J. Appl. Environ. Biol. Sci., 7(12)69-73, 2017 | ISSN: 2090-4274 |
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1,2,3,4,6Department of Information Technology, Hazara University Mansehra, Pakistan. 6Department of Information Technology University of Haripur, K-P, Pakistan Received: June 21, 2017 Accepted: October 8, 2017
Authenticated encryption enables the originator of a message to generate signature and encryption in a single step to reduce computations. There are some resource-constrained environments like: embedded systems, sensor networks and ubiquitous computing that need the security in three different modes such as authentication only mode, encryption only mode and authenticated encryption mode. Therefore, to provide these three modes we proposed a new scheme called public verifiable generalized authenticated encryption based on hyper elliptic curve. The proposed scheme is efficient because the hyper elliptic curve has very small parameters and key size. In addition, it meets up the security services of authenticated encryption e.g. confidentiality, integrity, authenticity, unforgeability and no-repudiation. Moreover, our scheme provides the extra security function of public verifiability. Thus, the proposed scheme also has low computational cost as compared to schemes available in literature. The proposed scheme is most appropriate to use the resource constrained devices like sensors, pager and smart phone etc. KEYWORDS: Authenticated Encryption, Generalized Authenticated Encryption, Public Verifiability, Hyper Elliptic Curve.
Any communication system needs Confidentiality and authenticity as the basic and major requirements. Generally to achieve these two basic requirements, encryption and signature are essential in public key cryptography. For authenticity and confidentiality the number of authentication schemes [1,2,3,4,5,6] and encryption schemes [7,8,9] were projected. Though these two are the basic cryptographic techniques can be pooled together in diverse ways like: sign-then-encrypt, encrypt-then-sign and sign and encrypt etc. to ensure privacy and authenticity in many applications simultaneously. Unfortunately, this technique is not appropriate because signature generations and then make encryption in two different steps need more computational efforts. Therefore, to remove this deficiency in 1997 Zheng [10] planned a novel scheme called “Signcryption” To enhance the efficiency, which achieves confidentiality and authenticity in a single logical step. Signcryption has fewer computational complexities, less communication cost efforts and fewer implementation efforts. Many applications like electronic transactions protocol, mobile agent protocol, key management and routing protocol, key management and routing protocol are made-on-hand by Signcryption. Deng and Bao [11] made a scheme to reduce computationcostupto16%andcommunicationcostupto85% as compared to Signature-Then-Encryption technique, but the scheme’s computation cost is more than Zheng [10]’s scheme, the scheme is not straightlypublic verifiable and needs zero knowledge collaborating protocol for verification of message via any third party. To provide message verification of signcrypted text Gamage et al. [7] Projected a scheme, This is an enhancement of Deng and Bao [11]’s signcryption scheme to provide solution to the problem of authentication of secure message by firewall without revealing message confidentiality, while in this scheme an effort is made to reduce computation cost up to 40 % as compared to customary tactic and its communication cost is correspondent to Zheng [10]. There the ciphertext is verified to protect confidentiality of message, which it is unable to provide forward secrecy. Numerous Signcryption schemes have been proposed based on RSA problem [12,13] or Diffie-Hellman problem. Baeket al. [14] was the first which formalized and defined security notionsin 2002 for signcryption. An identity based signcryption scheme was projected by Sharma et al. [15], this scheme suffers from public verifiability. Zhangand Imai [16] made an attempt to a first elliptic curve based signcryption scheme. Hwang et al. [17] made another attempt based on ECC signcryption provided public verifiability and forward. In the scheme [18] a lightweight ECC based signcryption was proposed that concluded verification and forward secrecy, the scheme bump into the lightweight requirements of store then forward and resource limited environment to reduce significant computation and communication costs, Nizamuddin et al. [19, 20] projected different signcryption attempts over HECC. But Nizamuddin et al.
[19] suffered direct public verifiability, while Nizamuddin et al. [20] scheme suffered the forward secrecy and public verifiability, where for verification zero knowledge protocol is essential. As in some cases the customary signcryption is proficient to provide both Signature and Encryption functions in chorus, while this might not be the case if we are in need of only one provision and if we need both of them in some cases at the same time. While its fact that not all time the data’s need both i.e secrecy and authenticity. Some messages might need to be signed only, while some others may need to be encrypted only. In this accordance, applications may essentially contain at least three cryptographic primitives (signcryption, signature, and encryption), which might be infeasible in some resource-constrained environments like: embedded systems, sensor networks and ubiquitous computing. As there it will not be affordable to accommodate all these three different schemes to achieve confidentiality and authenticity separately or simultaneously. Hence a new primitive called generalized signcryption, was projected by Yiliang Han [21] that provided a facility for encryption or signature or Signcryption, as it was the need. In other words deprived of any supplementary alteration and computation, it provides double functions when confidentiality and authenticity are both required simultaneously, and a discrete encryption or signature function when one of them is required. A leading global signcryption scheme named ECGSC (elliptic curve generalized signcryption) is based on ECDLP. Lal et al. [22] contribute firstly an id based generalized signcryption scheme. To provide a formal security model for Generalized Signcryption Wang et al. [23] suggested an attempt and then modified this attempt in [21]. HF Ji et al [24] tried an attempt in standard model to design Identity-based generalized approach. The same Secure and efficient generalized signcryption was projected by Zhang et al [25]. Yu et al. [26] expose that Lal et al. [22] is not completely secured. Gang Yu et al [27] also projected an identity based generalized signcryption scheme. They made a claim that their proposed scheme was Provable secure. For identity based generalized signcryption Prashant and Lal [28] also made an attempt in a scheme. Another identity based generalized signcryption was proposed by Shen et al [29]. All the above generalized signcryption schemes are based on RSA, Bilinear pairing and elliptic curve cryptography that needs more computations. Furthermore, these schemes are not suitable for fewer resource devices. Thus to solve this problem we proposed a new scheme on the basis of scheme in [30], called generalized signcryption based on hyper elliptic curve. The main drawbacks of RSA and elliptic curve cryptography are greater parameters size. The Hyper-elliptic curve cryptosystem (HECC) with 80 bits provide equal level of security as compared to other cryptosystems like RSA and bilinear pairing with 1024 bit, elliptic curve with 160 bit and other public key cryptosystems by using low resources. In addition, it meets up the security services of authenticated encryption e.g. confidentiality, integrity, authenticity, unforgeability and no-repudiation. Moreover, our scheme provides the extra security function of public verifiability. Thus, the proposed scheme also has low computational cost as compared to schemes available in literature. The proposed scheme is most appropriate to use the resource constrained devices like sensors, pager and smart phone etc.
Let a prime number Ϙ, where Ϙ→ Ϙ ≥ 2��.Suppose ℱϘis a finite order Ϙ field. Therefore, generalized elliptic curve GEC#ℱϘ# on ℱϘcan be defined by the following equation #A#.
GEC: Ƴ! + h#ҳ#Ƴ = ®#ҳ# mod q #A#
*Corresponding Author: Asad Mehmood, Department of Information Technology, Hazara University Mansehra, Pakistan. 69
Mehmood et al.,2017
Where h#ҳ# ℰ ℱ[ҳ] is polynomial anddegree is ≤while ®#ҳ# ∈ ℱ[ҳ] is a monic polynomialanddegree ®#ҳ# ≤ 2g+1 , points on EC, points on GEC is not form of a group. Divisor is finite formal sum of points on hyper elliptic curve and represented in Mum Ford form as:
∂ ∂∂∂
∂∂
D = $u#ҳ#, v#ҳ#∂ = ∂∂u∂ ҳ,∂vxҳ∏
∂
∂∂� ∂∂�
Divisor shape Abelian group which is called Jacobian group J*$F&∂ and order of the Jacobian group o#J#ℱϘ## is defined as
*
&$&q − 1∂!∂& ≤ o#J#ℱϘ## ≤ &$&q + 1∂!∂&
*
The section includes key generations, the basic notations, generalized authenticated encryption and generalized authenticated decryption.
Sender/Alice select their private key by choosing a random number α , &ℎ&&& 0 < α <Ϙ and calculate &= α.& is a public key. Also the receiver select their private key by choosing a random number γ , &ℎ&&& 0 < γ <Ϙ and calculate &= γ.& is a public key.
a) & is the cipher text b) & is the plain text/message c) && represents the secrete key d) & is the generated hash value e) && is the one way keyed hash function f) && represents decryption g) α is the private key of signcrypter/originator h) & is the public key of signcrypter/originator i) γ is the private key of unsigncrypter/verifier j) &is the public key of unsigncrypter/verifier k) & is the divisor of hyper elliptic curve
This algorithm take (mode, private key of sender, public keys of receiver, plaintext) means it work in three modes to generate signature only, encryption only and signcrypt text generation.
Algorithm
o select &
&
This algorithm take (mode, private key of receiver, public keys of sender, generalized authenticated encryptext) means it work in three modes to generate signature verification only, decryption only and un-signcrypt text generation.
Algorithm
o Accept only ℛ=&hold.
In this section we discuss about the different modes of our proposed generalized signcrypion scheme e.g. signature, encryption and signcryption only mode.
This algorithm takes the message and the private key of sender and public key of receiver group as an input to produce signature text.
Algorithm
a) if &=& b) select & c) compute & =&.DmodϘ d) Calculate & = ℎ #&||m#
&
e) ¶ = #¨¨α# mod Ϙ
f) Send ω = #¨, &, s)
Upon receiving the signature this algorithm take the public key of sender and private key of verifier to verify the signature.it also first verify the public key of sender.
Algorithm
a) &= ¶. #&+&.&#
J. Appl. Environ. Biol. Sci., 7(12)69-73, 2017
b) ℛ = && #& ||m#
c) ¨¨¨&¨¨¨¨ℛ= &
This algorithm takes the message and the public key of receiver as an input to produce to produce cipher text.
Algorithm
• Calculate &= Φ
Upon receiving the cipher text and the multi receiver secret key cipher text this algorithm take the public key of sender and private key of verifier to generate the secret keys.
Algorithm
• Accept only ℛ=& holds.
This algorithm takes the message and the private key of sender and public key of receiver as an input to produce signcryp text.
Algorithm
o select &
&
Upon receiving the signcrypted text and the multi receiver secret key cipher text this algorithm take the public key of sender and private key of verifier and decrypt the authenticated encryptext.
o Accept only ℛ=&hold.
This section briefly discusses the security services of a proposed scheme. All the security services are based on the hardiness of hyper elliptic curve discrete logarithm problem (HECDLP).
Definition: HECDLP
Let Ɗ the divisor of order n in the Jacobian groupJ*#ℱϘ#, find an integerҳ ∈ ℱϘ, such that: Ɗ∂ = ҳ.Ɗ
Our designed ℙ₢Ǣ scheme is said to be secured under cipher text attack if an attacker Å cannot get secrete key &Æ.Suppose an attacker wants to get the plain text & from the cipher text &, then Årequir the secrete key &Æ from (a). Therefore, for solving (a) Å needs & from (b).thus it is not possible for to get&, because it is equilents to solve hyper elliptic curve discrete logarithm problem. For this reason our designed scheme strongly provides the confidentiality of a plaintext.
&& = ℋ#&. &# #Æ#& = &.D #b#
In proposed ℙ₢Ǣ scheme the unsigncrypter/verifier can verify the message is modified or not and send by the originator or not. Hence in our ℙ₢Ǣ scheme before sending the message originator/sender first generates the hash value & = ℎ #&||m# of a message by using collision resistance one way hash function. When is required to change in cipher text & to &¢ , then automatically ¢ will be changed to ¢¢ and &into&¢.hence cannot be changed in & because hash function is one way. The unsigncrypter/verifier can verify the integrity of message by comparing ℛ=& if it holds then accepts otherwise rejected.
The proposed ℙ₢Ǣ scheme is secured from forgery of a signature. Suppose a forger ℱ wants to compute a forge signature ¶¢like (c).So to generates forge signature ¶¢ the forger ℱ must need & from (d) and α from (e).Thus it is hard for forger ℱ to solve two different hyper elliptic curve discrete logarithm problem.
¶¢= & #¨#
#& + α#& = &.D #d#& = α.& #&#
Mehmood et al.,2017
The designed ℙ₢Ǣ scheme provides sender public verifiability property. In our proposed ℙ₢Ǣ scheme when dispute occur between signcrypter and unsigncrypter/verifier the judge/third party can verify by using the following equations.
&= ¶. #&+&.&#
& &.& = .#α.&+&.&#= .#α+&#
#&+α# #&+α#
&. & = .#α+&#=&.D
#& + α#=& Proved
In our proposed ℙ₢Ǣ scheme the signcrypter/originator cannot deny transmitted cipher text .if the signcrypter/originator can deny from transmitted cipher text then judge perform the following steps to prove whether the cipher text from sender or not.
1. Verify the sender public key & from certificate authority.
2. ¨¢¢¨¢¨&&= ¶. #&+&.&# Note: If step no 2 is proved then the signcrypter/originator cannot deny/repudiate from cipher text.
In this section we compare the computational cost of our proposed generalized signcryption scheme with existing Generalized signcryption schemes [21,22,26,28,29]. The computational cost can be measure in terms of major operations. We investigate that the most cost operations in proposed generalized signcryption and existing Generalized signcryption schemes [21,22,26,28,29] is elliptic curve scaler multiplication, hyper or Generalized elliptic divisor multiplication, exponential operations and pairing operations. The Table 1 shows comparisons in term of major operations and we denote elliptic curve scaler multiplication by ԐՇ¢¢, hyper or generalized elliptic divisor multiplication by ¢ԐՇ¢¢, exponential operations by ¢¢¢¢ and pairing operations by ¢¢¢ . Table 2 shows the comparisons of proposed and existing schemes with respect to milli seconds. We briefly define the efficiency and comparisons of proposed and existing schemes in discussion phase.
Table 1: computational cost comparisons in terms of major operations
Schemes Y. Han [21] | Generalized Signcrypt ¢ ԐՇℙ¢ | Generalized Un-Signcrypt ¢ ԐՇℙ¢ | Total | ¢ ԐՇℙ¢ |
Lal et al [22] | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | |
Yu et al [26] | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | |
Prashant et al [28] | ¢ Ԑ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | |
Shen et al [29] | ¢ Ԑ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | ¢ Ԑ¢¢¢ + ¢ ¢¢¢ | |
Proposed scheme | ¢ ¢ԐՇℙ¢ | ¢ ¢ԐՇℙ¢ | ¢ ¢ԐՇℙ¢ |
Table 2: computational cost comparisons in terms of milli seconds
Schemes | Generalized Signcrypt | Generalized Un-Signcrypt | Total | Total reduction of proposed scheme in % from existing |
---|---|---|---|---|
Y. Han [21] | ¢ ∗ ¢. ¢¢ = ¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ = ¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ = ¢. ¢¢ ¢¢ | ¢. ¢¢ − ¢. ¢ ¢. ¢¢ = 50.51% |
Lal et al [22] | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢ = ¢¢. ¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢¢. ¢¢ − ¢. ¢ ¢¢. ¢¢ = 95.88 % |
Yu et al [26] | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢¢. ¢¢ − ¢. ¢ ¢¢. ¢¢ = 95.88 % |
Prashant et al [28] | ¢ ∗ ¢. ¢¢ = ¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢¢ ¢¢ | ¢¢. ¢¢ − ¢. ¢ ¢¢. ¢¢ = 93.77% |
Shen et al [29] | ¢ ∗ ¢. ¢¢ = ¢. ¢ ¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢¢¢ | ¢ ∗ ¢. ¢¢ + ¢ ∗ ¢¢. ¢¢ = ¢¢. ¢ ¢¢ | ¢¢. ¢ − ¢. ¢ ¢¢. ¢ = 97.15% |
Proposed scheme | ¢ ∗ ¢. ¢¢ = ¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ = ¢. ¢¢ ¢¢ | ¢ ∗ ¢. ¢¢ = ¢. ¢ ¢¢ | - |
We investigate that all the existing generalized signcryption schemes are based on RSA, Bilinear pairing and elliptic curve. These three cryptosystem needs heavy computational power which is not suitable for constrained resource devices. Thus to solve the problem we proposed Public Verifiable Generalized Authenticated Encryption (₢Ǣ) based on Hyper Elliptic Curve. The Hyper-elliptic curve cryptosystem (HECC) with 80 bits provide equal level of security as compared to other cryptosystems like RSA and bilinear pairing with 1024 bit, elliptic curve with 160 bit and other public key cryptosystems by using low resources. For efficiency we compare our proposed scheme with schemes [21, 22,26, 28, 29] in terms of computational cast. Moreover, we investigate that in proposed and existing schemes the most costly operations is elliptic curve scaler multiplication, hyper or generalized elliptic divisor multiplication, exponential operations and pairing operations. The Table 1 shows comparisons in term of major operations and we denote elliptic curve scaler multiplication by ԐՇ¶ℳ, hyper or generalized elliptic divisor multiplication by ℋԐՇ¢ℳ, exponential operations by ℰ¢&¢ and pairing operations by &¢&. Table 2 shows the comparisons of proposed and existing schemes [21,22,26,28,29] with respect to milli seconds. Therefore, to determine the primary cryptographic operations computation time in milli seconds such as ԐՇ¶ℳ, ℋԐՇ¢ℳ, ℰ¢&¢ Æ¢¢ &¢&. It is perceived from [31] by using Multi-precision Integer and Rational Arithmetic C Library (MIRACL)[32],Windows7 Home Basic64-bit operating system, Intel Core i7-4510UCPU running at 2.0GHz with 8GB of memory. The single ԐՇ¶ℳ take 0.97 milli seconds, ℰ¢&¢ needs 1.25 milli seconds and &¢& take 14.90 milli seconds. We assume that in this eminent if single ԐՇ¶ℳ take 0.97 milli seconds then ℋԐՇ¢ℳ take the half of ԐՇ¶ℳ 0.48 milli seconds. Therefore the generalized formula for
Æ¢∂¢¢∂¢∂ ¢*¢Æ¢Æ∂¢¢¢¢¢¢Æ¢ ¢*¢Æ¢Æ
reduction of computational cost is [33] . Thus it concluded from Table 2 that the proposed scheme reduced in
Æ¢∂¢¢∂¢∂ ¢*¢Æ¢Æ computational about 50.51% from Y. Han [21], about 95.88 % from Lal et al [22] and Yu et al [26], about 93.77% from Prashant et al [28] and about 97.15% from Shen et al [29]. In addition, it meets up the security services of authenticated encryption e.g. confidentiality, integrity, authenticity, unforgeability and no-repudiation. Moreover, our scheme provides the extra security function of public verifiability.
This desertion presents a new generalized authenticated encryption (ℙ₢Ǣ ) scheme based on hyper elliptic curve. The proposed generalized authenticated encryption (ℙ₢Ǣ ) scheme provides three different modes in single algorithm e.g. the signcryption mode, encryption mode and signature mode. It provides the security requirements like confidentiality, integrity, non-repudiation and public verifiability. Furthermore, the
J. Appl. Environ. Biol. Sci., 7(12)69-73, 2017
proposed scheme also has low computational cost as compared to schemes available in literature. The proposed scheme is most appropriate to use the resource constrained devices like sensors, pager and smart phone etc.
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