Output of a Linear Shift Register given Plaintext and intial state

Question

Given an initial state $S_0 = (0101)$ and plaintext $P = 1010101011$ I need to find the ciphertext.

The method to solve this was not taught in my coursework, and the information I have found online does not seem to apply to this.

Can anyone explain the steps needed to solve this?

Answer 1

Read the drawing assuming:

• each of the four (about) square boxes $a_j$ initially (at $t=0$) holds one of the four bits of $S_0$ (in reading order, unless otherwise stated);
• information follows arrows (almost) instantly, except when getting into a square box where that is delayed until $t$ has grown by $1$ (in time unit or clock period); notice this implies that the content of the left square box is also (with negligible delay) on the arrowhead going straight down and the one going right-down;
• there are three implicit short arrows between the four square boxes, oriented as the rightmost arrowhead;
• each $\oplus$ symbol performs the exclusive-OR of all the ingoing arrows, and outputs that result to outgoing arrow, with no significant delay;
• the plaintext bits are made available one by one (starting at $t=0$, and in reading order, unless otherwise stated) on the input side of the arrow marked "plaintext";
• the ciphertext bits are what's at the arrowhead marked "Cipher-text" (with the same convention as for plaintext).

You can now manually compute what there is in each of the boxes, arrowheads, "plaintext", "Cipher-text", at any integer value of $t\ge0$ (or, more precisely, shortly after). Assuming increasing time matches reading order, you might want to format this as a table with $t$ increasing from left to right, so that the lines with "plaintext" and "Cipher-text" will contain the plaintext and ciphertext in reading order.

Answer 2

If this can code can help you understanding the Linear Shift Register. (^ represents Exclusive OR)

        byte[] register = new byte[] { 0, 1, 0, 1 };
        byte[] plain = new byte[] { 1, 0, 1, 0, 1, 0, 1, 0, 1, 1 }; ;
        byte[] cipher = new byte[plain.Length];

        for (int i = 0; i < plain.Length; i++)
        {
            //plaintext ^ state a0
            cipher[i] = (byte)(plain[i] ^ register[0]);

            //Temporary storing Result of a0 ^ a3
            byte temp = (byte)(register[0] ^ register[3]);

            //Shifting states one index Left
            register[0] = register[1];
            register[1] = register[2];
            register[2] = register[3];

            //Setting new state a3
            register[3] = temp;
        }