$1.$ Implement $(2,3)$ Shamir$$s Secret Sharing $($SSS$)$ scheme $($both share preparation and reconstruction$)$ for images. You need to show the working of your program by creating shares of an image and show the image shares. Next, you will choose any two shares and show the reconstructed image as output. You need to operate on image data, leaving the header intact. You can use any BMP image of your choice. Note that the header length in BMP file format is $54$ bytes.

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$2.$ Next, you are required to demonstrate the homomorphic properties of SSS scheme using the image downscaling operation. In a homomorphic encryption scheme, computations performed on encrypted data, when decrypted, generates the same value as when the same computations are performed on the plaintext.$$

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The downscaling operation takes an input image and produces an image whose width and height are less than the input image. You need to implement downscaling by a factor of $2.$ This means that both the width and height of your output image will be exactly half of the input image.

You are required to do the following steps:

$1)$ Take an input image I of certain resolution and use the downscale method on this image to obtain the downscaled image Io$.$ In the downscale method, each pixel of Io is the average of four pixels of I. For example, pixel $(0,0)$ of Io is the average of pixels $(0,0),(0,1),(1,0),(1,1)$ of I; pixel $(1,0)$ of Io is the average of $(2,0),(2,1),(3,0),(3,1)$ of image I. As a result, the resolution of Io will be half of I in both x and y dimensions.

$2)$ Create $3$ shares $($denoted by I$1,$ I$2,$ I$3)$ of I using the SSS scheme.$$

$3)$ Perform the downscale method on all the three shares and obtain the downscaled shares $($denoted by Is$1,$ Is$2,$ Is$3).$

$4)$ Pick any $2$ downscaled shares, i$.$e$.,$ from Is$1,$ Is$2,$ Is$3,$ and reconstruct the downscaled plaintext image $($denoted by Is$).$

$5)$ Compute the mean average error between the two images, Io and Is$,$ using the following equation:

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MAE $=$j $=1$ to w$$h $(|$Io$($j$)$ Is$($j$)|)$

where w and h are image width and height.

$6)$Show the output of each step and report your observations.