Write a program that produces pseudo-random bits by simulating a linear feedback shift register, and then use it to implement a simple form of encryption for digital pictures.

LFSR review.  A linear feedback shift register is a register of bits that performs discrete step operations that

• Shift all of the bits one position to the left and
• Replaces the vacated bit by the exclusive or of the bit shifted off and the bit at a given tap position in the register.

A LFSR has three parameters that characterize the sequence of bits it produces: the number of bits N, the initial seed (the sequence of bits that initializes the register), and the the tap position tap. As in the example in Lecture 1, the following illustrates one step of an 11-bit LFSR with initial seed 01101000010 and tap position 8.

LFSR data type.  Your first task is to write a data type that simulates the operation of a LFSR by implementing the following API:

public class LFSR
-------------------------------------------------------------------------------------------------------------------------------
public             LFSR(String seed, int tap)  //  create LFSR with the given initial seed and tap
public         int step()                      //  simulate one step and return the least significant (rightmost) bit as 0 or 1
public         int generate(int k)             //  simulate k steps and return k-bit integer
public      String toString()                  //  return a string representation of the LFSR
public static void main(String[] args)         //  test all of the methods in LFSR

To do so, you need to choose the internal representation (instance variables), implement the constructor, and implement the three instance methods. These are interrelated activities and there are several viable approaches.

• Constructor. The constructor takes the initial seed as a String argument whose characters are a sequence of 0s and 1s. The length of the register is the length of the seed. We will generate each new bit by xoring the most significant (leftmost) bit and the tap bit. The position of the tap bit comes from the constructor argument. For example, the following code should create the LFSR described above.

  LFSR lfsr = new LFSR("01101000010", 8);
  

• String representation. The toString() method returns a string representation of the LFSR by concatenating the values in the registers. For example,

  LFSR lfsr = new LFSR("01101000010", 8);
  StdOut.println(lfsr);
  

  should output

  01101000010
  

• Simulate one step. The step() method simulates one step of the LFSR and returns the least significant (rightmost) bit as an integer (0 or 1). For example,

  LFSR lfsr = new LFSR("01101000010", 8);
  for (int i = 0; i < 10; i++) {
      int bit = lfsr.step();
      StdOut.println(lfsr + " " + bit);
  }
  

  should output

  11010000101 1
  10100001011 1
  01000010110 0
  10000101100 0
  00001011001 1
  00010110010 0
  00101100100 0
  01011001001 1
  10110010010 0
  01100100100 0
  

• Extracting multiple bits.  The method generate() takes an integer k as an argument and returns a k-bit integer obtained by simulating k steps of the LFSR. This task is easy to accomplish with a little arithmetic: initialize a variable to zero and, for each bit extracted, double the variable and add the bit returned by step(). For example, given the bit sequence 1 1 0 0 1 the variable takes the values 1, 3, 6, 12, 25, ending with the binary representation of the bit sequence. For example,

  LFSR lfsr = new LFSR("01101000010", 8);
  for (int i = 0; i < 10; i++) {
      int r = lfsr.generate(5);
      StdOut.println(lfsr + " " + r);
  }
  

  should output

  00001011001 25
  01100100100 4
  10010011110 30
  01111011011 27
  01101110010 18
  11001011010 26
  01101011100 28
  01110011000 24
  01100010111 23
  01011111101 29
  

  Implement the generate() method by calling the step() method k times and performing the necessary arithmetic.

A client to encrypt and decrypt images.  Your final task is write a LFSR client PhotoMagic.java that can encrypt and decrypt pictures, by implementing the following API:

public class PhotoMagic
-----------------------------------------------------------------------------------------------------------------------------
public static Picture transform(Picture picture, LFSR lfsr)  //  transform picture using lfsr
public static    void main(String[] args)                    //  read in the name of a picture and the description of an LFSR
                                                             //  from the command-line and encrypt the picture with the LFSR

• Transform method. The transform() method takes a Picture (see pp. 324–337 in the textbook) and an LFSR as arguments and returns a new picture that is the result of transforming the argument picture using the linear feadback shift register as follows: For each pixel (x, y), in raster-scan order—(0, 0), (0, 1), (0, 2), ... —extract the red, green, and blue components of the color (each component is an integer between 0 and 255). Then, xor the red component with 8 newly generated bits. Do the same for the green (using another 8 newly generated bits) and, finally, the blue. Create a new color using the result of the xor operations, and set the pixel in the new picture to that color. Note: Although the LFSR parameter changes during the execution of transform, the Picture parameter is to be treated as if it were immutable.

• Main method. The main() method takes three command-line arguments, a picture name, a binary password (the initial LFSR seed), and an integer (the tap number). It should display the transformed picture on the screen (using the show() method in the Picture class). For example, typing

  % java PhotoMagic pipe.png 01101000010100010000 16
  

  takes as input the picture pipe.png (left) and displays as output the picture on the right. You can save the right-hand picture as Xpipe.png by selecting File -> Save -> Xpipe.png from the menu in the window where the image is displayed.

  Now, here's the magic: running the same program with the same binary password and tap on the transformed picture recovers the original picture! For example, typing

  % java PhotoMagic Xpipe.png 01101000010100010000 16
  

  takes as input the picture Xpipe.png (left) and displays as output the picture on the right.

  Anyone knowing this password and tap can get the original picture back, but another password won't work. If you're not convinced, try it. Thus, for example, you can post the transformed picture on the web, but only friends who have the password (and your program) can see the original.

Submission.   Submit LFSR.java and PhotoMagic.java. Also, submit a readme.txt file and answer the questions.

Extra credit.  Using a short binary password is weak protection and using a long one is inconvenient. For extra credit, write a client PhotoMagicDeluxe.java with the same API as PhotoMagic.java, but use an alphanumeric password instead of a binary one. Assume that the password contains only characters from the 64-character alphabet:

String base64 = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";

Challenge for the bored.  Write a program BreakPhotoMagic.java that the name of an encrypted picture and the number of bits N in the password as command-line arguments, tries all possible binary passwords of length N and all possible taps, and decrypts the picture.

Hint: all but the correct seed and tap will produce pseudo-random colors, so you can track the frequencies of each 8-bit value and pick the seed and tap that results in the frequencies that deviate the most from 128.

Warning: this program can take a very long time to run; we recommend debugging it using pictures transformed with binary passwords of length 5 or 6.