Hash-enhanced elliptic curve bit-string generator for medical image encryption

Abstract

Bit-string generator (BSG) is based on the hardness of known number theoretical problems, such as the discrete logarithm problem with the elliptic curve (ECDLP). Such type of generators will have good randomness and unpredictability properties as it is challenged to find a solution regarding this mathematical dilemma. Hash functions in turn play a remarkable role in many cryptographic tasks to accomplish different security levels. Hash-enhanced elliptic curve bit-string generator (HEECBSG) mechanism is proposed in this study based on the ECDLP and secure hash function. The cryptographic hash function is used to achieve integrity and security of the obtained bit-strings for highly sensitive plain data. The main contribution of the proposed HEECBSG is transforming the x-coordinate of the elliptic curve points using a hash function H to generate bit-strings of any desirable length. The obtained pseudo-random bits are tested by the NIST test suite to analyze and verify its statistical and randomness properties. The resulted bit-string is utilized here for encrypting various medical images of the vital organs, i.e. the brain, bone, fetuses, and lungs. Then, extensive evaluation metrics have been applied to analyze the successful performance of the cipherimage, including key-space analysis, histogram analysis, correlation analysis, entropy analysis and sensitivity analysis. The results demonstrated that our proposed HEECBSG mechanism is feasible for achieving security and privacy purposes of the medical image transmission over unsecure communication networks.

Keywords

Elliptic curve cryptography hash function bit-string generator medical image encryption security analysis

1 Introduction

A pseudo-random bit generator (PRBG) is an algorithm for producing a string of random bits to represent almost similar properties of bit-strings produced from a truly random bit generator. The obtained bit series from the PRBG is estimated by a comparative set of known initial values, called the PRBG’s seed [1]. Although PRBG strings are closed to truly random sequences, they could be generated by using hardware random bit-string generators. Therefore, PRBGs are implemented in real world applications for speeding-up the generated bits, and also for the reproducibility. PRBGs are widely used in many recent applications such as electronic games, numerical simulations, statistical research, randomized algorithms, cryptography, and lottery.

In 1985, elliptic curve cryptography (ECC) were introduced by Neal Koblitz [2] and Victor Miller [3]. By substituting the subgroup of the multiplicative group $ℤ_{p}^{*}$ with the group of points on an elliptic curve (EC) over a finite field $𝔽$ , these cryptosystems could be considered as EC-analogues of the traditional discrete logarithm ones. The security beyond these EC-cryptosystems is proven by the computational intractability support of the elliptic curve discrete logarithm problem. Since this problem tends to be substantially more difficult than the ordinary discrete logarithm problem, the strength-for-key-bit is essentially greater in elliptic curve-based systems than in classical discrete logarithm-based systems. Parameters with small sizes are therefore used for ECC rather than for discrete logarithm systems with nearly equal security levels [4]. Many advantages could be acquired from the use of such small parameters include smaller key sizes, fast computations, and secured data transmission. Such features are important in environments where storage space, power of processing, and/or power consumption are bounded. EC-based pseudo-random bit generator logically can be represented in two successive steps as follow. First step generates a sequence of EC-points and the second step to extract a bit-string from the point sequence as a source of pseudo-random bits of the uniform distribution [5].

Security schemes of medical imaging are needed to achieve high levels of protection to different attack modes without compromising the diagnostic quality of these images [6]. Changes made to the images during processing may lead to irreversible false diagnostic consequences [7]. Therefore, the algorithms of medical image encryption are not always suitable for certain types of images because of some intrinsic features including high redundancy, large data capacity and high neighboring pixels correlations [8]. Accordingly, EC-based encryption schemes have recently been widely suggested as presented in [9 –11].

In this study, we propose a hash-enhanced elliptic curve bit-string generator (HEECBSG) method for generating strings of binary bits of any length quickly. The proposed HEECBSG is based on an elliptic curve points operations over finite fields ( $𝔽_{p}$ ), which is secured and integrated by a hash function. The statistical properties of obtained bit-strings can be significantly improved by applying our secured mechanism. Moreover, the HEECBSG mechanism is more suitable in producing pseudo-random bits for the encryption of medical images to accomplish safe transmission of patients’ data in unsecured networks.

The remainder of this article is divided into the following sections. The preliminaries of EC and cryptographic hash function are introduced in Section 2. In Section 3, the related work is presented. In Section 4, the HEECBSG mechanism is proposed. Experimental and NIST test results are given in Section 5. In Section 6, an encryption of medical images with various security analysis is discussed and finally conclusions are given in Section 7.

2 Preliminaries

This section presents a detailed description of EC over $𝔽_{p}$ and the main operations of EC-points. In addition, main characteristics of cryptographic hash function are introduced to give the required background of our proposed security mechanism.

2.1 Elliptic curves over $𝔽_{p}$

Consider an elliptic curve; namely E; over finite prime field $𝔽_{p}$ , for p prime satisfy that p > 3. E can be defined by the affine Weierstrass equation of the following form: $E : y^{2} = x^{3} + ax + b,$ (1) where coefficients a and b belong to the field $𝔽_{p}$ such that the discriminant satisfy the equation 4a³ + 27b² ≠ 0. The $𝔽_{p}$ is a set of all the points $(x, y), x \in 𝔽_{p}, y \in 𝔽_{p}$ , which fulfill the defining Eq.(1), together with a special point O called the point at infinity [2].

2.2 EC-point operations

The addition operation of two elliptic curve points P and Q results a third point R on the same curve, using the chord-and-tangent rule. With this addition operation, the set of all points of $E (𝔽_{p})$ forms a group with O serving as its identity. The formed group is used in the EC-cryptosystem structure. Consider that P = (x₁, y₁) and Q = (x₂, y₂) are two featured points on an elliptic curve E. The sum of these points P and Q, computed as P + Q = R = (x₃, y₃), is calculated as following. A line is drawn through P and Q points; this line intersects the EC itself in a third R point as depicted in Figure 1. The R point is considered as the reflection of that operation on the x-axis [12].

Fig.1

Description of two EC-points addition: P + Q = R.

Fig.2

Description of one EC-point doubling: P + P = 2P = R.

Consider the point P = (x₁, y₁), doubling of P point which is calculated via 2P = R = (x₃, y₃), is obtained as following. The tangent line with EC at point P is drawn firstly. The mentioned line is intersected with the EC itself in another point. As a result, the R point is considered as the reflection of this operation on the x-axis as shown in Figure 2. Sum of two points operation and also doubling of one point is deduced from algebraic description as the following:

Note that P + O = O + P all of $P \in E (𝔽_{p})$ .

For point $P = (x, y) \in E (𝔽_{p})$ , so that (x, y) + (x, - y) = O, the point (x, - y) is indicate as -P which is represented the negative of P; notice that -P is in fact a point on the curve E itself.

Addition of two points P = (x₁, y₁) and Q = (x₂, y₂) both belong to $E (𝔽_{p})$ and satisfy that P ≠ ± Q. The sum is P + Q = R = (x₃, y₃), where $\begin{matrix} x_{3} & = {(\frac{y_{2} - y_{1}}{x_{2} - x_{1}})}^{2} - x_{1} - x_{2} \\ and \\ y_{3} & = (\frac{y_{2} - y_{1}}{x_{2} - x_{1}}) (x_{1} - x_{3}) - y_{1} \end{matrix} .$ (2)

Doubling of point $P = (x_{1}, y_{1}) \in E (𝔽_{p})$ , with P ≠ - P. The operation P + P = 2P = (x₃, y₃), where $\begin{matrix} x_{3} & = {(\frac{3 x_{1}^{2} + a}{2 y_{1}})}^{2} - 2 x_{1} \\ and \\ y_{3} & = (\frac{3 x_{1}^{2} + a}{2 y_{1}}) (x_{1} - x_{3}) - y_{1} \end{matrix} .$ (3)

EC-operation of the two points P and Q in $E (𝔽_{p})$ needs combining of other arithmetic operations such as addition, subtraction, multiplication and inversion in the $𝔽_{p}$ field.

2.3 Hash function

Recently, pseudo-random bit generators can be established based on a non-invertible or one-way hash function. To achieve different security levels, a suitable hash function is utilized, and sufficient entropy is gained for the seed value. In general, for any hash function H with X input data, the hash value h of X is defined by: $h_{i} = H (X_{i}) i = 1, 2, . . .$ (4)

Calculation of H (X) is considered computationally fast operation and not possible to reverse the obtained hash value h, i.e. the hash function acts as one-way function [13]. The proposed HEECBSG method is flexible and designed to allow the use of any suitable secure hash function, as defined in Eq.(4).

3 Related work

EC-based generators have been widely applied in the authentication and encryption operations of medical images. Mustapha et al. [9] showed a comparative analysis between chaos and ECC-based encryption schemes. The comparison confirmed that both techniques have good security features. Yin et al. [10] analyzed the traditional ECC districts and modified them by combining the homomorphic encryption for the application of medical images. Their experimental results showed that the key space of the modified algorithm is improved with better encryption effect and higher key sensitivity. Singh et al. [11] achieved a new finding in the ElGamal encryption scheme, such that a separate computation is removed for encoding plain message to elliptic curve coordinate. The enhanced algorithm is designed to encrypt medical images where the problem of data expansion is solved, and the execution time is decreased. A hybrid, multi-layered EMOTE encryption system for managing medical images has been also suggested based on binary curves [14]. It is a stable machine-friendly binary system. Password authentication accompanied by multimodal biometric fusion using finger vein and finger knuckle to ensure the user security to access the database [15]. Additionally, authentication is reinforced by encrypting the fused image. Jinasena et al. [16] proposed a technique for implementing a dynamic and flexible access control mechanism to ensure the access rights of sensitive medical data in a collaborative mobile-based clinical discussion. A secure electronic medical record (EMR) service system, named the ECC-based secure EMR system is proposed by Tsai et al. in [17]. The system employs a smart card, a cloud database, an ECC integration unit, and portable devices to support users with a safe environment for EMR transmission. A secure scheme is designed based on Rives–Shami–Adleman (RSA) encryption and Shamir’s secret sharing schemes [18]. It can effectively ensure the data confidentiality and check the integrity of data. A new mathematical scheme is proposed in [19] to encrypt and decrypt grayscale and colored images. It combined the utilization of RSA algorithm elements and projective transformations. Two encryption algorithms are used item-by-element and two-elements. Reasonable results had been obtained in both cases. An implementation for encryption/decryption based on the RSA cryptosystem is tested on images in the medical field [20]. However, the RSA protocol software remains slow because the medical images are large, and the key sizes are within the range of (1024 - 2048) bits and tend to increase in the future. The proposed scheme in [21] presented the encryption and authentication of well-known format of medical images; called digital imaging and communications in medicine (DICOM) using RSA and advanced encryption standard (AES) algorithms. It showed that AES algorithm has more security capabilities than RSA algorithm, but the AES algorithm requires secure cipher key transmission. A new encryption-decryption approach for grayscale and color images was proposed using a simple RSA algorithm and additional use of binary bitwise operations [22].

Fig.3

HEECBSG generator mechanism.

Fig.4

HEECBSG backtracking resistance.

4 The proposed HEECBSG mechanism

The ECDLP hardness based on a finite field is a major key to the security of all cryptographic elliptic curve schemes [23]. Therefore, the HEECBSG scheme is based mainly on the hardness of ECDLP. The problem assumed that for a given two points P and Q on an EC of order n, it is considered intractable to get a value such that Q = aP. The main steps of proposed HEECBSG are depicted in Figure 3.

The instantiation stage of the HEECBSG mechanism involves selecting a suitable EC and a secret point P on that curve for the desired level of security. The seed value is used to locate the initial value (s₀) of the HEECBSG, including enough bits of entropy with adequate security length. In the initial state, the value of t is accounted for the seedlen-bit number, so let’s consider that t = s₀ in that case. The HEECBSG offers security level according to the security strength of the desired used curve. The main key point for using the H hash function is to ensure that the entropy is distributed in the extracted bits if it is verifiably random. Backtracking resistance in this mechanism is deep-seated, even in the case that the internal state is vulnerable to exposure as shown in Figure 4.

The HEECBSG generates a seed value for each step as represented by Eqs.(1) and (2).

$s_{i} = φ (x ([s_{i - 1}]) P), i = 1, 2, . . .,$ (5)

$h_{i} = φ (H (s_{i})), i = 1, 2, . . .,$ (6)

where $s_{0} \in E (𝔽_{p})$ is the “initial value". The HEECBSG method represents an EC scalar multiplication in which the x-coordinate is extracted from the resulting points s_i. Then, the random hash output h_i is calculated with truncation operation to obtain the pseudo-random bit-string. Normal operation is to follow a line in the same direction of the arrow. Inverting direction reveals the ability to solve the ECDLP for that nominated curve. In Figure 4, the ability of an opponent to invert the arrow means the opponent has solved the ECDLP for that used elliptic curve. Backtracking resistance is established with the mechanism design. Therefore, knowledge of s₁ does not allow an adversary to determine s₀ (and so forth) unless the adversary is able to solve the ECDLP for that specific curve. Additionally, knowledge of h₁ does not allow the determining of s₁ (and so on) by an adversary unless that adversary has the ability to solve the ECDLP for that used curve.

The HEECBSG generates pseudo-random bit-string through extracting bits from obtained hash values h. The internal state of the HEECBSG is a secret value s₀ that represents the x-coordinate of a secret point P on the used EC. Output bits are produced by first computing s to be the x-coordinate of the point multiplication operation [s] P, and then extracting low order bits from the hashcode output h of the computed values of H (s).

5 Experimental setup

The implementation of the HEECBSG mechanism should include an approved curve. Once the designer chooses the security level required by a given application, he can then start the implementation of an EC that most NIST SP 800-90A [1] appropriately meets this requirement.

5.1 Implementation example

The HEECBSG algorithm allows an exhaustion application to instantiate using a prime curve. In accordance security key strengths of 112, 128, 192 and 256-bits may then be required. In this experiment, the implementation process used the following EC equation: $E : y^{2} = x^{3} + 4 x + 1$ (7) where the selected E parameters values a = 4, b = 1 over $𝔽_{p}$ with prime p = 503. So, the cardinality of E is computed as $N = # E (𝔽_{503}) = 516$ . Also choosing generator point G = (283, 315) of order ℓ=129 for the considered HEECBSG mechanism. The secure hash algorithm (SHA)-256 is chosen as the hash function. SHA-256 function generates an almost-unique, fixed size 256-bit hash and considered secure in most real-world-crypto applications.

Table 1

Test results for 1048576 bit-strings

Test-name	P-value	Result
Block Frequency (m = 100)	0.046169	Succeed
Frequency	0.681211	Succeed
Cusum (Forward)	0.878529	Succeed
Cusum (Reverse)	0.674391	Succeed
Long Runs of Ones	0.128851	Succeed
Spectral DFT	0.149590	Succeed
Rank	0.638151	Succeed
Lempel Ziv Complexity	1.000000	Succeed
Overlapping Templates (m = 9)	0.120402	Succeed
NonOverlapping Templates (m = 9)	0.197506	Succeed
Approximate Entropy (m = 10)	0.681211	Succeed
Universal (L = 7, Q = 1280)	0.051599	Succeed
Random Excursions (x = +1)	0.297235	Succeed
Serial (m = 16)	0.343750	Succeed
Random Excursions Variant (x = +1)	0.050388	Succeed
Runs	0.499889	Succeed
Linear Complexity (M = 500)	0.703017	Succeed

Fig.5

HEECBSG based image encryption schema.

5.2 NIST randomness tests

The NIST [24] test suite is a statistical package that consisting of up to 15 tests. It is developed for testing the randomness of bit-strings obtained by either hardware or software based cryptographic pseudo-random and random bit generators. The 15 tests focused on a variety of different non-randomness types that could exist in bit-strings. The proposed mechanism produces a very random bit-strings as reflected by the high p-values of 1048576 bit-string length as shown in Table1.

6 Encryption and security analysis

The security of digital images need pseudo-random bit-string that have pretty good randomness properties and also high periodicity. Recently, several EC-based works for classical and medical images encryption have been presented in the literature such as [14 , 26]. In this study, the obtained bit-string from the HEECBSG method is used as a key-stream for the encryption of five 256 × 256 grayscale medical images which are bone, brain, echo, front and top test images. For each pixel has a 8-bit value of between 0 and 255 of that images and each plainimage block is divided into 128-bit sub-blocks. Each sub-block is scrambled by bit-permutation operation to hide image pixels information with the encryption algorithm. The pseudo-random bit-string in turn divided into sub-blocks of 128-bit each. Next, bitwise XOR operation is carried on every bit of the 128-bit sub-blocks. The resulted bit blocks then grouped together to obtain the cipherimage, as shown in Figure 5. The decryption process is done vice-versa and the following security analysis is carried out.

Fig.6

Histogram of different medical images and the corresponding cipherimages with the HEECBSG mechanism.

6.1 Key space security

The key space includes different keys that can participate in the encryption process. The key space is usually expected to be greater than 2¹⁰⁰ as mentioned in [27]. The initial values of the HEECBSG proposed method are obtained by choosing a proved curve with a prime p of length at least 192-bit. In case of using hash function SHA-256 for integrity, the key space is equals to 2¹⁹² × 2²⁵⁶ = 2⁴⁴⁸. In this case, the total key space is greater than 2¹⁰⁰, which is sufficient to withstand exhaustive attack.

6.2 Histogram analysis

The histograms for the five plain medical images and their corresponding cipherimages are estimated. All plain medical images histograms contain large spikes while the histogram of their corresponding cipherimages is almost flat and uniform as depicted in Figure 6. It denotes equal probability of occurrence of each pixel. Histogram of the cipherimages is remarkably different from the respective plainimages. Therefore, no evidence to classify established statistical attacks on the image encryption process is provided [28].

6.3 Entropy analysis

The entropy H (m) of a message source m is calculated from the equation: $H (m) = - \sum_{k = 0}^{255} P (m_{k}) \log_{2} P (m_{k})$ (8)

where P (m_k) represents the probability of message m_k [28]. The various entropy values for the five medical plainimages and the encrypted images (see Figure 6) are illustrated in Table 2. The entropy of the encrypted images is very closed to the theoretical value of 8. Obviously, all pixels in the encrypted five images occur with approximately equal probability. Therefore, the proposed HEECBSG is secure against the entropy-based attack and the information leakage is negligible.

6.4 PSNR and SSIM

Peak signal-to-noise ratio (PSNR) is mainly used in image processing area as a consistent image quality metric [29] and the greater PSNR, the better the output image quality as given by Eq.(9). $PSNR = 10 \log_{10} (\frac{R^{2}}{MSE})$ (9) where the mean-squared error (MSE) is calculated using the following equation: $MSE = \frac{\sum_{M, N} [I_{1} (m, n) - I_{2} (m, n)]^{2}}{M * N}$ (10) where M and N are the number of rows and columns in the input images I₁ and I₂.

Table 2

Entropy of five test medical images and their cipherimages

Test image	Plainimage	Cipherimage
Bone	6.16874	7.99752
Brain	4.65599	7.99714
Echo	3.04892	7.99741
Front	7.47160	7.99722
Top	6.97803	7.99728

Table 3

Evaluation metrics test results

Scheme	PSNR	SSIM	MAE
Bone	32.29	0.00545	97.84
Brain	30.89	0.00513	104.41
Echo	36.09	0.00188	117.39
Front	27.45	0.01085	72.65
Top	26.74	0.00918	88.57

Table 4

NPCR and UACI test results

Scheme	NPCR	UACI
Bone	99.61	38.37
Brain	99.62	40.94
Echo	99.66	46.04
Front	99.59	28.49
Top	99.63	34.74

Table 5

Correlation analysis test results for five medical images

	Plainimage			Cipherimage
Image	Horizontal	Vertical	Diagonal	Horizontal	Vertical	Diagonal
Bone	0.988057	0.992602	0.980819	0.002175	0.000677	-0.000883
Brain	0.964582	0.977760	0.949311	-0.001832	0.007736	0.005951
Echo	0.968350	0.975245	0.945333	-0.005632	0.009044	0.002944
Front	0.993502	0.995227	0.989938	-0.006848	-0.001852	-0.000342
Top	0.957377	0.964132	0.933977	0.003979	-0.000738	0.007344

Also, structural similarity (SSIM) measures the similarity between plainimages and cipherimages [30]. The SSIM is derived from Eq.(11): $SSIM = \frac{(2 μ_{x} μ_{y}) (2 σ_{xy} + c_{2})}{((μ_{x})^{2} + (μ_{y})^{2} + c_{1}) ((σ_{x})^{2} + (σ_{y})^{2} + c_{2})}$ (11) where c₁ and c₂ are constants, μ_x and μ_y are the averages of x and y. The covariance of x and y is σ_xy and the variance of x and y is σ_x and σ_y respectively.

The performance of the resulted encrypted images is estimated on the basis of PSNR and SSIM. Table 3 illustrates the obtained test values of these two measures. These values clearly showed that the HEECBSG mechanism is well suited for many types of image encryption operations.

6.5 Mean absolute error

The significant difference between the cipherimage and corresponding plainimage can be measured by mean absolute error (MAE) [28]. MAE values are calculated by using the following equation: $MAE = \frac{1}{W * H} \sum_{j = 1}^{H} \sum_{i = 1}^{W} | (P_{ij} - C_{ij}) |$ (12)

where parameters W and H are the width and height of the considered image. P_ij and C_i,j are the gray level of the pixel in the plainimage and cipherimage, respectively. In Table 3, the MAE test demonstrated high values which then guarantee the resistance of the HEECBSG mechanism against differential attacks.

Fig.7

Correlation test of Bone, Brain, Echo and Front images and the cipherimages with the HEECBSG method.

Fig.8

Correlation test of Top image and its cipherimage with the HEECBSG method.

6.6 Correlation analysis

The correlation coefficient maximum value is 1 and the minimum value is 0 as mentioned in [32]. Horizontal, vertical and diagonal directions values are obtained as given in Table 5 and also shown in Figures 7 and 8 for the five plain medical images and corresponding ciphered images. The results indicated that there is a marginal correlation in the cipherimage between the two adjacent pixels, verifying that the HEECBSG scheme can strongly defense against statistical attacks.

6.7 Sensitivity analysis

Two joint measures are used to assure sensitivity properties: Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) [31]. These are used to evaluate the strength of images encryption cipher with respect to differential attacks. They are defined by the following two equations: $NPCR = \frac{\sum_{i, j} D (i, j)}{M \times N} \times 100 %$ (13) $UACI = \frac{1}{M \times N} \frac{\sum_{i, j} | C_{1} (i, j) - C_{2} (i, j) |}{L} \times 100 %$ (14) where C₁ (i, j) and C₂ (i, j) are the pixel values in position (i, j) of two cipherimages C₁and C₂ respectively. L represents the number of gray levels. D (i, j) is mathematically defined as: $D (i, j) = {\begin{matrix} 0, & C_{1} (i, j) = C_{2} (i, j) \\ 1, & otherwise \end{matrix}$

The two test results shown in Table 4 demonstrated that the average percentage values of pixels in cipherimage changed to be greater than 37.72% for UACI and 99.63% for NPCR for the pseudo-random bit-string generator. It means that the HEECBSG approach works completely and precisely in relation to minor changes in the plainimage pixels. Consequently, the known-plaintext attack loses its efficacy and becomes essentially useless.

Table 6

Comparison of correlation coefficients with other methods

Scheme	Horizontal	Vertical	Diagonal
Ref. [33]	0.0743	0.0687	0.0639
Ref. [14]	-0.0072	0.0202	–
Ref. [11]	0.0035	0.0079	0.0010
Ref. [10]	0.0041	0.0022	0.0018
Ref. [9]	0.0003	-0.0043	0.0004
HEECBSG	0.0021	0.0006	-0.0008

Table 7

Comparative execution times for encryption and decryption of 256 × 256 grayscale images

Method	Encryption (sec)	Decryption (sec)
Ref. [11]	0.09375	0.09375
Ref. [34]	0.29	0.30
Ref. [35]	1.2615	1.2615
Ref. [36]	0.22	0.22
Ref. [37]	1.1708	–
Ref. [38]	0.23	0.25
HEECBSG	0.03956	0.03956

Table 8

Comparison of several tests with other methods

Scheme	Entropy	PSNR	SSIM	NPCR	UACI
Ref. [33]	7.9746	7.05	0.00745	99.53	33.46
Ref. [14]	7.3769	7.7939	0.00490	–	–
Ref. [11]	7.9993	27.53	0.00616	99.60	33.42
Ref. [10]	7.97	–	–	99.18	38.59
Ref. [9]	7.9975	–	–	99.99	33.42
HEECBSG	7.9975	32.29	0.00545	99.61	38.37

6.8 Results comparison

A comparison of results is done between the proposed HEECBSG and previous studies. The comparison is for medical image encryption schemes provided in [9–11 , 33]. The performance assessment of our proposed mechanism by performing multiple tests focused on image quality and other criteria for evaluation. Comparison of correlation coefficient values with other methods are shown in Table 6 while comparison of entropy, PSNR, SSIM, NPCR and UACI are listed in Table 8. For real-time encryption and decryption, execution times were estimated for grayscale medical images of size 256 × 256, as illustrated in Table 7. As a result, the proposed HEECBSG scheme outperforms many existing methods and it is definitely a dependable option to encrypt and decrypt classical and medical images.

7 Conclusion

This paper presented a new HEECBSG mechanism for generating pseudo-random bit-string. The HEECBSG scheme is based mainly on two mathematical algorithms; which are elliptic curve cryptography and hash function; to achieve high data integrity and security levels. Moreover, medical image encryption application based on cipher bit-string stream was examined and various evaluation metrics of the cipherimages are reported. The performance analysis and security results showed that the obtained bit-string have high periodicity and good randomness properties. Therefore, it is suitable for security and privacy purposes and usage in medical image transmission. Based on the performance results, it should be noted that the proposed mechanism can be applied to any images for the purpose of privacy and security issues.

Future studies will be conducted towards the use of the proposed scheme in a cloud storage system. In addition, the crypto-stability of the HEECBSG and improving its running time over different platforms are going to be further evaluated.

Compliance with ethical standards

All the authors declare that they have no conflict of interest.

Acknowledgments

We would like to thank the editors and anonymous reviewers for their valuable comments and suggestions to improve the readability of the manuscript.

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Hash-enhanced elliptic curve bit-string generator for medical image encryption

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Elliptic curves over 𝔽 p

5.1 Implementation example

6 Encryption and security analysis

6.2 Histogram analysis

6.3 Entropy analysis

6.7 Sensitivity analysis

7 Conclusion

Compliance with ethical standards

Acknowledgments

References

2.1 Elliptic curves over $𝔽_{p}$