Can someone please help me with the question below?
Explain why 0 ⊕ x ≠ 1 ⊕ x, whatever the value of the bit x.
Hence, explain why flipping a bit in the plaintext produces a predictable change in the ciphertext when using a stream cipher that XORs the plaintext with the keystream to produce a ciphertext.
What is the name of the circle in the middle ?
The symbol of the circle with the + in it is one of many symbols for exclusive-or. XOR, EOR, EXOR, ⊻, ⊕, ↮, and ≢. Binary OR is true when either input is true; binary XOR is true when exactly one input is true. If both inputs are true, the XOR result is false.
One property of this is that if either input bit flips, the output bit will also flip. That's sort of the definition of XOR, so you'll have to figure out how to reword that for your stream cipher question.
A XOR B Truth Table
Input | Output
A B |
________|_______
0 0 | 0
0 1 | 1
1 0 | 1
1 1 | 0
0 = FALSE
1 = TRUE
It's exclusive or (or XOR), corresponds to $x \oplus y := x+y \pmod 2$ for single bits, i.e., scalars. It is sometimes used for binary vectors as well, whereby two bitvectors of length $n$ $$ \mathbf{x}=(x_1,\ldots,x_n)$$ and $$ \mathbf{y}=(y_1,\ldots,y_n)$$ result in $$ \mathbf{x}\oplus \mathbf{y}=(x_1\oplus y_1,\ldots,x_n \oplus y_n) $$ i.e., a bitwise operation.
