Question

You are given a text file of size \(60 \mathrm{MB}\), which is to be hidden using steganography in the low-order bits of each color in an image file. What size image would be required in order to encrypt the entire file? What size would be needed if the file were first compressed to a third of its original size? Give your answer in pixels, and show your calculations. Assume that the images have an aspect ratio of \(3: 2\), for example, \(3000 \times\) 2000 pixels.

Short answer

For the original file, an image of about 15870x10580 pixels is needed. For the compressed file, an image of about 9159x6107 pixels is needed.

Step-by-step solution

Calculate Bytes Required to Store the File

The initial file size is given as 60 MB. First, we convert this size into bytes. Since 1 MB is equal to 1024 KB and 1 KB is equal to 1024 bytes, we have:\[60 \text{ MB} = 60 \times 1024 \times 1024 \text{ bytes} = 62,914,560 \text{ bytes}\]

Determine Bits Needed for Storage

Each byte consists of 8 bits. Therefore, the total number of bits required to store the file is:\[62,914,560 \text{ bytes} \times 8 \text{ bits/byte} = 503,316,480 \text{ bits}\]

Calculate Image Pixels Required

In a color image with 3 color channels (Red, Green, and Blue), each pixel can store 3 bits using the least significant bit from each channel. Therefore, the required number of pixels to store the file is:\[\frac{503,316,480 \text{ bits}}{3 \text{ bits/pixel}} = 167,772,160 \text{ pixels}\]

Consider Aspect Ratio for Image Dimensions

The image aspect ratio given is 3:2. Therefore, let the width be \(3x\) and the height be \(2x\). Then the area equation in terms of \(x\) is:\[3x \times 2x = 6x^2 = 167,772,160\]Solving for \(x^2\):\[x^2 = \frac{167,772,160}{6} = 27,962,027\]\[x = \sqrt{27,962,027} \approx 5290\]Thus, the image dimensions are approximately 3x = 15870 pixels by 2x = 10580 pixels.

Calculate Requirements for Compressed File

If the file is compressed to a third of its original size, the new size in bytes is:\[\frac{60 \text{ MB}}{3} = 20 \text{ MB} \]Converting this to bytes:\[20 \times 1024 \times 1024 = 20,971,520 \text{ bytes}\]Thus, the required bits are:\[20,971,520 \times 8 = 167,772,160 \text{ bits}\]

Calculate Pixels for Compressed File

Since each pixel stores 3 bits, the number of pixels needed is:\[\frac{167,772,160 \text{ bits}}{3 \text{ bits/pixel}} = 55,924,053 \text{ pixels}\]

Consider Aspect Ratio for Compressed Image Dimensions

Assuming a 3:2 aspect ratio, we apply the same method:\[6x^2 = 55,924,053\]Solving for \(x^2\):\[x^2 = \frac{55,924,053}{6} = 9,320,675\]\[x = \sqrt{9,320,675} \approx 3053\]Thus, the approximate dimensions for the compressed image are 3x = 9159 pixels by 2x = 6107 pixels.

Key concepts

Image Pixels Calculation

When hiding a text file in an image using steganography, one important step is calculating the number of pixels required to store the data. Each pixel in a color image can typically hide 3 bits of information, utilizing the least significant bit of the Red, Green, and Blue channels. To determine how many pixels are needed, you begin with the total bits needed for the file.

For example, if the text file size is 60 MB, first convert this to bytes, and then convert the bytes to bits. Since 1 MB is equal to 1,024 KB and 1 KB equals 1,024 bytes, a 60 MB file sums up to approximately 62,914,560 bytes. Since each byte consists of 8 bits, multiply the number of bytes by 8, resulting in 503,316,480 bits of data that need to be stored.

To find how many pixels are necessary, divide the total number of bits by 3, because each pixel can store 3 bits. This calculation shows that 167,772,160 pixels are necessary to hide the data in its entirety in the least complex scenario. These steps are essential to correctly estimating the image size needed for successful steganography.

File Compression

One way to reduce the number of pixels required for hiding a file is to compress the file before embedding it into an image. Compression reduces the file size, thereby reducing the number of bits required to store the file. Imagine compressing the file to one-third of its original size.

For our example, a 60 MB file, when compressed to a third, is reduced to 20 MB. Convert the compressed file size into bytes to get approximately 20,971,520 bytes, and then convert those bytes into bits, resulting in a total of 167,772,160 bits.

After compression, use the same method to calculate the number of pixels needed. Divide the 167,772,160 bits by 3, yielding about 55,924,053 pixels. This demonstrates the significant savings in pixel demand achieved through compression. Not only does this make embedding more efficient, but it also minimizes the risk of noticeable changes in the image quality.

Aspect Ratio

After calculating the total number of pixels required to store a file using steganography, the next step is determining suitable image dimensions. Images often have a given aspect ratio; in this example, it's 3:2, which means the width is 1.5 times the height.

To find the image dimensions when the image area is 6, start by expressing the dimensions in terms of a single variable. Assume the width is \(3x\) and the height is \(2x\). The total area, or number of pixels, can be expressed in this form: \(6x^2\). Solving it with the pixel calculation from previous sections provides the values for \(x\), which you use to calculate the width and height.

With 167,772,160 pixels required, solve for \(x^2\): \[6x^2 = 167,772,160\] \[x^2 = 27,962,027 \\] \[x \approx 5290\] \[\text{Width} = 3 \times 5290 = 15870 \text{ pixels} \text{Height} = 2 \times 5290 = 10580 \text{ pixels}\] For a compressed variant with fewer pixels, the same calculations provide adjusted, smaller dimensions while respecting the 3:2 aspect ratio.