Is a polynomial of degree not larger than g and f(x) โ K[x] is a monic polynomial of degree 2g + 1 [7]. From this definition it follows that **elliptic** curves are **hyper**-**elliptic** curves of genus 1. In **hyper**-**elliptic** **curve** cryptography K is often a finite field. The Jacobian of C, denoted J(C), is a quotient group, thus the elements of the Jacobian are not points, they are equivalence classes of divisors of degree 0 under the relation of linear equivalence. This agrees with the **elliptic** **curve** case, because it can be shown that the Jacobian of an **elliptic** **curve** is isomorphic with the group of points on the **elliptic** **curve**. The use of **hyper**-**elliptic** curves in cryptography came about in 1989 from Neal Koblitz [14]. Although introduced only 3 years after ECC, not many cryptosystems implement **hyper**-**elliptic** curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on **elliptic** curves or factoring (RSA). The efficiency of implementing the arithmetic depends on the underlying finite field K, in practice it turns out that finite field of characteristic 2 are good choices for hardware implementations while software is usually faster in odd characteristic.

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Some researcherโs interests have recently fall on **Hyper**- **Elliptic** **Curve** because it offers equal security for a smaller key size. Since, HECC is a typically fast public key cryptosystem and it has superiority and more application efficiency. There are some superiorities of **Hyper**-**Elliptic** **Curve** cryptosystem such as high efficiency, short key length as compared with other public key cryptosystems and they are as follows:

In this thesis we have proposed explicit formulae for group operation such as addition and doubling on the Jacobians of **Hyper** **Elliptic** Curves genus 2, 3 and 4. The Cantor Algorithm generally involves to perform arithmetic operations in the polynomial ring . The explicit method performs the arithmetic operation in the integer ring of . Significant improvement has been made in the explicit formulae algorithm proposed here. Other explicit formulae used Montgomery trick to derive efficient formulae for faster group computation. The method used in this thesis to develop an efficient explicit formula was inspired by the geometric properties in the **hyper** **elliptic** curves of genus and by keeping the Jacobian variety **curve** constant. This formulae take Mumford coordinates as input. The explicit formulae here performs the computation in affine space of genus 2, 3 and 4 of **Hyper** **Elliptic** Curves in general form, which can be used to develop **Hyper** **Elliptic** **Curve** Cryptosystem.

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In the study of cryptosystem, **hyper**-**elliptic** **curve** has recently attracted some researchersโ interests because it gives The main attraction of combining EC with HEC is that it appears to offer equal security for a far smaller key size, thereby saving the processing overhead because it gives the same security level with a smaller key length as compared to cryptosystems using **elliptic** curves. From the fact it is expected to be possible to use **hyper**-**elliptic** curves to factor integers, since **elliptic** **curve** method exploits the property of the Abelian groups in the same way as the cryptosystems.

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In [24], authors have presented a paper on enhanced level of security using DNA computing technique with **hyper** **elliptic** **curve** cryptography. DNA based **elliptic** **curve** cryptographic technique require larger key size to encrypt and decrypt the message resulting in increased processing time, more computational and memory overhead. To overcome the above limitations, DNA strands are used to encode the data to provide first level of security and HECC encryption algorithm is used for providing second level of security. HECC is better than the existing public key cryptography technique such as RSA, DSA, AES and ECC in terms of smaller key size. DNA cryptography is a next generation security mechanism, storing almost a million gigabytes of data inside DNA strands. Hence this proposed integration of DNA computing based HECC provides higher level of security with less key size of HECC-80 bits than ECC-160 bits and with less computational and memory overhead.

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Gamage extended the concept of proxy signature and proposed a new cryptography primitive called proxy signcrypion [3]. Proxy signcrypion scheme allows the original user (Alice) to transfer the signing capability to the trusted entity known as proxy signcrypter. The proxy signer signcrypt the message on behalf of the original user and sends to the recipient (Bob) . Proxy Signcrypion schemes based on discrete logarithm problem (DLP) and **elliptic** **curve** discrete logarithm problem (ECDLP) were proposed. The common problems of these schemes are high computational cost and more communication overhead. In this paper, proxy signcrypion based on **hyper** **elliptic** **curve** is proposed, whose security relies on the hardness of the **hyper** **elliptic** **curve** discrete logarithm problem (HECDLP). In order to achieve the same security level, NIST has recommended key length of 80 bits for HECC, 1024 bits for RSA and 160 bits for ECC. **Hyper** **elliptic** **curve** is of public key cryptosystem with small key length. The proposed scheme saves the computational and communication cost due to its shorter key size. The scheme achieves the security properties.

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The signature will be the pair (x, y) The signature is used to authenticate between application components so that data transmitted only to known parties. For Securing API and Interfaces, the shared secret key used to ensure strong authentication and access controls with encrypted transmission. The client can access the cloud by applying their shared secret key, which is monitor by the cloud environment. They can demand the cloud environment according to their choice and need. The signed message encrypted using ElGamal cryptosystem based on **elliptic** **curve**. The client selects a private integer named ๐ข๐ ๐๐ ๐๐๐๐ฃ . This private number used to generate a public key name ๐ข๐ ๐๐ ๐๐ข๐ for encryption process. This public key represents a point on the **curve**. The encryption based on Elgamal algorithm generates two points on the **curve** named point c1 and point c2. ๐๐๐๐๐ก ๐1 = ๐ข๐ ๐๐ ๐๐๐๐ฃ โ ๐ ๐๐๐ฆ (๐๐๐ ๐) (7)

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From table (2a), figure (2a) and (2b) we can see that bit size does not affect our algorithms encryption and decryption time. Comparison As our proposed of algorithm requires mapping, we compared our algorithm with those algorithm which requires mapping. Implementation of **Elliptic** **Curve** Cryptography on Text and Image [7] and Implementation of Text based Cryptosystem using **Elliptic** **Curve** Cryptography [6] simulation was performed using Lenovo ideapad Z510 laptop with system configuration of i7 processor @2.20GHz and 8GB Ram [6]. Our simulation is done by using Toshiba Satellite C600 with system configuration dual core processor @2.30 GHz and 2 GB Ram. Word length=409, Text Length: 2325. We have done our comparison with only those methods where mapping is required.

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a speedup factor around 3, depending on various details of field arithmetic. The disadvantage of genus 2 is that each group operation requires many more field operations; but for Gaudryโs [24] Kummer-surface formulas this loss factor is only slightly above 2. Even better, 24% of Gaudryโs field multiplications are multiplications by **curve** parameters that can be chosen to be small; a secure small-parameter genus-2 **curve** was announced by Gaudry and Schost [27] after a massive point-counting computation. A further advantage of genus 2, exploited in a very recent paper [4] by Bernstein, Chuengsatiansup, Lange, and Schwabe, is a synergy between the structure of Gaudryโs formulas and the availability of vector operations in modern CPUs. One can speed up genus 1 using โnon-constant-timeโ addition chains. However, non-constant- time computations are a security problem; see, e.g., the attacks cited in [4, Section 1.2].

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In the paper we have presented the implementation technique of image encryption/decryption and inclusion of digital signature to the cipher image to provide authenticity and integrity to the received image. We have performed our operation by grouping the pixel and explained how many pixels can be grouped according to the ECC parameters. Pairing of the grouped pixel value was performed instead of mapping those values to **Elliptic** **curve** coordinate. It helps to ignore the used of reference mapping table for encryption and decryption. Our algorithm generates a low correlated cipher image even with a image which is made up of same pixel value. We have also analysed our technique to support the strength of the algorithm.

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This paper presents the design and implementation of a complete near-field communication (NFC) tag system that supports high-security features. The tag design contains all hardware modules required for a practical realization, which are: an analog 13.56-MHz radio-frequency identification (RFID) front-end, a digital part that includes a tiny (programmable) 8-b microcontroller, a framing logic for data transmission, a memory unit, and a crypto unit. All components have been highly optimized to meet the fierce requirements of passively powered RFID devices while providing a high level of flexibility and security. The tag is fully compliant with the NFC Forum Type-4 specification and supports the ISO/IEC 14443A (layer 1โ4) communication protocol as well as block transmission according to ISO/IEC 7816. Its security features include support of encryption and decryption using the Advanced Encryption Standard (AES- 128), the generation of digital signatures using the **elliptic** **curve** digital signature algorithm according to NIST P-192, and several countermeasures against common implementation attacks, such as side-channel attacks and fault analyses. The chip has been fabricated in a 0.35-ฮผm CMOS process technology, and requires 49 999 GEs of chip area in total (including digital parts and analog front-end).

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Data security is an important requirement for many appli- cations in our daily life. Especially internet applications such as e-commerce need to transmit secret data via inse- cure communication channels. Therefore, various crypto- graphical methods exist that allow to protect sensitive data. Asymmetric cryptography, which is also known as public- key cryptography, does not only provide algorithms for en- cryption and decryption of data, but also for digital signa- tures and authentication. Recently asymmetric cryptography based on **elliptic** curves is gaining interest. Compared to traditional asymmetric techniques, e.g. the RSA algorithm, the **elliptic** **curve** cryptography (ECC) achieves an equivalent level of security with smaller key sizes. Using **elliptic** **curve** cryptography therefore results in memory as well as band- width savings.

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Suppose that we had a way of masking the contents of messages or other information traffic so that an attacker, even if he or she captured the message, would be unable to extract the information from the message. The common technique for doing masking is encryption. The encryption is done by using Symmetric key or public key Algorithms. The most commonly used public key algorithms are 1. Rivest Shamir Adelman(RSA) and 2 **Elliptic** **Curve** cryptography

However we had another criteria in mind which was to exploit the Granger-Scott idea [5] for fast modular arithmetic. This guided our choice of the modulus p. We also wanted a **curve** that would be implemented just as conveniently on both 32-bit and 64-bit architectures. Whereas a nice fit to a 64-bit architecture facilitates record-setting, in our view a good fit to a 32-bit architecture is probably more important in practise. As a bonus we wanted our implementation to be e๏ฌcient even when written in a portable high-level language.

When designing Goldilocks, I hypothesized that timings within the same implementation strat- egy should roughly follow a power law t โ b k , where b is the bits in the field size and k is some exponent, and therefore that b k /t might be described as an โefficiency scoreโ. I evaluated designs with Karatsubaโs k = 1 + log 3/ log 2, which works surprisingly well even though not all the designs use Karatsuba multiplication. In arbitrary units 2 , Ed448-Goldilocks and E-521 score 3.3 and 3.1 by this metric. ECCLibโs ed-384-mers and ed-512-mers both score 2.3, and the optimized secp256r1 scores 1.4. By this metric, Goldilocks is about 40% more efficient than either ed-mers **curve** on Haswell.

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This algorithm can be applied to achieve multiple reading simultaneously at the same time and by using the identity of a particular patient the reading can be accessed. To setup ECC, we first select a particular **elliptic** **curve** E over GF (K), where k is a big prime number. We also denote G as the base point of E and q as the order of K, where q is also a big prime. We then pick a secret key x, and the corresponding public key y, where y = x ยทG, and a Message Authentication Code hash function. Finally, we have the secret key x and public parameters (Y, K, k, q, h (.) G). Encrypting a message m using public key Y as EciesEncrypt (m, Y). The **Elliptic** **curve** integrated encryption standard technique consists of reading the data from the sensors which is then encrypted in the sensors using Algorithm 1.The Algorithm 2 is used to decrypt the data. The Algorithm 3 is used to encrypt the data using a public key i.e. the identity of the patient. The Algorithm 4 is used by the doctor to decrypt the data using the identity of the patient. The Algorithm 6 is used to provide a certificate for both the identity of the patient and the doctor so that the identities of both are stored as PKG in the storage site which is then used to verify the doctor and decrypt the reading. The resulting ciphertext is denoted by c. The decryption of ciphertext c using the secret key x is given as EciesDecrypt(c, x). The recipient is assumed to have a long-term public /private key pair (Y, x) where

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Elliptic curve arithmetic can be used to develop a variety of elliptic curve cryptography (ECC) schemes, including key exchange, encryption, and digital signature. For pur[r]

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email: sdixit_dr@rediffmail.com Abstract - The true impact of any public-key cryptosystem can only be evaluated in the context of a security protocol. Secure communication is an intrinsic requirement in any transaction. It is very important to implement cryptosystems securely against the attacks. **Elliptic** **Curve** Cryptography (ECC) is coming forth as an attractive public key cryptosystem for mobile/wireless environments compared to traditional cryptosystems like RSA and DSA. ECC fits well for an efficient and secure encryption scheme because it provides better security with smaller key sizes, which results in faster computations, lower power consumption, as well as memory and bandwidth savings. However, the digital signature is the indispensable way to ensure the security of web services and has great significance in practical applications. By using a digital signature algorithm we can provide authenticity and validation to the electronic document. ECDSA and ECDH use the concept of ECC. In this article we present ECC and most popular algorithms such as RSA, ECDH, ECDSA and ECPP and a comparative study of all these algorithms have been done.

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In this paper we discuss existing methods for personalized, short-lived ECC parameter gener- ation. Even with current technology, each end-user can in principle refresh and recertify his or her ECC parameters on a daily basis (cf. Section 2): โin principleโ because user-friendly interfaces to the required software are not easily available to regular users. But it allows arbitrary, personalized choices โ within the restrictions of ECC of course โ in such a way that no other party can control or predict any of the newly selected parameters (including a **curve** parameterization and a finite field that together define an **elliptic** **curve** group, cf. below). Personalization isolates each user from attacks against other users, and using keys for a period of time that is as short as possible reduces the potential attack pay-off. Once personalized, short-lived ECC (public, private) key pairs are adopted at the end-user level, certifying parties may also rethink their sometimes decades-long key validities.

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