the processor costs 463k gates and requires 0.028 ms for one SM. Our results indicate that the ECC processor is superior to other state-of-the-art designs reported in the literature in terms of speed and area-time product metrics.

An efficient scalar multiplication algorithm on Koblitz curves using τ<SUP align="right">3</SUP>-NAF

Elliptic curve cryptography (ECC) is an efficient and widely used public-key cryptosystem. It uses relatively shorter keys compared to conventional cryptosystems hence offering faster computation. The efficiency of ECC relies heavily on the efficiency of scalar multiplication which internally depends on the representation of the scalar value. Based on the representation, the number of point additions and point doublings varies. Koblitz curves are binary elliptic curves defined over F2 and also known as anomalous binary curves. Scalar multiplication algorithms on these curves can be designed without any point doublings. In τ-NAF representation, we need 0.333 m point additions whereas in τ2-NAF it is 0.215 m. This paper proposes a method to improve the efficiency of scalar multiplication on Koblitz curves using τ3-NAF representation that further reduces the point additions to 0.143 m.

Improved Software Implementation for Montgomery Elliptic Curve Cryptosystem

The last decade witnessed rapid increase in multimedia and other applications that require transmitting and protecting huge amount of data streams simultaneously. For such applications, a high-performance cryptosystem is compulsory to provide necessary security services. Elliptic curve cryptosystem (ECC) has been introduced as a considerable option. However, the usual sequential implementation of ECC and the standard elliptic curve (EC) form cannot achieve required performance level. Moreover, the widely used Hardware implementation of ECC is costly option and may be not affordable. This research aims to develop a high-performance parallel software implementation for ECC. To achieve this, many experiments were performed to examine several factors affecting ECC performance including the projective coordinates, the scalar multiplication algorithm, the elliptic curve (EC) form, and the parallel implementation. The ECC performance was analyzed using the different factors to tune-up them and select the best choices to increase the speed of the cryptosystem. Experimental results illustrated that parallel Montgomery ECC implementation using homogenous projection achieves the highest performance level, since it scored the shortest time delay for ECC computations. In addition, results showed that NAF algorithm consumes less time to perform encryption and scalar multiplication operations in comparison with Montgomery ladder and binary methods. Java multi-threading technique was adopted to implement ECC computations in parallel. The proposed multithreaded Montgomery ECC implementation significantly improves the performance level compared to previously presented parallel and sequential implementations.

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

New Regular Sliding Window Algorithms for Elliptic Curve Scalar Point Multiplication

Machine Learning Attacks and Countermeasures on Hardware Binary Edwards Curve Scalar Multipliers

Machine Learning techniques have proven effective in Side Channel Analysis (SCA), enabling multiple improvements over the already-established profiling process of Template Attacks. Focusing on the need to mitigate their impact on embedded devices, a design model and strategy is proposed that can effectively be used as a backbone for introducing SCA countermeasures on Elliptic Curve Cryptography (ECC) scalar multipliers. The proposed design strategy is based on the decomposition of the round calculations of the Montgomery Power Ladder (MPL) algorithm and the Scalar Multiplication (SM) algorithm into the underlined finite field operations, and their restructuring into parallel-processed operation sets. Having as a basis the proposed design strategy, we showcase how advanced SCA countermeasures can be easily introduced, focusing on randomizing the projective coordinates of the MPL round’s ECC point results. To evaluate the design approach and its SCA countermeasures, several simple ML-based SCAs are performed, and an attack roadmap is provided. The proposed roadmap assumes attackers that do not have access to a huge number of leakage traces, and that have limited resources with which to mount Deep Learning attacks. The trained models’ performance reveals a high level of resistance against ML-based SCAs when including SCA countermeasures in the proposed design strategy.

Online Template Attacks: Revisited

An online template attack (OTA) is a powerful technique previously used to attack elliptic curve scalar multiplication algorithms. This attack has only been analyzed in the realm of power consumption and EM side channels, where the signals leak related to the value being processed. However, microarchitecture signals have no such feature, invalidating some assumptions from previous OTA works.In this paper, we revisit previous OTA descriptions, proposing a generic framework and evaluation metrics for any side-channel signal. Our analysis reveals OTA features not previously considered, increasing its application scenarios and requiring a fresh countermeasure analysis to prevent it.In this regard, we demonstrate that OTAs can work in the backward direction, allowing to mount an augmented projective coordinates attack with respect to the proposal by Naccache, Smart and Stern (Eurocrypt 2004). This demonstrates that randomizing the initial targeted algorithm state does not prevent the attack as believed in previous works.We analyze three libraries libgcrypt, mbedTLS, and wolfSSL using two microarchitecture side channels. For the libgcrypt case, we target its EdDSA implementation using Curve25519 twist curve. We obtain similar results for mbedTLS and wolfSSL with curve secp256r1. For each library, we execute extensive attack instances that are able to recover the complete scalar in all cases using a single trace.This work demonstrates that microarchitecture online template attacks are also very powerful in this scenario, recovering secret information without knowing a leakage model. This highlights the importance of developing secure-by-default implementations, instead of fix-on-demand ones.

A Large Scalar Multiplication Algorithm using Modified Pell Numbers for Key Generation

Cryptographic algorithms consist of two parts, a key and an algorithm, to encrypt and decrypt data. The key is an essential part that works with the algorithm. The security of encryption schemes depends on the key size (key length) and the longer the key, the better the security it provides. Applying an elliptic curve has for key agreement provides a high-performance architecture and high security. The main process for calculating key points in Elliptic Curve Cryptography (ECC) is called scalar multiplication, which relates to point addition and point doubling. An efficient algorithm, proposed as the Large Scalar Multiplication Algorithm using Modified Pell Numbers (LSMA-MPN), was introduced to speed up the calculation of points on elliptic curves during large scalar multiplications. This system also reduced computation time by applying Modified Pell numbers in a 22 matrix representation. The experimental results showed that computation time was reduced by approximately 67% in comparison with the computation time required by a general algorithm.

Efficient scalar multiplication of ECC using SMBR and fast septuple formula for IoT

In order to solve the problem between low power of Internet of Things devices and the high cost of cryptography, lightweight cryptography is required. The improvement of the scalar multiplication can effectively reduce the complexity of elliptic curve cryptography (ECC). In this paper, we propose a fast formula for point septupling on elliptic curves over binary fields using division polynomial and multiplexing of intermediate values to accelerate the computation by more than 14%. We also propose a scalar multiplication algorithm based on the step multi-base representation using point halving and the septuple formula we proposed, which significantly reduces the computational cost. The experimental results show that our method is more efficient over binary fields and contributes to reducing the complexity of ECC.

Survey on performance and security problems of countermeasures for passive side-channel attacks on ECC

This paper presents an analysis on the state of the art of several proposals for algorithmic countermeasures to prevent passive side-channel attacks (SCA) on elliptic curve cryptography (ECC) defined over prime fields. One of the main applications of ECC is in the field of Internet of Things, where the interconnection of devices requires public-key technology with small key sizes and high security levels. Since the secure implementation of ECC on embedded devices involves many challenges for security and efficiency, this work evaluates the trade-offs between security and performance of side-channel attack countermeasures for scalar multiplication algorithms without precomputation. The main contribution is to present a panorama of explicit solutions that may be used for the implementation of ECC mechanisms suitable for embedded devices. In addition to the security problems, some countermeasures are also analyzed.

Efficient Septuple Formula for Elliptic Curve and Efficient Scalar Multiplication Using a Triple-Base Chain Representation

Because of shorter key and higher security, elliptic curve cryptosystem has attracted people’s attention and it is widely used in various fields, such as wireless sensor networks. Scalar multiplication is one of the most basic and critical links in the realization of elliptic curve public key systems, and its operation efficiency directly affects the implementation efficiency of the entire cryptographic system. To improve speed up the efficiency of the scalar multiplication algorithm, we express $k $ for scalar multiplication $kP $ by using triple-base chain representation of the scalar using $\{2,3,7\} $ as basis of the triple-base chain in this paper. However, the efficiency of scalar multiplication is not only related to the length of representation but also the numbers and costs of doubling, tripling, septupling and addition. Therefore, we improve septuple formula of the elliptic curve by using Co_Z operation, which costs decreased by 8.3%. Due to the high redundancy of the triple-base chain representation, the algorithm can resist side channel attacks. The experimental results show that the proposed algorithm compared with that of other scalar multiplication algorithms, it requires less cost.