Pseudo-random scalar multiplication based on group isomorphism

This work presents an evaluation of different software implementations of algorithms to compute the most demanding operation in cryptographic schemes based on Elliptic Curve Cryptography (ECC), the scalar multiplication. Five different methods were studied, including traditional and more sophisticated methods for elliptic curve cryptography defined over prime and binary fields. For evaluation, the key generation algorithm in ECC consisting on a single scalar multiplication was implemented on a P500h LG smarthphone, which includes an ARM processor running at 600MHz. Both execution time and memory usage was evaluated. It was found that in general, scalar multiplication runs 8 times faster for ECC defined over prime fields, being the NAF the best performer method. For ECC over binary fields, the best performer method was wNAF. The results presented in this work could help a designer to select the most appropriate method when implementing ECC-based cryptographic schemes such as encryption or digital signatures on mobile devices like smartphones, meeting implementation requirements in terms of execution time and memory usage.

Elliptic curve cryptography (ECC) can ensure an equivalent security with much smaller key sizes. Elliptic curve scalar multiplication (ECSM) is a fundamental computation used in ECC. This paper focuses on ECSM resisting simple power attack and safe error attack of side-channel attack specifically. Elliptic curve complete addition (CA) formulae can achieve secure ECSM algorithms but are inefficient from memory and computational cost perspectives. Another secure ECSM, which uses (extended) affine, is more efficient for both memory and computational costs. However, it scans input scalars from right to left. In this paper, our developed scalar multiplication algorithms also use their extended affine, but scan from left to right (LR). We also prove the security of our LR ECSM algorithms and analyze them both theoretically and experimentally. Our new LR ECSM algorithms can reduce the amount of memory by \(37.5\%\) and reduce the computational time by more than \(40\%\) compared to Joye’s regular 2-ary LR algorithm with CA formulae.

Elliptic curve cryptography (ECC) has shorter key length than other asymmetric cryptography algorithms such as RSA with the same security level. Existing faults in cryptographic computations can cause faulty results. If a fault occurs during encryption, false information will be sent to the destination, in which case channel error detection codes are unable to detect the fault. In this paper, we consider the error detection in elliptic curve scalar multiplication point, which is the most important operation in ECC. Our technique is based on non-linear error detection codes. We consider an algorithm for scalar multiplication point proposed by Microsoft research group. The proposed technique in our methods has less overhead for additions (36.36%) and multiplications (34.84%) in total, compared to previous works. Also, the proposed method can detect almost 100% of injected faults.

Background: Today’s developing era data and information security plays an important role in unsecured communication between Internet of Things (IoT) elements. As the technologies are developing huge amount of information are exchanged with each other which increases the demand of a secure encryption standards. Objectives: To protect the information transmission between nodes in the perceptual layer of IoT, security is to be enhanced such that the Men in Middle (MIM) attack of information by intruders in the communication link is not possible. Methods/Statistical analysis: The security method is implemented using the concept of symmetric, asymmetric and hash algorithms between the sender and receiver. Elliptic Curve Cryptography (ECC) is performed for encrypting and sharing the Advanced Encryption Standard (AES) secret key between the sender and receiver. Elliptic Curve digital signature are added with the encrypted data for validation in transmission. Secured Hash Algorithm (SHA) is considered for generating Elliptical Curve Digital Signature Algorithm (ECDSA). All the algorithms are combined to enhance the security. Findings: Elliptic Curve Cryptography is generally applied for encrypting and sharing the secret key between the sender and receiver when Advanced Encryption Standard is used for encryption/decryption of data. A highly efficient architectures for asymmetric key encryption is applied with symmetric key encryption for encrypting the data transmitted between the devices present in the perceptual layers in the IoT. Due to security issues NIST selected SHA-3 Keccak-f[1600] algorithm. An authentication process based on ECC is applied to perceptual layer involving initialization and authentication phase apart from encryption and signature generation. Sender combines the encrypted data and key with signature together before transmitting to the receiver. The receiver verifies the signature and decrypts the key and obtains original information. Hence it is very difficult for the man in middle to access the information exchanged between the Perceptual Layer devices. Application/Improvements: This generally doesn’t require an innovative technique but an approach to develop the method implemented successfully in the perceptual layer of Internet of Things. The key is encrypted and decrypted using a novel ECC algorithm with improved ECDSA.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

The security of Internet of Things (IoT) devices is one of the most important problems to be addressed by the cryptographers and security engineers. The processing ability of IoT devices is limited, therefore light-weight and secure cryptographic tools are necessary for security of them. This paper shows the implementation of 256-bit Elliptic Curve Cryptography (ECC) on an 8-bit microcontroller. The proposed implementation applies towering technique for extension field of degree 32 with a certain 8-bit prime characteristic instead of the 256-bit prime characteristic. It enables to execute 256-bit ECC operations without complicated multiple-precision arithmetic on small computers like 8-bit microcontrollers. This approach efficiently realizes the scalability of the ECC encryption strength. In addition, the authors use a twisted Montgomery curve with a Montgomery ladder technique which enables fast calculations without inversions referring to Curve25519. It is considered resistant to the Side Channel Attack (SCA) since it applies the Montgomery ladder technique for scalar multiplication (SCM). This ECC implementation on Arduino UNO, an 8-bit microcontroller board, can be utilized for a key agreement protocol among IoT devices.

With the advent of Elliptic curve cryptography (ECC); it grew unusually quickly because it had a smaller key size compared with that of the RSA cryptography system. Many researches are being done nowadays to make this type of cryptography more efficient. These investigations are about improving efficiency of both algorithm and method of implementations. This paper studies and analyzes some software libraries which support elliptic curve computations. Scalar multiplication is a major operation in ECC. Therefore, this study implements some methods of scalar multiplication by two software libraries LiDIA and MIRACL and compares the results at the end.

This paper presents a new approach to precompute points [3]P, [5]P, ..., [2k-1]P, for some k ≥ 2 on an elliptic curve over $\mathbb{F}_p$. Those points are required for the efficient evaluation of a scalar multiplication, the most important operation in elliptic curve cryptography. The proposed method precomputes the points in affine coordinates and needs only one single field inversion for the computation. The new method is superior to all known methods that also use one field inversion, if the required memory is taken into consideration. Compared to methods that require several field inversions for the precomputation, the proposed method is faster for a broad range of ratios of field inversions and field multiplications. The proposed method benefits especially from ratios as they occur on smart cards.

We present a high-speed public-key cryptoprocessor that exploits three-level parallelism in Elliptic Curve Cryptography (ECC) over GF(2 n ). The proposed cryptoprocessor employs a Parallelized Modular Arithmetic Logic Unit (P-MALU) that exploits two types of different parallelism for accelerating modular operations. The sequence of scalar multiplications is also accelerated by exploiting Instruction-Level Parallelism (ILP) and processing multiple P-MALU instructions in parallel. The system is programmable and hence independent of the type of the elliptic curves and scalar multiplication algorithms. The synthesis results show that scalar multiplication of ECC over GF(2163) on a generic curve can be computed in 20 and 16 μs respectively for the binary NAF (Non-Adjacent Form) and the Montgomery method. The performance can be accelerated furthermore on a Koblitz curve and reach scalar multiplication of 12 μs with the TNAF (τ-adic NAF) method. This fast performance allows us to perform over 80,000 scalar multiplications per second and to enhance security in wireless mobile applications.

A SM2 based efficient and lightweight batch verification approach for IC cards

Elliptic curve cryptography involves numerous scalar multiplications, incurring high operational costs. In view of this, fast endomorphism is used to represent scalar multiplications, kP on elliptic curves. In the past, techniques such as Gallant-Lambert-Vanstone (GLV) method and Integer Sub-Decomposition (ISD) method have been proposed to reduce the cost of scalar multiplication on elliptic curves by using fast endomorphism. The GLV method employs a single-layer decomposition, breaking k into k1 and k2, while the ISD method uses a bilayer decomposition. The existence of fast endomorphisms which are constructed based on the concept of isogeny increase the computational efficiency of the GLV approach and reduce the operation count on the ISD method. This paper embeds the fast endomorphisms in the scalar multiplications on one of the family of elliptic curves with j-invariant 0, E0, which is the 192-bit Koblitz curve (Secp192k1). The performance of the ISD method in computing certain scalar multiplications on Secp192k1 in conjunction with fast endomorphisms and other various techniques such as binary representation, NAF representation, w-NAF and sliding windows are computed. The results demonstrated that the ISD method together with fast endomorphism, yields the most promising outcomes. This underscores the advantages of using fast endomorphisms in the ISD method on E0. .

We propose to represent a scalar in the form of d (1/2)(superscript a)3(superscript b), where a and b are integers and d is an odd integer belonging to a given set. This representation is a combination of the extended double-base number system (DBNS) and the double-base chain representation using powers of 1/2 and 3. Experimental results show that our approach leads to a shorter DBNS expansion and a lower complexity in elliptic curve scalar multiplication, with the cost of only a few pre-computations and storages. This contributes to the efficient implementation of elliptic curve cryptography (ECC).

Wireless sensor networks (WSNs) pose a number of unique security challenges that demand innovation in several areas including the design of cryptographic primitives and protocols. Despite recent progress, the efficient implementation of Elliptic Curve Cryptography (ECC) for WSNs is still a very active research topic, and techniques to further reduce the time and energy cost of ECC are eagerly sought. This paper presents an optimized ECC implementation that we developed from scratch to comply with the severe resource constraints of 8‐bit sensor nodes such as the MICAz and IRIS motes. Our ECC software uses Optimal Prime Fields as underlying algebraic structure and supports two different families of elliptic curves, namely, Weierstraß‐form and twisted Edwards‐form curves. Due to the combination of efficient field arithmetic and fast group operations, we achieve an execution time of 5.3·106 clock cycles for a full 160‐bit scalar multiplication on an 8‐bit ATmega128 microcontroller, which is more than three times faster than the widely used TinyECC library. Our implementation also shows that the energy cost of scalar multiplication on a MICAz (or IRIS) mote amounts to just 17.34mJ when using a twisted Edwards curve over a 160‐bit Optimal Prime Field. This result further demonstrates the advantage of special family of elliptic curves for resource‐constrained environments. Copyright © 2015 John Wiley & Sons, Ltd.

Wireless Sensor Networks (WSNs) pose a number of unique security challenges that demand innovation in several areas including the design of cryptographic primitives and protocols. Despite recent progress, the efficient implementation of Elliptic Curve Cryptography (ECC) for WSNs is still a very active research topic and techniques to further reduce the time and energy cost of ECC are eagerly sought. This paper presents an optimized ECC implementation that we developed from scratch to comply with the severe resource constraints of 8-bit sensor nodes such as the MICAz and IRIS motes. Our ECC software uses Optimal Prime Fields (OPFs) as underlying algebraic structure and supports two different families of elliptic curves, namely Weierstraß-form and twisted Edwards-form curves. Due to the combination of efficient field arithmetic and fast group operations, we achieve an execution time of 5.8 · 106 clock cycles for a full 158-bit scalar multiplication on an 8-bit ATmega128 micro-controller, which is 2.78 times faster than the widely-used TinyECC library. Our implementation also shows that the energy cost of scalar multiplication on a MICAz (or IRIS) mote amounts to just 19~mJ when using a twisted Edwards curve over a 160-bit OPF. This result compares fairly well with the energy figures of two recently-presented hardware designs of ECC based on twisted Edwards curves.

The data everywhere was being transferred for each second using domain of Internet of Things (IoT). To secure data using the Internet of Things (IoT) is a tedious task. When the data is being transferred using Internet of Things (IoT), more security can be provided using elliptic curve cryptography. Asymmetric cryptography is used by most of the applications for providing secure communication between two parties (Weber in Comput Law Secur Rev 26:23–30, 2010). The purpose of this type of cryptography is the requirement of huge amount of computation and storage. This is where the use of elliptic curve cryptography comes into picture, as it needs less storage and can be used in small computational devices. ECC needs smaller key sizes and provides stronger encryption compared to various asymmetric cryptographic algorithms like RSA. The usage of power required is low but the performance of the devices using ECC is high for different types of devices like IoT, sensors, etc. This paper shows how ECC must be implemented strongly for providing communication securely to encode the data on an elliptic curve, in IoT devices. The encryption of the data must be done securely while mapping the data on to the elliptic curve. This paper shows how the data is encrypted and mapped on to the curve securely. The work shows how the data can be encrypted with ECC and how it can be visible to only authorized users. In the field of cryptography, ECC is a method for asymmetric cryptography which is dependent on the algebraic structure of an elliptic curve on the finite field.