How Do You Hide a Picture Inside Another?

Play with the photo above.

• a.Start at bit 8 and slide to the right, one bit at a time. At which bit does the temple stop being easy to see?
• b.At bit 1, what do you see? Now press Without. What changes in the right picture, and what changes in the photo on the left?
• c.Look at the readouts with the secret in. About how many of the photo’s colors were changed to hide the leopard?

A color is a byte

In Grade 7 you met colors from numbers: every pixel is three lights, red, green and blue, and each light’s brightness is a number from 0 to 255. That number is one byte: eight bits, worth 128, 64, 32, 16, 8, 4, 2 and 1. Here is a red of 202:

128 + 64 + 8 + 2 = 202. The bits are not equally important. If bit 8 flips, the color jumps by 128, from a bright red to a dark one. If bit 1 flips, it moves by 1, from 202 to 203. In Grade 7 you saw that a screen’s 16.7 million colors are “enough that the steps between neighboring colors are too small for your eyes to see”. One step is exactly what a change in bit 1 makes. Bit 8 is called the most significant bit, and bit 1 the least significant bit.

Hiding one bit

The last bit of a number is the remainder when you divide it by 2: 0 for an even number, 1 for an odd one. To hide a secret bit in a color, make the color even or odd to match:

• take away the remainder, which makes it even: 203 → 202, and 202 stays 202;
• then add the secret bit: 0 keeps it even, 1 makes it odd.

As one rule: new color = color − (remainder of color ÷ 2) + secret bit.

Every color moves by 1 at most, and about half don’t move at all: their last bit already matched. And nothing ever goes below 0 or above 255.

To get the secret back, the person who receives the photo only needs the remainder of each color ÷ 2. That is the whole trick.

The leopard in the widget is a black-and-white picture of 160 × 120 squares, each square doubled to cover 2 × 2 of the photo’s pixels. A white square is a 1, a black square a 0. Each bit is hidden in all three colors of its pixel, red, green and blue, so the right picture shows the leopard in plain black and white.

With blocks

Here is the rule as a block program. It asks for a color and a secret bit, and says the new color.

A photo is millions of colors: a program in a loop does them all in a moment.

In Python (from Grade 8): the bits of a number

// divides and throws away the remainder, and % gives the remainder. So (color // worth) % 2 is the bit worth worth: 202 // 8 is 25, and 25 % 2 is 1.

color = 202worth = 128for place in range(8):    bit = (color // worth) % 2    print('worth', worth, ':', bit)    worth = worth // 2

worth 128 : 1worth 64 : 1worth 32 : 0worth 16 : 0worth 8 : 1worth 4 : 0worth 2 : 1worth 1 : 0

Reading down: 11001010, the table above.

A photo as a list of rows

The programs below use a small gray photo of the same temple, made from the same photo as the widget’s: 30 rows of 40 gray values, from 0 (black) to 255 (white), in the file nyatapola.py. A gray photo has one number a pixel instead of three, which keeps the programs short. They also use flag.py, our secret: Nepal’s flag drawn in 30 rows of 40 squares, '#' for the flag and '.' for the rest.

This program draws the photo, then three of its bit planes: one bit of every pixel, white where the bit is 1 and black where it is 0. A loop inside a loop goes through every pixel: the outer one takes each row, the inner one each value in it.

from nyatapola import photofrom pixels import show_grid def bit_plane(picture, worth):    plane = []    for row in picture:        new_row = []        for color in row:            new_row.append((color // worth) % 2 * 255)        plane.append(new_row)    return plane show_grid(photo)show_grid(bit_plane(photo, 128))show_grid(bit_plane(photo, 8))show_grid(bit_plane(photo, 1))

Picture

Even in a photo this small, bit 8 is the temple and bit 1 is noise.

Hiding the flag

Now hide the flag in bit 1. hide() makes a new photo, pixel by pixel, with the rule from above. Then the program counts how many pixels changed, and by how much.

from nyatapola import photofrom flag import flagfrom pixels import show_grid def hide(photo, picture):    new_photo = []    for r in range(len(photo)):        new_row = []        for c in range(len(photo[r])):            color = photo[r][c]            if picture[r][c] == '#':                bit = 1            else:                bit = 0            new_row.append(color - color % 2 + bit)        new_photo.append(new_row)    return new_photo secret_photo = hide(photo, flag) changed = 0biggest = 0for r in range(30):    for c in range(40):        change = abs(secret_photo[r][c] - photo[r][c])        if change > 0:            changed = changed + 1        biggest = max(biggest, change)print(changed, 'of 1200 pixels changed')print('The biggest change:', biggest)show_grid(photo)show_grid(secret_photo)

562 of 1200 pixels changedThe biggest change: 1

Picture

Put your finger between the two pictures. Can you find one square that is different?

Getting it back

The person who receives the photo needs only the remainders. This program hides the flag and then finds it again, using a palette: 0 is drawn white and 1 crimson, the flag’s color.

from nyatapola import photofrom flag import flagfrom pixels import show_grid def hide(photo, picture):    new_photo = []    for r in range(len(photo)):        new_row = []        for c in range(len(photo[r])):            color = photo[r][c]            if picture[r][c] == '#':                bit = 1            else:                bit = 0            new_row.append(color - color % 2 + bit)        new_photo.append(new_row)    return new_photo def reveal(photo):    found = []    for row in photo:        new_row = []        for color in row:            new_row.append(color % 2)        found.append(new_row)    return found secret_photo = hide(photo, flag)show_grid(reveal(photo), palette=['white', 'crimson'])show_grid(reveal(secret_photo), palette=['white', 'crimson'])

Picture

The photo’s own last bits are noise; the second picture is the flag, back exactly.

• 562 of the 1,200 pixels changed, and none by more than 1. The other 638 already had the right last bit. The photo with the flag in it looks exactly like the photo without it.
• The flag came back perfectly, every square of it, from nothing but the remainders.
• The last bit of a photo is noise, so a secret there doesn’t stand out in the photo.

It works because the bits of a number are not equal: the top bit carries half of a color’s brightness, the last bit 1 part in 255.

Now try the opposite. Change one line of hide() so the flag goes into bit 8:

from nyatapola import photofrom flag import flagfrom pixels import show_grid def hide_on_top(photo, picture):    new_photo = []    for r in range(len(photo)):        new_row = []        for c in range(len(photo[r])):            color = photo[r][c]            if picture[r][c] == '#':                bit = 1            else:                bit = 0            new_row.append(color % 128 + bit * 128)        new_photo.append(new_row)    return new_photo show_grid(hide_on_top(photo, flag))

Picture

color % 128 keeps bits 7 to 1 and drops bit 8; bit * 128 puts the secret there. Same flag, same photo, still one bit a color, but now the flag shouts. Where you hide a bit matters more than how many you hide.

How much could a real photo hold? A phone photo of 12 million pixels has three colors a pixel:

pixels = 12000000colors = pixels * 3letters = colors // 8print(colors, 'last bits to hide in')print(letters, 'letters, at 8 bits a letter')

36000000 last bits to hide in4500000 letters, at 8 bits a letter

Four and a half million letters: more than a whole thick book, inside one holiday photo.

Hide more, see more. Hide a secret in the last two bits instead of one. Each pixel can then keep a number from 0 to 3: four shades of gray, enough for a recognizable photo, not just black and white. How big is the biggest change now? Challenge ★★ asks you to write it. With four bits, you can hide a whole gray photo inside another; at what bit does the cover photo start to look spoiled?

Text in a photo. A letter is a number (Grade 7’s ASCII: H is 72), and a number is 8 bits, so 8 pixels can hold one letter. Challenge ★★★ reads a message hidden this way.

A secret is easy to break. We tried saving the widget’s photo, with the leopard in it, as a JPEG file at quality 90, the usual way a photo file is squeezed smaller (Grade 8). The JPEG keeps the photo looking the same, but it changes most colors by a few steps, and 48 out of every 100 hidden bits came back wrong. That is no better than tossing a coin: the leopard was gone. Saved as a PNG file, which keeps every number exactly, not one bit changed. So a hidden picture must travel as a file that is never squeezed.

★ Read a hidden letter (blocks)

Eight color numbers are typed, one at a time. Their last bits, the first worth 128 and the last worth 1, make one letter’s code. Finish the program so it says the code. For 200, 151, 88, 64, 203, 90, 40, 18 it says 72, the letter H.

★★ Hide in two bits (Python, from Grade 8)

Write hide_two(photo, picture). photo is a list of rows of numbers from 0 to 255, and picture a list of rows of the same size whose numbers are 0, 1, 2 or 3. Give back a new photo with each number’s last two bits replaced by the picture’s number.

def hide_two(photo, picture):    # each picture number is 0 to 3    return photo print(hide_two([[200, 201, 202]], [[3, 0, 1]]))

★★★ Read the hidden message (Python, from Grade 8)

A message is hidden in a list of color numbers: each letter is 8 last bits in a row, the first worth 128, and the message ends where 8 bits make 0 (or where the numbers run out). Write read_message(colors) that gives back the message. chr(72) turns the code 72 into 'H'.

def read_message(colors):    # 8 last bits a letter; the code 0 ends the message    return '' print(read_message([200, 151, 88, 64, 203, 90, 40, 18]))

• Grade 7, How Computers Store Everything: colors as three numbers from 0 to 255, and binary.
• Grade 8, Photos, Sound and Video: pixels, file sizes and why JPEG files are small.
• Grade 8, Data, Privacy and You: the other data that travels inside your photos.
• Grade 9, Lists, Strings and Dictionaries: more programs on the same gray photo of the Nyatapola.
• Which Card Was Flipped?: extra bits that find and fix a mistake, instead of hiding a message.