A robust image encryption scheme based on chaotic system and elliptic curve over finite field

Abstract

The paper proposes a robust image encryption scheme based on chaotic system and elliptic curve over a finite field. The sender and receiver agree on an elliptic curve point based on Diffie-Hellman public key sharing technique. The logistic map is used to generate a chaotic sequence with initial conditions derived from the shared elliptic curve point. The chaotic sequence is converted to integers and the point multiplication is performed with the shared elliptic curve point. The resulting elliptic curve points are converted to byte values to generate a random sequence. The image to be encrypted is scrambled using Arnold’s transform where the number of scrambling rounds is derived from the shared elliptic curve point. The scrambled image pixels value is XOR with the random sequence to generate the cipher image. Statistical, performance, security and robustness analyses show that the proposed scheme is a robust encryption scheme with the ability to resist from different types of attacks.

Appendices

Appendix A: Basic of elliptic curve cryptography

Given an elliptic curve equation over a finite field E _p .

$$ E_{p}:y^{2} = x^{3} + ax +b \text{ mod } p $$

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where,

• a and b are constant with 4a ³ + 27b ² ≠ 0.

• p: prime number.

• Coordinates (x, y) ∈ E _p follows certain additive abelian properties.

1.1 Identity

A point P(x, y) on addition with a point at infinity $\infty$ is the point itself.

$$ P + \infty = \infty + P = P \forall P \in E_{p} $$

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1.2 Negatives

Negative of a point P = (x, y) is given as − P = (x, −y).

1.3 Point addition

Two points P(x ₁, y ₁) and Q(x ₂, y ₂) ∈ E _p , adds up to produce a third point R(x ₃, y ₃) ∈ E _p . Mathematically the coordinate of R is computed as:

$$ x_{3}=\{\lambda^{2} - x_{1} - x_{2}\} \text{ mod } p $$

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$$ y_{3}=\{\lambda(x_{1} - x_{3}) - y_{1}\} \text{ mod } p $$

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where,

$$ \lambda = \frac{y_{2} - y_{1}}{x_{2}-x_{1}} \text{ mod } p $$

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If points P and Q are same, R(x ₃, y ₃) ∈ E _p is computed as:

$$ x_{3}=\{\lambda^{2} - 2x_{1}\} \text{ mod} p $$

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$$ y_{3}=\{\lambda(x_{1}-x_{3})-y_{1}\} \text{ mod } p $$

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where,

$$ \lambda = \frac{3{x_{1}^{2}} + a}{2y_{1}} \text{ mod } p $$

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1.4 Point at infinity

If x ₁ = x ₂ but y ₁ ≠ y ₂ in point addition case, the two point is said to meet at infinity.

$$ P + Q = \infty $$

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If the y coordinate is 0 in point doubling case, the point doubling operation is said to meet at infinity.

$$ P + P = \infty $$

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1.5 Point multiplication

Point multiplication is computed using a series of point addition operation. Combining point addition and point doubling can help in reducing the number of computations required to obtain the point multiplication coordinate.

$$ kG = G + G + G + \dots{} k \text{ times } $$

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where, G is known as seed value or generator and it is one of the coordinates that satisfies the elliptic curve equation. Point multiplication can be reduced by combining point addition and point doubling operation. To compute k G with the reduced method, the following steps are performed.

• 1. Begin $a = k, B=\infty , C = G$.

• 2. If a is even, a = a/2, B = B and C = 2C.

• 3. If a is odd, a = a − 1, B = B + C and C = C.

• 4. If a ≠ 0, go back to Step 2.

After completion B = k G.

Appendix B: Base conversion

The value of x _{p m} and y _{p m} coordinate obtained after point multiplication in (10) are large value in the range of 512 bits. If these coordinates are base converted using 256 as base, a list of integers (64 at maximum) in the range of 0 to 255 will be produce. Table 15 shows the values obtained after performing base conversion on a 100 bits integer value 668393009581810503045091707312 with 256 as base.

$$668393009581810503045091707312 \,=\, \{8,\! 111,\! 177,\! 189, 53, 34, 220,\! 117,\! 162, 238,\! 164,\! 73,\! 176\}_{256}$$

Table 15 Base conversion

Each coordinate in an elliptic curve point of 512 bits can generate upto 64 values after base conversion using 256 as base. So, every L[i] generated using Logistic map can later generate 128 byte values after performing point multiplication with secretly shared elliptic curve point k G. So in Step 3 of proposed encryption scheme i ranges 1 to $\frac {m \times m \times cc}{128}$ instead of 1 to m × m × c c. Base conversion operation helps in reducing the number of point multiplication operation.

Appendix C: PSNR and SSIM

$$ \text{PSNR}=20 \times log_{10}\frac{\text{MAX}}{\sqrt{\text{MSE}}} $$

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For gray-scale image,

$$ \text{MSE}=\frac{1}{m \times n}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{n-1}[I(i,j)-K(i,j)]^{2} $$

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For color image,

$$ \text{MSE}=\frac{1}{m \times n \times c}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{n-1}[I(i,j)-K(i,j)]^{2} $$

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where,

• MAX: Maximum supported pixel value.

• MSE: Mean squared error.

• m × n: Size of the image.

• c: Number of color channel.

• I(i, j): Original image pixel value at location (i, j).

• K(i, j): Received image pixel value at location (i, j).

$$ \text{SSIM}(x,y)=\frac{(2\mu_{x} \mu_{y} + c_{1})(2\sigma_{xy} + c_{2})}{({\mu_{x}^{2}} + {\mu_{y}^{2}} + c_{1})({\sigma_{x}^{2}} + {\sigma_{y}^{2}} +c_{2})} $$

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where,

• μ _x = Mean of x.

• μ _y = Mean of y.

• σ _x = Standard deviation of x.

• σ _y = Standard deviation of y.

• σ _{x y} = Covariance of x and y.

• c ₁ = (k ₁ L)².

• c ₂ = (k ₂ L)²

• k ₁ = 0.01 and k ₂ = 0.03.

• L = 2^{No. of bits per pixel}-1.