1. Introduction
In recent years, remote access technology and device control have become an imperative requirement. Particularly, this is evident nowadays due to the increased spread of the coronavirus pandemic (COVID-19), which necessitated the imposition of some restrictions, including mandatory social distancing in many countries. Therefore, it became clear how developing remote access communication technology is of immense importance depending on the Internet service, especially research in improving the technology of the Internet of things (IoT). The IoT is the technology to assemble devices that need to be monitored, linked, and interacted [
1]. IoT is associated with great prospects of physical objects with the cyber world, such as healthcare devices, intelligent transportation systems, home appliances, sensors, and environmental monitoring [
2]. Connected devices to the IoT are exponentially increasing [
3], which add more security challenges that must be taken into consideration [
4]. Cryptosystems based on asymmetric keys play vital roles in the security of diverse communication systems. Cryptanalysis techniques motivate researchers to develop novel signature schemes to dominate the growth in security attacks [
5]. The financial field of Bitcoin has become one of the most required research areas for security from cyber-attacks. The blockchain concept succeeded in achieving that, as it provides reliable and secure decentralized solutions [
6]. The elliptic curve cryptography (ECC)-based digital signature algorithm (DSA) is used for data signature and verification in wireless devices. Identity-based authentication and access control in wireless network devices help to protect from illegitimate access and preserve the security issues of the wireless nodes [
7]. DSA is a robust tool in data authentication and privacy. Since the emergence of public key cryptography in 1970, many schemes have been developed, such as the efficient ECC technique [
8]. The cyclic group of order
p was also developed
, which is isomorphic to the additive group of ((
Z/pZ)
∗,.), where
Z is the set of all integers, and
p is a prime number.
The ECC algorithm has demonstrated a considerable effectiveness on public key cryptography [
9]; as a result, an efficient digital signature approach was proposed in [
10]. The strength of the scheme in [
10] is its dependence on the discrete logarithm problem (DLP). On the contrary, traditional schemes are challenged by more complicated and effective attacks. This paved the way for the robust security schemes previously represented, while the era of the traditional techniques is expired [
11].
To ensure a strong and efficient cryptosystem, it is necessary to achieve Shannon properties, where the permutation process is an important operation. Logistic mapping and cyclic groups are very important steps to ensure the randomness of performance. However, for a robust algorithm that can withstand different attacks, it is important to achieve confusion and diffusion properties [
12]. Another goal, in addition to robustness, is to minimize execution time to be applicable in real-time applications. Logistic mapping and cyclic groups are used to generate S-Box, which is important to guarantee the cryptographic strength, such as nonlinearity, bijection, strict avalanche criterion, output bits independence criterion (BIC), and equiprobable input/output XOR distribution [
13].
This work proposes a novel signature scheme based on the integration of the ECC algorithm with the Ong–Schnorr–Shamir (OSS) scheme. The robustness of the proposed approach relies on using a reversible key matrix of 4×4 as a portion of the OSS signature equation with decoding modification on ECC algorithm. This consolidation increases the degree of complexity and thus increases the confidentiality of the data. To prove the novelty and credibility of the presented technique, the new scheme was tested and demonstrated robustness against other approaches.
The rest of this manuscript is sequenced as follows.
Section 2 introduces the previous work related to the proposed pipeline.
Section 3 details the proposed methodology and the employed schemes (i.e., the ECC and the OSS signature schemes).
Section 4 outlines the details of the proposed OSS–ECC digital signature technique. Robustness measures of the proposed scheme are demonstrated and discussed in
Section 5, and conclusions are given is
Section 6.
5. Experimental Results and Discussion
The quantitative/qualitative performance evaluation of both the traditional and proposed technique could be measured using various parameters, including (i) timing analysis, (ii) security analysis, and (iii) robustness to security attacks. In particular, timing analysis is among the critical points in any digital signature’s schemes, namely, the encryption speed plays a vital role in real-time applications. The longer the encryption/decryption process is, the less suitable the method for some applications, such as video conferencing and live streaming. Thus, a comparison of the execution time of the proposed scheme and different digital signature schemes (signature generation) was conducted.
Figure 8 summarizes the comparison with the approach proposed by Al-Sewadi et al. [
36], where their authentication algorithm based on NIST-DSA is performed. As readily seen in the figure, the resulting data highlight the high speed of applying the ECC with the OSS scheme with a key length of 1024. However, the proposed method is slightly faster than OSS–ECC as an additional level of S-Box is added to the total processing time. It is worth mentioning that for the RSA algorithm to achieve the same level of security as the ECC technique, it basically requires an increased key length. Although RSA introduces simple computations, the ECC depends on DLP with a lesser key length [
37].
Table 2 summarizes the comparative results between RSA, OSS–ECC, and the modified OSS–ECC in signature computation time with respect to key length, which achieved the same security level for all compared schemes. Moreover,
Table 2 shows the signature time between RSA, OSS–ECC, and modified OSS–ECC. It is clear that the proposed modification cost is slightly higher than OSS–ECC, as an additional step of S-box and reversible matrix is added to the traditional OSS–ECC scheme.
Table 3 summarizes the performance comparison between RSA, EC–DSA, OSS–ECC, and the proposed digital signature. As can be seen, the OSS–ECC and the modified OSS–ECC have the best performance.
In addition to the timing analysis, the strength of the proposed S-Box was measured using two strong cryptanalytic attacks: differential and linear attacks. Typically, differential cryptanalysis aims to detect the “difference” between related encrypted plaintexts. The plaintexts may vary by a few bits. It attacks depending on a chosen plaintext: the attacker chooses the plaintext to be encrypted without the key, and then encrypts the related plaintexts [
38]. The difference distribution tables of cyclic group sub-bytes were constructed, a worst case assumption was made in our consideration, which has a probability of 22/256 with input data difference
= {0, 1, 1, 1, 1, 0, 0, 0}, as illustrated in
Table 4, except the cases of a one bit input/output difference, which are considered to be impossible (probability is zero). On the other hand, linear cryptanalysis exploits the high probability occurrences in bits of plaintext, ciphertext, and sub-key [
39]. Such an expression takes the form:
where
x represents the input
x = [
x1,
x2…] and y represents corresponding output y = [y
1, y
2…]. Equation (18) represents the exclusive-OR “sum” of
u input bits and
v output bits [
40]. If the scheme shows a tendency to hold with high probability or not for Equation (18), this illustrates failure in randomization abilities [
41,
42]. The main factor that assesses the efficiency of the scheme is the linear probability bias, which is the amount of probability of a linear expression deviating from
. The higher the magnitude of the probability bias, |
P
L|, the worse the security is [
36]. In linear cryptanalysis, the relations between two bits of cyclic group sub-bytes can be found; the probability (
P) of these relations is restricted by
as illustrated in Algorithm 1.
Finally, the performance of the proposed technique can be evaluated by its robustness to security attacks, such as (i) brute-force, (ii) password sniffing, (iii) man-in-the-middle, and (iv) replay attacks. The former is the most popular public key attack, which tries to derive the private key from the know public key [
43]. A given system is said to be secured against this type of attack if its key length is ≥70 bits with a probability of 2
70 [
44]. In the proposed scheme, the key length is considered to be the summation of the ECC key length with a probability of 2
163 bits in addition to 2
7 generators of the proposed S-Box. Thus, the robustness of the proposed scheme implies that if a third party has the main parameters of EC function, {
a,
b,
p,
G} and {
PA,
k}, which are the public keys, it is not possible to estimate the signature {x, y} using trial and error with a probability of 2
(163+7) bits. Second, the password sniffing attack eavesdrops the network to intercept keys or passwords by capturing passing data. Attackers analyze data to predict keys. Encryption algorithms are the best way to be resilient against sniffing attack. The proposed scheme basically depends on the ECC, which works as a firewall against that types of attack.
Third, Pollard and Schnorr [
18] succeeded in forging the signature without solving the quadratic equation. Their scheme approves its strength, as the signatures
x and
y are a 4
× 4 matrix with a large key length, which makes it hard to estimate. If a third party tried to estimate the second part of signature
y where the attacker intercepts and/or modifies the data in transit, a constant value of
x could be assumed. Here is a quadratic equation with a complex term. It is hard to calculate the square root to obtain
y, while estimating
x and fixing the second part
y equals factoring
n. Such quadratic equation is difficult to solve, which can be explained in the following algorithm (Algorithm 1):
Algorithm 1 Steps for calculating quadratic equation roots |
1. Given n = P × Q, where {P, Q} are unknown prime numbers; |
2. Choose w and Compute w−1(mod n); |
3. Scheme A: requires signatures of random messages (m) to run, and m must be signed using the private key w−1; |
4. Recall Scheme (A) using the signatures and the public keys {k, n}; |
5. w′ is computed by scheme (A) as follows: |
6. With a probability of, the are the two prime numbers {P, Q}; |
7. According to the previous steps, if, then choose another (w) and reiterate all steps; |
8. After (n) rounds, the possibility of computing the factorization is. |
Last, in replay attacks, the attackers try to intercept and record the plaintext. The captured data are used another time to try and recreate authentication. The hybrid ECC is used in the declared scheme, as the main parameters a, b, p, and G are generated randomly for each iteration, i.e., it has completely different signatures in each round. Therefore, the modified ECC–OSS approved its strength against this type of attack.