# An Efficient Conformable Fractional Chaotic Map-Based Online/Offline IBSS Scheme for Provable Security in ROM

Authors: Chandrashekhar Meshram; Rabha W. Ibrahim; Rafida M. Elobaid
Journal: Complexity
 (2022)

---

## Abstract

Chaos distributes with a covert method to condense the dynamic of complexity and satisfies the security requirements of a cryptographic system. This study gives an ability online/offline (O/O) ID-based short signature (IBSS) scheme using conformable fractional chaotic maps. Furthermore, we establish its security under IBSS existential unforgeability of identity-based short signature (IBSS) under chosen message attack (EUF-IBSS-CMA) in the random oracle model (ROM). Some of the stimulating preparations of obtainable processes are that they give a multiperiod application of the offline storage, which licenses the agent to recycle the offline pre-registered data in time series (especially the polynomial time), rather than one-period usage in all past IBSS processes.


---

## Body

## 1. Introduction

Newly, the time-fractional difference [1] provides a robust concept for discrete (not continuous) fractional display. It has a limited fractional alteration formula, which rests on the change consequences of all the past figurines. This attribute can show the disconnected arrangements long historical properties or long interactions. In the meantime, chaos definitions, formulas, ideas, and chaos synchronization have wide uses [2–5]. Discrete maps can produce chaotic signatures. Therefore, they rewarded much care in all areas of mathematical sciences. The logistic map idea (is a well-known repeated record founded on the first-order nonlinear alteration equation) and other types of maps have converted straightforward representations. Nevertheless, fewer works utilized the fractional discrete arrangements, which clamp compound chaotic dynamics. This action presents the disconnected memory, which occurs in the chaotic records. Then, chaos and harmonization of the fractional logistic record are specified. The diverse fractional powers yield different chaotic ranges so that the chaotic activities will take extra problematical [6, 7]. Discrete maps are used regularly in disconnected natural phenomena. The standing fractional disconnected arrangements (equations, inequalities, and inclusions) are typically joined with two techniques: mathematical discretization (the process of changing continuous functions, simulations, variables, and equations into discrete complements) of time-fractional differential equations and fractional time-difference equations. The former one is a numerical formulation of fractional continuous simulations and the Grünwald–Letnikov difference usually accepted in the numerical action. In this study, we shall use the fractional Caputo difference operator. Our aim is to use a new fractional calculus, called fractional conformable calculus, to generalize the Chebyshev polynomials [8].The inquiry into chaotic constructions and their possible cryptographic structures has been the subject of considerable interest in research over the past few years. Chaotic systems are clearly characterized by their delicate reliance on the initial conditions and random surrounding operations, both of which are fundamentally similar to the behavior of some cryptographic primitives [9]. Even et al. [10] introduced the concept of the O/O sign in 1989. In an O/O system, a message signing is broken into two phases with (i) more computational and time-consuming phase being executed offline in advance and (ii) much faster phase being conducted online at the point of signing the message. Even et al. created a general construction that could turn any digital signature scheme into an O/O sign structure [10]; nevertheless, the system is not very practical as it lengthens a quadratic variable in each signature. Instead, in 2001, as a response to the impracticality of Even et al.’s 1989 scheme, Shamir and Tauman [11] introduced a new theoretical idea called “hash-sign-switch” to more effectively transform any signature scheme into an O/O sign structure. This hash-sign-switch process converts signature schemes into O/O sign operations irrespective of the types as a generalized tool. In order to address certain types of signature schemes specialized in specific applications, some researchers have proposed their own designs [12–16], among which [15] is the most efficient [16], while the work of Kurosawa and Schmidt-Samoa included the possibility of creating O/O sign operations without random oracles [12]. Nonetheless, all the above schemes concentrated on the public-key-based standard setting without participating in identity-based settings. Several identities-based sign structures using pairing have been published in [17–20] since about a decade ago. On the other hand, for a discrete log setting that does not require pairing, Galindo and Garcia [21] adapted Schnorr’s sign to construct an identity-based sign structure. Xu et al. [22] imposed an idea of O/O-IBSs and multisign in 2006. Xu et al. proposed the O=O IBS sign structure and then transformed it to O/O-ID-based multisign structure. Xu et al.’s [22] sign structure can be extended to different routing protocols using the pairing technique. Later on, however, Li et al. [23] showed that the sign structure of Xu et al. [22] was in fact weak against the attack on the forgery and thus deficient in security. In addition, a truly secure O/O–IBSS scheme has yet to be found in the relevant literature. On the other hand, since the 1990s [9, 24, 25], chaotic cryptography has been used in the development of secure communication techniques. Chaotic records are now essential for different methods of symmetric encryption [26–30], hash functions [31, 32], and S-boxes [33]. Recently, numerous chaotic methods were released on key convention models [34–38], authentication protocols [39, 40], and telecommute medicine information systems [41–43].As given above, we suggest the well-organized conformable chaotic map (CCM)-based O/O–IBSS arrangement. The existing scheme allows the reusability of the offline data. Consequently, the agent is not dynamic to device the offline technique-assigning stopping when he/she requests to indicate other communication. In addition, in view of circumstances, such as an extensive section of the present O/O signs (non-ID-based), an obtainable offline passing scheme does not require any private signer information [44–58]. Thus, some favorite associates covering the private key group (PKG) can build it. A new ID-based setting requires no approval of confirmation attached to the sign, which is displayed in ROM. The suggested O/O–IBSS structure is locked when the selected communication occurs in the logic of the ability of IBSS, assuming that the CCM theory grips in the ROM with less ranked cost [56–58].Recently, Meshram et al. [59] developed a subtree-centric paradigm for cryptosystems in cloud-based environments using chaotic maps. Meshram et al. [58] also used chaos theory to develop a level online/offline subtree-based short signature framework that is both efficient and secure. A new chaotic system with the hyperbolic sinusoidal function was presented by Mobayen et al. [60]. This chaotic system introduces a novel type of chaotic flow that allows for a better understanding of chaotic attractors. In the presence of external disturbances and Lipchitz nonlinearities, Karami et al. [61] developed the observer-based state feedback stabiliser design for a class of chaotic systems. For the robust synchronization of uncertain delayed chaotic systems, Mofid et al. [62] developed the disturbance observer-based Sliding mode control (SMC) scheme. More studies can be located in [63–66].We present a detailed literature review of the existing identity-based online/offline short signature schemes. Unfortunately, most schemes are built on difficult problems like the elliptic curve and pairing and pose huge computational and communication costs. It is also worth noting that most of the schemes have not been thoroughly tested with Scyther, AVISPA, and other high-end security validation tools. As a result, small devices with limited processing resources find it difficult to manage such schemes.The main contributions of this study can be stated in the following aspects:(i)
Under the security of the random oracle model, we propose a secure and efficient conformable fractional chaotic map-based online/offline identity-based short signature scheme.(ii)
The proposed scheme is secure with unforgeability under chosen message attack (UF-IBSS-CMA) in the random oracle model.(iii)
Unlike past identity-based online/offline signature schemes, the proposed scheme’s design allows the signer to enter the offline storage numerous times to reuse the offline preinformation in polynomial time.(iv)
The presented scheme has the lowest computational cost amid six competing schemes.(v)
The proposed scheme is a separated signing method that does not call all types of secluded key data. It can be recorded by an insulated key group with offline data equally being employed. It consistently assumes the very slightest process in every practice.(vi)
The proposed scheme is an astonishing promising model in wireless sensor network circumstances as the detached data may be complex-inserted into the sensor hub in the collecting or procedure position.The article is organized as follows: Materials and Methods involve essential mathematical initiations offered in Section2. Section 3 deals with the results, including the O/O–IBSS by using CCM. Security examination and other discussion can be seen in Section 4. Finally, Section 5 concludes the suggested algorithm.


## 2. Materials and Methods

In the analysis of arbitrary calculus, a part of the major experiments is to discover appropriate algorithms, which are correctly given by difference derivatives with the history of this measure. The Chebyshev polynomials are one of the greatest valuable polynomials, which are appropriate in numerical analysis comprising polynomial approximation. We briefly explain the fractional Chebyshev polynomial-2 (second type), generalize chaotic records, and develop it as a sector of the suggested technique, correspondingly.


### 2.1. Chebyshev Polynomials-2

Chebyshev polynomials-2 of degreen on the interval [−1, 1] is formulated in terms of sinϑ,ϑ=arccost.(1)Υnt=sinn+1ϑsinϑ,ϑ∈0,π.This satisfies (see Figure1)(2)Υ0t=1Υ1t=2t⋮Υnt=2tΥn−1t−Υn−2t,n=2,3,...Figure 1
CCP for different values ofν and κ1v,t=1−ν/Γν+1,κ0ν,t=ν/Γν+1. It pretenses to the historical illustrations for Υnt (see [67]).The polynomials of the second type fulfill(3)Υn−1cosϑ⋅sinθ=sinnϑ,or(4)Υncosϑ=sinn+1ϑsinϑ.This is systemically likewise to the Dirichlet kernel:(5)σnt=sin2n+1t/2sint/2=Υ2ncost2.The polynomialsΥn are orthogonal on [−1,1] connecting with the inner products:(6)Υnt,Υqt=∫−11ΥntΥqt1−t2dt=π2δnq,where 1−t2 is the weight function and δnq is the well-known Kronecker function. The analytic representation takes the following summation:(7)Υnt=∑m=0n/2−1mΓn−m+12n−2mΓn−2m+1Γm+1tn−2m,where n/2 is the integer part of n/2. And the first derivative is(8)DΥnt≔Υn′t=2∑m=0n−1m+1Υmt.As a special case (shifted second-type Chebyshev polynomial), when the variable is2t−1, we have the following formula:(9)Υn2t−1≔Υn∗t=22t−1Υn−1∗t−Υn−2∗t,n=2,3,....with Υ0∗t=1 and Υ1∗t=22t−1. Moreover, it achieves the analytic representation formula:(10)Υn∗t=∑m=0n−1mΓ2n−m+222n−2mΓm+1Γ2n−m+2tn−m.Clearly,Υn∗0=−1n and Υn∗1=2. In addition, the orthogonal representation is of the following form:(11)Υn∗t,Υq∗t=∫01Υn∗tΥq∗tt−t2dt=π8δnq.The first derivative of the shifted second-type Chebyshev polynomial is given by [8](12)DΥn∗t≔Υn∗′=4∑m=0n−1m+1Υm∗t.


### 2.2. Conformable Calculus

Recently, connected to the arbitrary calculus field, Khalil et al. [44] formulated a “conformable fractional derivative” definition of a given real-valued function ϕ:0,∞⟶ℝ as follows:(13)Dνϕt=limϵ⟶0ϕt+ϵt1−ν−ϕtϵ,for all t>0 and a fractional power ν∈0,1. If ϕ is ν-differentiated in the interval 0,ϱ,ϱ>0 and limt⟶0+ϕνt occurs and then ϕν0=limx⟶0+ϕνt is introduced.More generalization criteria for differential operators to be a real-valued conformable fractional derivative was recently proposed by Anderson and Ulness (see [45]).Definition 1.
Conformable differential operator.
Letν be a fractional power, such that ν∈0,1. A differential functional Dν is called conformable ⇔D0 is the identity function and D1 is the ordinary derivative function. Particularly, Dν is conformable ⇔ for differentiable function ϕt.(14)D0ϕt=ϕtandD1ϕt=ddtϕt=ϕ′t.
In general, for two continuous functionsκ0,κ1:0,1×ℝ⟶0,∞, we obtain(15)Dνϕt=κ1ν,tϕt+κ0ν,tϕ′t,such that κ1ν,t≠−κ0ν,t,(16)limν⟶0κ1ν,t=1,limν⟶1κ1ν,t=0,κ1ν,t≠0,∀t,ν∈0,1.(17)limν⟶0κ0ν,t=0,limν⟶1κ0ν,t=1,κ0ν,t≠0,∀t,ν∈0,1.
In [46], the authors noted that in the theory of control systems, a comparative controller for supervisory result μ at the variable t with two correction factors has the following process:(18)μt=κpΞt+κdddtΞt,where κp is the comparative gain, κd is the changing gain, and Ξ is the slip between the formal variable and the practice variable.


### 2.3. Conformable Chebyshev Polynomials-2 (CCP)

In this section, we employ the concept of conformable derivative to obtain the CCP (of the second type). By using equations (1) and (3) in (8), we have the following CCP:(19)DνΥnt=κ1ν,tΥnt+κ0ν,tΥn′t.Moreover, the shifted CCP can be formulated by applying (5) and (6) in (8) to obtain(20)DνΥn∗t=κ1ν,tΥn∗t+κ0ν,tΥn∗′t,where κ1 and κ0 are given in Definition 1. In our discussion, we shall select one of the following formulas of κ1 and κ0:(21)κ1ν,t=1−νtν,κ0ν,t=νt1−ν,t∈0,∞,κ1ν,t=1−νtν,κ0ν,t=νt1−ν,κ1ν,t=cosνπ2tν,κ0ν,t=sinνπ2t1−ν,t∈0,∞.Or,(22)κ1ν,t=cosνπ2|t|ν,κ0ν,t=sinνπ2t1−νt∈ℝ0.Furthermore, constant functions can be realized by using the gamma function as follows:(23)κ1ν,t=1−νΓν+1,κ0ν,t=νΓν+1.Obviously, for finite casem=n−1 in relations (3) and (6), respectively, we have the qualities(24)Υn′t=2nΥn−1t,Υn∗′=4nΥn−1∗t.Consequently, by utilizing the definition ofDνΥnt and DνΥn∗t, we have the CCP-2 and its shifted polynomial:(25)DνΥnt=κ1v,tΥnt+2nγtκ0v,tΥn−1t,where(26)γt=1+2t+4t2−1+4t2t2−1+...+︸n−1−times(27)DνΥn∗t=κ1v,tΥn∗t+4nγ∗tκ0v,tΥn−1∗t,where(28)γ∗t=1+22t−1+...+︸n−1−times.In the sequel, we putΥνt≔DνΥnt and Υ∗νt≔DνΥn∗t. Then, we have the following constructions.Proposition 1.
The CCP satisfies the following recurrent relations:(29)Υnvt=2tκ1v,t+2nγtκ0v,tΥn−1t−κ1v,tΥn−2t.(30)Υn∗νt=2κ1v,t2t−1+4nγ∗tκ0v,tΥn−1∗t−κ1v,tΥn−2∗t.From (1) and (25), we conclude that(31)Υnvt=Υntκ1v,t+2nγtκ0v,tΥn−1t=2tΥn−1t−Υn−2tκ1v,t+2nγtκ0v,tΥn−1t=2tκ1v,t+nγtκ0v,tΥn−1t−κ1v,tΥn−2t.Now, in view of (9) and (27), we obtain(32)Υn∗νt=Υn∗tκ1v,t+4nγ∗tκ0v,tΥn−1&lowast;t=22t−1Υn−1∗t−Υn−2∗tκ1v,t+2nγ∗tκ0v,tΥn−1t=2κ1v,t2t−1+2nγ∗tκ0v,tΥn−1∗t−κ1v,tΥn−2∗t.Remark 1.
It is clear that whenν⟶0, we have the original case that was proved in [47].Proposition 2.
The semigroup possessions clamps for CCP positioned on interval (−∞,∞).We takeT≔tκ1v,t+nγtκ0v,t, where ν⟶0 implies T=t (the original case). From Proposition 1, we have(33)Υn+2vt=2TΥn+1t−κ1Υnt.The previous formula implies an alteration equation (disconnected equation) which has a typical formula(34)λ2−2Tλ+κ1=0,with the two roots λ1,2=T±T2−κ1 satisfying λ1+λ2=2T and λ1λ2=κ1. Then, for a positive integer n, we obtain the conclusion(35)Υnvt=λ1n+λ2n/2=T−T2−κ1n+T+T2−κ1n2=∑m=0n/2nmTn−2mT2−κ1m.Following the proof in [47] on the above summation, we obtain(36)ΥℓvΥ♭vt=ρ1ℓ+ρ2ℓ2,ρ1+ρ2=2Υ♭v,ρ1ρ2=κ1.Then, in general, we obtain the following relation for positive integersℓ and ♭:(37)Υℓ♭vt=ΥℓvΥ♭vt=Υ♭vΥℓvtmodN,where N is large, as much as data set size.Remark 2.
It is clear that whenν⟶0, we have the main theorem (see Figure 1).


## 2.1. Chebyshev Polynomials-2

Chebyshev polynomials-2 of degreen on the interval [−1, 1] is formulated in terms of sinϑ,ϑ=arccost.(1)Υnt=sinn+1ϑsinϑ,ϑ∈0,π.This satisfies (see Figure1)(2)Υ0t=1Υ1t=2t⋮Υnt=2tΥn−1t−Υn−2t,n=2,3,...Figure 1
CCP for different values ofν and κ1v,t=1−ν/Γν+1,κ0ν,t=ν/Γν+1. It pretenses to the historical illustrations for Υnt (see [67]).The polynomials of the second type fulfill(3)Υn−1cosϑ⋅sinθ=sinnϑ,or(4)Υncosϑ=sinn+1ϑsinϑ.This is systemically likewise to the Dirichlet kernel:(5)σnt=sin2n+1t/2sint/2=Υ2ncost2.The polynomialsΥn are orthogonal on [−1,1] connecting with the inner products:(6)Υnt,Υqt=∫−11ΥntΥqt1−t2dt=π2δnq,where 1−t2 is the weight function and δnq is the well-known Kronecker function. The analytic representation takes the following summation:(7)Υnt=∑m=0n/2−1mΓn−m+12n−2mΓn−2m+1Γm+1tn−2m,where n/2 is the integer part of n/2. And the first derivative is(8)DΥnt≔Υn′t=2∑m=0n−1m+1Υmt.As a special case (shifted second-type Chebyshev polynomial), when the variable is2t−1, we have the following formula:(9)Υn2t−1≔Υn∗t=22t−1Υn−1∗t−Υn−2∗t,n=2,3,....with Υ0∗t=1 and Υ1∗t=22t−1. Moreover, it achieves the analytic representation formula:(10)Υn∗t=∑m=0n−1mΓ2n−m+222n−2mΓm+1Γ2n−m+2tn−m.Clearly,Υn∗0=−1n and Υn∗1=2. In addition, the orthogonal representation is of the following form:(11)Υn∗t,Υq∗t=∫01Υn∗tΥq∗tt−t2dt=π8δnq.The first derivative of the shifted second-type Chebyshev polynomial is given by [8](12)DΥn∗t≔Υn∗′=4∑m=0n−1m+1Υm∗t.


## 2.2. Conformable Calculus

Recently, connected to the arbitrary calculus field, Khalil et al. [44] formulated a “conformable fractional derivative” definition of a given real-valued function ϕ:0,∞⟶ℝ as follows:(13)Dνϕt=limϵ⟶0ϕt+ϵt1−ν−ϕtϵ,for all t>0 and a fractional power ν∈0,1. If ϕ is ν-differentiated in the interval 0,ϱ,ϱ>0 and limt⟶0+ϕνt occurs and then ϕν0=limx⟶0+ϕνt is introduced.More generalization criteria for differential operators to be a real-valued conformable fractional derivative was recently proposed by Anderson and Ulness (see [45]).Definition 1.
Conformable differential operator.
Letν be a fractional power, such that ν∈0,1. A differential functional Dν is called conformable ⇔D0 is the identity function and D1 is the ordinary derivative function. Particularly, Dν is conformable ⇔ for differentiable function ϕt.(14)D0ϕt=ϕtandD1ϕt=ddtϕt=ϕ′t.
In general, for two continuous functionsκ0,κ1:0,1×ℝ⟶0,∞, we obtain(15)Dνϕt=κ1ν,tϕt+κ0ν,tϕ′t,such that κ1ν,t≠−κ0ν,t,(16)limν⟶0κ1ν,t=1,limν⟶1κ1ν,t=0,κ1ν,t≠0,∀t,ν∈0,1.(17)limν⟶0κ0ν,t=0,limν⟶1κ0ν,t=1,κ0ν,t≠0,∀t,ν∈0,1.
In [46], the authors noted that in the theory of control systems, a comparative controller for supervisory result μ at the variable t with two correction factors has the following process:(18)μt=κpΞt+κdddtΞt,where κp is the comparative gain, κd is the changing gain, and Ξ is the slip between the formal variable and the practice variable.


## 2.3. Conformable Chebyshev Polynomials-2 (CCP)

In this section, we employ the concept of conformable derivative to obtain the CCP (of the second type). By using equations (1) and (3) in (8), we have the following CCP:(19)DνΥnt=κ1ν,tΥnt+κ0ν,tΥn′t.Moreover, the shifted CCP can be formulated by applying (5) and (6) in (8) to obtain(20)DνΥn∗t=κ1ν,tΥn∗t+κ0ν,tΥn∗′t,where κ1 and κ0 are given in Definition 1. In our discussion, we shall select one of the following formulas of κ1 and κ0:(21)κ1ν,t=1−νtν,κ0ν,t=νt1−ν,t∈0,∞,κ1ν,t=1−νtν,κ0ν,t=νt1−ν,κ1ν,t=cosνπ2tν,κ0ν,t=sinνπ2t1−ν,t∈0,∞.Or,(22)κ1ν,t=cosνπ2|t|ν,κ0ν,t=sinνπ2t1−νt∈ℝ0.Furthermore, constant functions can be realized by using the gamma function as follows:(23)κ1ν,t=1−νΓν+1,κ0ν,t=νΓν+1.Obviously, for finite casem=n−1 in relations (3) and (6), respectively, we have the qualities(24)Υn′t=2nΥn−1t,Υn∗′=4nΥn−1∗t.Consequently, by utilizing the definition ofDνΥnt and DνΥn∗t, we have the CCP-2 and its shifted polynomial:(25)DνΥnt=κ1v,tΥnt+2nγtκ0v,tΥn−1t,where(26)γt=1+2t+4t2−1+4t2t2−1+...+︸n−1−times(27)DνΥn∗t=κ1v,tΥn∗t+4nγ∗tκ0v,tΥn−1∗t,where(28)γ∗t=1+22t−1+...+︸n−1−times.In the sequel, we putΥνt≔DνΥnt and Υ∗νt≔DνΥn∗t. Then, we have the following constructions.Proposition 1.
The CCP satisfies the following recurrent relations:(29)Υnvt=2tκ1v,t+2nγtκ0v,tΥn−1t−κ1v,tΥn−2t.(30)Υn∗νt=2κ1v,t2t−1+4nγ∗tκ0v,tΥn−1∗t−κ1v,tΥn−2∗t.From (1) and (25), we conclude that(31)Υnvt=Υntκ1v,t+2nγtκ0v,tΥn−1t=2tΥn−1t−Υn−2tκ1v,t+2nγtκ0v,tΥn−1t=2tκ1v,t+nγtκ0v,tΥn−1t−κ1v,tΥn−2t.Now, in view of (9) and (27), we obtain(32)Υn∗νt=Υn∗tκ1v,t+4nγ∗tκ0v,tΥn−1&lowast;t=22t−1Υn−1∗t−Υn−2∗tκ1v,t+2nγ∗tκ0v,tΥn−1t=2κ1v,t2t−1+2nγ∗tκ0v,tΥn−1∗t−κ1v,tΥn−2∗t.Remark 1.
It is clear that whenν⟶0, we have the original case that was proved in [47].Proposition 2.
The semigroup possessions clamps for CCP positioned on interval (−∞,∞).We takeT≔tκ1v,t+nγtκ0v,t, where ν⟶0 implies T=t (the original case). From Proposition 1, we have(33)Υn+2vt=2TΥn+1t−κ1Υnt.The previous formula implies an alteration equation (disconnected equation) which has a typical formula(34)λ2−2Tλ+κ1=0,with the two roots λ1,2=T±T2−κ1 satisfying λ1+λ2=2T and λ1λ2=κ1. Then, for a positive integer n, we obtain the conclusion(35)Υnvt=λ1n+λ2n/2=T−T2−κ1n+T+T2−κ1n2=∑m=0n/2nmTn−2mT2−κ1m.Following the proof in [47] on the above summation, we obtain(36)ΥℓvΥ♭vt=ρ1ℓ+ρ2ℓ2,ρ1+ρ2=2Υ♭v,ρ1ρ2=κ1.Then, in general, we obtain the following relation for positive integersℓ and ♭:(37)Υℓ♭vt=ΥℓvΥ♭vt=Υ♭vΥℓvtmodN,where N is large, as much as data set size.Remark 2.
It is clear that whenν⟶0, we have the main theorem (see Figure 1).


## 3. Proposed CCM-Based IBSS

In this section, we demonstrate an efficient conformable chaotic map (CCM) found by the online/offline IBSS method. It covers the supplementary five parts. Figure2 depicts the proposed online/offline IBSS scheme’s configuration.Figure 2
Block diagram of the proposed scheme.


### 3.1. Setup

(i)
Pick out a large enough primeq1 and general parameter t∈Zq1∗(ii)
Adopt a random parametera←Zq1∗ and infer u←Υavtmodq1(iii)
Opt a functionℏ (chaotic hash function) such that ℏ achieves ℏ:0,1∞⟶Zq1∗(iv)
The appearance of keys can be seen by the formalmpk=q1,t,ℏ,u (master public key) and msk=a (master secret key)


### 3.2. Extract

Assumed customer an individualityid, the PKG performs the following:(i)
Opts inordinatek←RZq1∗(ii)
AssessesA←Υkvtmodq1 and v←ℏA,id(iii)
Considersx←av∗kmodq1(iv)
The specific key of mediators is obtainable by the ordered pairA,x


### 3.3. Offline Signing

Any agent that performs the uniting is shown in the following steps:AssessesQi′←Υ2ivtmodq1, ∀i∈0,q1−1


### 3.4. Online Signing

In the recent step, to symbol a missiveM∈−∞,+∞, tiring A,x the agent stays as occurs subsequently:(i)
Selectl←Zq1∗ randomly such that li is the i-th bit of l(ii)
AssessQ←Πi=1p1Qi−1'(iii)
Assessb←ℏA,M,Q(iv)
Assessw←l∗bxmodq1; consequently, deliver η←Υwvtmodq1(v)
The signσ of any missive M assumed by: σ←A,Q,w


### 3.5. Verification

A confirmation of any signσ on the set of M with the identity of the agent id remains as receipts subsequently:(i)
Calculateη′←QΥbvAΥbvvumodq1(ii)
If we arrive at the stateη=η′, then in this situation, the sign is agreeable; otherwise, it is ostracized


### 3.6. Reliability of the Process

The private key needs to attain the effectiveness with fairness:(38)Υxvtmodq1=AΥvvumodq1.To prove the reliability of the procedure, we utilizeQ=Υlvtmodq1 to compute the following:(39)QΥbtvAmodq1Υbvvumodq1=ΥlvtΥkbvtΥabvvtmodq1=Υwvtmodq1.


## 3.1. Setup

(i)
Pick out a large enough primeq1 and general parameter t∈Zq1∗(ii)
Adopt a random parametera←Zq1∗ and infer u←Υavtmodq1(iii)
Opt a functionℏ (chaotic hash function) such that ℏ achieves ℏ:0,1∞⟶Zq1∗(iv)
The appearance of keys can be seen by the formalmpk=q1,t,ℏ,u (master public key) and msk=a (master secret key)


## 3.2. Extract

Assumed customer an individualityid, the PKG performs the following:(i)
Opts inordinatek←RZq1∗(ii)
AssessesA←Υkvtmodq1 and v←ℏA,id(iii)
Considersx←av∗kmodq1(iv)
The specific key of mediators is obtainable by the ordered pairA,x


## 3.3. Offline Signing

Any agent that performs the uniting is shown in the following steps:AssessesQi′←Υ2ivtmodq1, ∀i∈0,q1−1


## 3.4. Online Signing

In the recent step, to symbol a missiveM∈−∞,+∞, tiring A,x the agent stays as occurs subsequently:(i)
Selectl←Zq1∗ randomly such that li is the i-th bit of l(ii)
AssessQ←Πi=1p1Qi−1'(iii)
Assessb←ℏA,M,Q(iv)
Assessw←l∗bxmodq1; consequently, deliver η←Υwvtmodq1(v)
The signσ of any missive M assumed by: σ←A,Q,w


## 3.5. Verification

A confirmation of any signσ on the set of M with the identity of the agent id remains as receipts subsequently:(i)
Calculateη′←QΥbvAΥbvvumodq1(ii)
If we arrive at the stateη=η′, then in this situation, the sign is agreeable; otherwise, it is ostracized


## 3.6. Reliability of the Process

The private key needs to attain the effectiveness with fairness:(38)Υxvtmodq1=AΥvvumodq1.To prove the reliability of the procedure, we utilizeQ=Υlvtmodq1 to compute the following:(39)QΥbtvAmodq1Υbvvumodq1=ΥlvtΥkbvtΥabvvtmodq1=Υwvtmodq1.


## 4. Security Investigation and Discussion

To demonstrate the security of our new O/O–IBSS utilizing CCM, we apply the security proofs contributed in [54].Theorem 1.
The suggested IBSS isϵ,t,Qℏ,Qs,QE secure in the knowledge of unforgeability of IBSS based on the chosen message attack (UF-IBSS-CMA) in the ROM, implementing the (ϵ',t′)–CCM hypothesis in Zq1∗, where(40)ϵ′=q1−1/q1q1−QℏQE+Qs/q1Qℏϵ,t′=t+τOQs+QE.And,Qℏ-hashing queries, Qs-signing queries, and QE-extraction queries are the quantity of chaos. Here, τ is the period of a function of exponentiation.Suppose that∃ is a foe F. We develop an algorithm A depending on the utilization of F to solve CCM. The algorithm A is provided with a zq1∗ that comes with a variable D∈zq1∗ and a general parameter t. Algorithm A is tested to check β∈Zq1∗ in such a method that Υβvtmodw=D. We apply the same approach in [54].Setting AlgorithmA takes ℏ (chaotic hash function), which is similar to a ROM behavior. A is liable for the model of this reformation method. A assigns a variable u←D that yields the general argument p1,t,ℏ,u to F.Removal advice investigation:F is allowed to inquire for id in the extraction device. A recreates the oracle. It requires random c,d∈Zq1∗ and sets(41)Aν←Υdvt/Υcvumodq1,x←d,ℏAν,id←c.A yields Aν,t as a private key for id and stores the consistency evaluation Aν,x,ℏAν,id,id in the list. Note that, when ν⟶0, we have the case in [56].Indication oracle requests F make an inquiry for and sign a message. The algorithmA discovers whether id has been requested for the device ℏ or the extraction device in the past. If yes, it will only improve the list Aν,x,ℏAν,id,id. Then, algorithm A utilizes these estimates to indicate the missive by performing the passing procedures. It generates the signature Aν,Q,w on the message and maintains the list of Aν,Q,w for reliability in the chaotic hash construction. If id is not requested to extract the oracle, then A starts the removal advice simulation procedure by shattering the secretive key to symbolize the missive.Productivity calculation: eventually,F produces a fake sign σ1∗=Q∗,A∗,w1∗ on id∗ and M∗. The algorithm Av changes F to the view that it makes an inquiry ℏA∗,M∗,Q∗ and provides another value to the justified. Foe F produces a few other signatures σ2∗=Q∗,A∗,w2∗. Algorithm A rehashes again and obtains σ3∗=Q∗,A∗,w3∗. It is well known that Q∗ and A∗ must inevitably be likewise. We put n1,n2,n3 in order to be created three times in a row from the arbitrary advice investigations ℏQ∗,A∗,M∗.For eachk,a,l∈Zq1∗, we now project CCM of Aν,u, and Q separately, i.e.,(42)Aν=Υkvtmodq1,u=Υavtmodq1,and Q=Υlνtmodq1. From equation (18), we obtain the following facts:(43)wj∗=anj×ℏA∗,id×l×knj,modq1j=1,2,3.Onlyk,a, and l are unfamiliar with A in these mathematical examinations. For the estimates of the overhead linear autonomous mathematical proclamations, the algorithm Aν estimates for j=1,2,3 and generates a as the solution of the CCM.Ranked cost test: the simulation practicability with removal oracle failures presupposes that the consignmentℏAν,id of the random oracle is irregular, suggesting a combined probability of no less than Qℏ/q1 . Accordingly, the simulation procedure is effective (Qs+QE) times (ensued from the consideration that ℏA,id also can furthermore be requested in the sign advice, if id is not requested in the removal advice) with the probability as follows:(44)1−Qℏq1Qs+QE≥1−Qs+QEQℏq1.Because of the arbitrary advice’s ideal mediation, an inquiryℏA∗,M∗,Q∗ occurs with a probability of no less than 1−1/q1 Algorithm Av guesses that it is exactly because of the rewind, at least as a possibility of 1/Qℏ. The overall possibility of success is as follows:(45)q1−1q1q1−QℏQE+Qsq1Qℏϵ.The period density of procedureAv is determined by the exponentiation achieved in sign and removal procedures which is equivalent to t+τOQs+QE.


### 4.1. Discussion

In this position, we discuss the analysis and the performance of the CCM-based IBSS model by comparing it with some competing O/O identity-based sign (IBS) models and non-O/O models.


#### 4.1.1. Contrast with Other O/O-IBS Models

Here, we contrast the performance of six well-designed O/O identity-based signature (IBS) schemes including Shamir and Tauman’s scheme [11], Xu et al.’s model [22], Kar’s scheme [55], Gao et al.’s model [16], Meshram et al.’s 2016 model [53], Meshram et al.’s 2019 model [57], Meshram et al.’s 2020 model [58], and our new CCM based IBSS model in view of the ranked cost. We note that, among the models, Xu et al.’s work[22] has no multiperiod adaptation to it, so the multiperiod evaluation for it was carried out by linking the same type of technique together. However, it is not possible to apply Shamir and Tauman’s technique [11] for a multiperiod performance test.The ranked costCζ of operation ζ is estimated by the bits of ξ. Besides that, ρ,μ,m, and η, which stands for the pairing operation, the multiplication function (similar to point addition in ECC) in the group, the modular multiplication operation in Zq1∗, and exponentiation function (similar to scalar multiplication in ECC) in a group, respectively, are all included in the evaluations. Other operations such as addition in Zq1∗ and representative hashing are negligible and are therefore ignored.Table1 indicates the outcomes of performance evaluations in the form of rank cost. The recent method depends on Υν,ν=0.998. Therefore, when ν⟶0 yields κ0=0 (the case in [55]). C1: the ranked cost of offline (multistep); C2: the ranked cost of online (multistep); C3: the ranked cost of offline (single-step); C4: the ranked cost of online (single-step); C5: the ranked cost in confirmation step. The rows are the outcomes of [11, 16, 22, 55, 56] and the proposed method, respectively. For example, h is a Chameleon-hash function, which needs the minimum of one η computation; ℏ stands for a chaotic hash operation; σV and σG represent usual verification and signature creation, respectively, and each requires no less than one η computation. Similarly, CV is the operation of one record confirmation, which requires no less than one η computation.Table 1
Contrast in terms of the ranked cost.
C1C2C3C4C5——Ch+CσGmCσV+Ch+CCVη|q|−13m+μOq−1ηq−12mμ+ρ2η|q|m+2μOq2η+mmμ+2η+ρ0m+μON20m2η+μ0m+μOq10m2μμq−13m+μO|q|−1ηq−13m3μ+2ρ+η0m + µO(|.|)0mµO(|⋅ |)κ0νm+κ0ν+μOq1κ0νm+ κ0ν2μ+κ0ν


#### 4.1.2. Estimation with Other Methods

As with IBS models (O/O), we also contrast the planned model with some well-established IBS schemes (non-O/O) recognized by ISO/IEC including Cha and Cheon’s design [56], Guillou and Quisquater’s scheme [57], Hess’s formula [17], Meshram et al.’s system [55], and Meshram et al.’s system [58]. The full outcomes can be located in Table 2 with the same notations used as in Table 1. Notably, non-O/O models may not run very smoothly in wireless sensor networks since the lightweight wireless sensors probably will be overwhelmed by the operation demand. For example, both Hess’s design [17] and Cha and Cheon’s model [56] inevitably require operation ρ (pairing) in the verification phase and operation η in the signing phase, which amounts to too much of a burden for lightweight gadgets.Table 2
Contrast in virtue of the ranked cost and sign measure forν⟶1.
Cost techniquesCost in verification stageCost in signing stageSize of signature (bits)[17]η+2ρμ+3η320[55]μ+2ηm480[56]2μm480[56]μ+2η+ρ2η320[57]2μ+2η2m+η2048[58]μm480Proposed method2μ+κ0νm+κ0ν480


### 4.2. Vision on the Recent Method

This section deals with the following visions on the suggested method.In the present study, we have discussed the new generalization of Chebyshev polynomials using the fractional calculus that is called fractional conformable calculus. Also, we have discussed the necessary properties such as semigroup property and chaotic property of conformable fractional chaotic maps, which are very useful for designing of a new cryptography scheme. This paper introduces an online/offline ID-based short signature (IBSS) scheme that uses conformable fractional chaotic maps for secure communication. The presented scheme is secure under an existential enforce-ability of identity-based short signature (IBSS) under chosen message attack (EUF-IBSS-CMA) in the random oracle model (ROM). We have used less rigorous operations to carry out signing and verification procedures, similar to human signing on valid documents and then verifying them as per witness. The proposed online/offline ID-based short signature (IBSS) offers a better security assurance than currently established signature schemes.The use of conformable fractional chaotic maps increased the security of ID-based short signature under the probabilistic polynomial time and decreased the computational cost. The key advantage of the presented online/offline ID-based short signature (IBSS) scheme that uses conformable fractional chaotic maps is that at the verification stage and signing period, it takes less computation; it retains the degree of protection. Therefore, the presented scheme indicates less bandwidth for storage, communication, and computing resources, particularly applicable to wireless devices and smart cards.As a real-world application, wireless sensor networks (WSNs) have made rapid progress in recent years and have been widely applied by various works, such as healthcare centers, institutes of ecological and environmental research, and government and military organizations. In WSNs, the sensor nodes can collect their own raw data, process the data locally, and jointly send information to one or more collection points (base stations). As the data collected by the sensor nodes and transmitted through WSNs are most sensitive, confidential, or personal, secure information transmission is a critical challenge and one of the most significant security requirements is authentication. The online/offline ID-based short signature (IBSS) plays a key role in ensuring data integrity, authentication, and nonrepudiation.


## 4.1. Discussion

In this position, we discuss the analysis and the performance of the CCM-based IBSS model by comparing it with some competing O/O identity-based sign (IBS) models and non-O/O models.


### 4.1.1. Contrast with Other O/O-IBS Models

Here, we contrast the performance of six well-designed O/O identity-based signature (IBS) schemes including Shamir and Tauman’s scheme [11], Xu et al.’s model [22], Kar’s scheme [55], Gao et al.’s model [16], Meshram et al.’s 2016 model [53], Meshram et al.’s 2019 model [57], Meshram et al.’s 2020 model [58], and our new CCM based IBSS model in view of the ranked cost. We note that, among the models, Xu et al.’s work[22] has no multiperiod adaptation to it, so the multiperiod evaluation for it was carried out by linking the same type of technique together. However, it is not possible to apply Shamir and Tauman’s technique [11] for a multiperiod performance test.The ranked costCζ of operation ζ is estimated by the bits of ξ. Besides that, ρ,μ,m, and η, which stands for the pairing operation, the multiplication function (similar to point addition in ECC) in the group, the modular multiplication operation in Zq1∗, and exponentiation function (similar to scalar multiplication in ECC) in a group, respectively, are all included in the evaluations. Other operations such as addition in Zq1∗ and representative hashing are negligible and are therefore ignored.Table1 indicates the outcomes of performance evaluations in the form of rank cost. The recent method depends on Υν,ν=0.998. Therefore, when ν⟶0 yields κ0=0 (the case in [55]). C1: the ranked cost of offline (multistep); C2: the ranked cost of online (multistep); C3: the ranked cost of offline (single-step); C4: the ranked cost of online (single-step); C5: the ranked cost in confirmation step. The rows are the outcomes of [11, 16, 22, 55, 56] and the proposed method, respectively. For example, h is a Chameleon-hash function, which needs the minimum of one η computation; ℏ stands for a chaotic hash operation; σV and σG represent usual verification and signature creation, respectively, and each requires no less than one η computation. Similarly, CV is the operation of one record confirmation, which requires no less than one η computation.Table 1
Contrast in terms of the ranked cost.
C1C2C3C4C5——Ch+CσGmCσV+Ch+CCVη|q|−13m+μOq−1ηq−12mμ+ρ2η|q|m+2μOq2η+mmμ+2η+ρ0m+μON20m2η+μ0m+μOq10m2μμq−13m+μO|q|−1ηq−13m3μ+2ρ+η0m + µO(|.|)0mµO(|⋅ |)κ0νm+κ0ν+μOq1κ0νm+ κ0ν2μ+κ0ν


### 4.1.2. Estimation with Other Methods

As with IBS models (O/O), we also contrast the planned model with some well-established IBS schemes (non-O/O) recognized by ISO/IEC including Cha and Cheon’s design [56], Guillou and Quisquater’s scheme [57], Hess’s formula [17], Meshram et al.’s system [55], and Meshram et al.’s system [58]. The full outcomes can be located in Table 2 with the same notations used as in Table 1. Notably, non-O/O models may not run very smoothly in wireless sensor networks since the lightweight wireless sensors probably will be overwhelmed by the operation demand. For example, both Hess’s design [17] and Cha and Cheon’s model [56] inevitably require operation ρ (pairing) in the verification phase and operation η in the signing phase, which amounts to too much of a burden for lightweight gadgets.Table 2
Contrast in virtue of the ranked cost and sign measure forν⟶1.
Cost techniquesCost in verification stageCost in signing stageSize of signature (bits)[17]η+2ρμ+3η320[55]μ+2ηm480[56]2μm480[56]μ+2η+ρ2η320[57]2μ+2η2m+η2048[58]μm480Proposed method2μ+κ0νm+κ0ν480


## 4.1.1. Contrast with Other O/O-IBS Models

Here, we contrast the performance of six well-designed O/O identity-based signature (IBS) schemes including Shamir and Tauman’s scheme [11], Xu et al.’s model [22], Kar’s scheme [55], Gao et al.’s model [16], Meshram et al.’s 2016 model [53], Meshram et al.’s 2019 model [57], Meshram et al.’s 2020 model [58], and our new CCM based IBSS model in view of the ranked cost. We note that, among the models, Xu et al.’s work[22] has no multiperiod adaptation to it, so the multiperiod evaluation for it was carried out by linking the same type of technique together. However, it is not possible to apply Shamir and Tauman’s technique [11] for a multiperiod performance test.The ranked costCζ of operation ζ is estimated by the bits of ξ. Besides that, ρ,μ,m, and η, which stands for the pairing operation, the multiplication function (similar to point addition in ECC) in the group, the modular multiplication operation in Zq1∗, and exponentiation function (similar to scalar multiplication in ECC) in a group, respectively, are all included in the evaluations. Other operations such as addition in Zq1∗ and representative hashing are negligible and are therefore ignored.Table1 indicates the outcomes of performance evaluations in the form of rank cost. The recent method depends on Υν,ν=0.998. Therefore, when ν⟶0 yields κ0=0 (the case in [55]). C1: the ranked cost of offline (multistep); C2: the ranked cost of online (multistep); C3: the ranked cost of offline (single-step); C4: the ranked cost of online (single-step); C5: the ranked cost in confirmation step. The rows are the outcomes of [11, 16, 22, 55, 56] and the proposed method, respectively. For example, h is a Chameleon-hash function, which needs the minimum of one η computation; ℏ stands for a chaotic hash operation; σV and σG represent usual verification and signature creation, respectively, and each requires no less than one η computation. Similarly, CV is the operation of one record confirmation, which requires no less than one η computation.Table 1
Contrast in terms of the ranked cost.
C1C2C3C4C5——Ch+CσGmCσV+Ch+CCVη|q|−13m+μOq−1ηq−12mμ+ρ2η|q|m+2μOq2η+mmμ+2η+ρ0m+μON20m2η+μ0m+μOq10m2μμq−13m+μO|q|−1ηq−13m3μ+2ρ+η0m + µO(|.|)0mµO(|⋅ |)κ0νm+κ0ν+μOq1κ0νm+ κ0ν2μ+κ0ν


## 4.1.2. Estimation with Other Methods

As with IBS models (O/O), we also contrast the planned model with some well-established IBS schemes (non-O/O) recognized by ISO/IEC including Cha and Cheon’s design [56], Guillou and Quisquater’s scheme [57], Hess’s formula [17], Meshram et al.’s system [55], and Meshram et al.’s system [58]. The full outcomes can be located in Table 2 with the same notations used as in Table 1. Notably, non-O/O models may not run very smoothly in wireless sensor networks since the lightweight wireless sensors probably will be overwhelmed by the operation demand. For example, both Hess’s design [17] and Cha and Cheon’s model [56] inevitably require operation ρ (pairing) in the verification phase and operation η in the signing phase, which amounts to too much of a burden for lightweight gadgets.Table 2
Contrast in virtue of the ranked cost and sign measure forν⟶1.
Cost techniquesCost in verification stageCost in signing stageSize of signature (bits)[17]η+2ρμ+3η320[55]μ+2ηm480[56]2μm480[56]μ+2η+ρ2η320[57]2μ+2η2m+η2048[58]μm480Proposed method2μ+κ0νm+κ0ν480


## 4.2. Vision on the Recent Method

This section deals with the following visions on the suggested method.In the present study, we have discussed the new generalization of Chebyshev polynomials using the fractional calculus that is called fractional conformable calculus. Also, we have discussed the necessary properties such as semigroup property and chaotic property of conformable fractional chaotic maps, which are very useful for designing of a new cryptography scheme. This paper introduces an online/offline ID-based short signature (IBSS) scheme that uses conformable fractional chaotic maps for secure communication. The presented scheme is secure under an existential enforce-ability of identity-based short signature (IBSS) under chosen message attack (EUF-IBSS-CMA) in the random oracle model (ROM). We have used less rigorous operations to carry out signing and verification procedures, similar to human signing on valid documents and then verifying them as per witness. The proposed online/offline ID-based short signature (IBSS) offers a better security assurance than currently established signature schemes.The use of conformable fractional chaotic maps increased the security of ID-based short signature under the probabilistic polynomial time and decreased the computational cost. The key advantage of the presented online/offline ID-based short signature (IBSS) scheme that uses conformable fractional chaotic maps is that at the verification stage and signing period, it takes less computation; it retains the degree of protection. Therefore, the presented scheme indicates less bandwidth for storage, communication, and computing resources, particularly applicable to wireless devices and smart cards.As a real-world application, wireless sensor networks (WSNs) have made rapid progress in recent years and have been widely applied by various works, such as healthcare centers, institutes of ecological and environmental research, and government and military organizations. In WSNs, the sensor nodes can collect their own raw data, process the data locally, and jointly send information to one or more collection points (base stations). As the data collected by the sensor nodes and transmitted through WSNs are most sensitive, confidential, or personal, secure information transmission is a critical challenge and one of the most significant security requirements is authentication. The online/offline ID-based short signature (IBSS) plays a key role in ensuring data integrity, authentication, and nonrepudiation.


## 5. Conclusion

Here, we established an effective CCM based on the IBSS method. Our new design does not involve a record devoted to the sign for confirmation, and there is no pairing operation involved either in the signature obstetrics phase or in the verification phase. It is secure in the ROM with unfeasibility based on EUF-IBSS-CMA-secure. Our new scheme provides multistep use of offline storage, enabling the agent to reuse the data set in a polynomial format contrasting to the single-attempt inconvenience in most other processes. In our new method, a preregistration procedure can be carried out with a private key and then no private key is needed in the offline phase. In such a design, we only need the least operations in each procedure. This structure does not request any type of record, registration, or verification related to the sign for assertion and does not demand any combination in both stage’s confirmation and signature group. Our presented disconnected signing method does not request all kinds of secluded key data. It can -record by an isolated key group with offline data equally be utilized. It correspondingly consumes the very slightest operation in every procedure. This is an astonishing promising model in wireless sensor network circumstances as the disengaged data may be complex-inserted into the sensor hub in the accumulating or arrangement position. This is a notably desirable feature for wireless sensor network applications; for this way, the offline information in the setup or configuration stage can be complex-inserted into the device hub. Our performance analysis reveals that the proposed method has the lowest ranked cost among competing methods. In future work, we will develop a new efficient online/offline ID-based aggregation short signature scheme using conformable fractional chaotic maps under fuzzy user data sharing for wireless sensor networks by using the presented scheme.



---

*Source: 6302328-2022-03-31.xml*