What does a circled plus mean?
MathCryptographyMatrixMath Problem Overview
I cannot understand the calculation "66 ⊕ fa = 9c". The sum is clearly over "ff", so I am confused. The topic is simple encryption algorithm.
What does a circled plus mean?
Math Solutions
Solution 1 - Math
People are saying that the symbol doesn't mean addition. This is true, but doesn't explain why a plus-like symbol is used for something that isn't addition.
The answer is that for modulo addition of 1-bit values, 0+0 == 1+1 == 0, and 0+1 == 1+0 == 1. Those are the same values as XOR.
So, plus in a circle in this context means "bitwise addition modulo-2". Which is, as everyone says, XOR for integers. It's common in mathematics to use plus in a circle for an operation which is a sort of addition, but isn't regular integer addition.
Solution 2 - Math
This is not an plus, but the sign for the binary operator XOR
a b a XOR b
0 0 0
0 1 1
1 0 1
1 1 0
Solution 3 - Math
It's not an addition, but an exclusive OR operation. At least the output confirms to the same.
Solution 4 - Math
That's the XOR operator, not the PLUS operator
XOR works bit by bit, without carrying over like PLUS does
1 XOR 1 = 0
1 XOR 0 = 1
0 XOR 0 = 0
0 XOR 1 = 1
Solution 5 - Math
Hope this layout works, take it to the binary representation with an XOR:
66h = 102 decimal = 01100110 binary
FAh = 250 decimal = 11111010 binary
------------------------------------
10011100 binary <------ that's 9Ch/156 decimal
- XOR rules are basically:
- 1 XOR 1 = 0 false
- 1 XOR 0 = 1 true
- 0 XOR 0 = 0 false
Solution 6 - Math
The plus-symbol in a circle does not denote addition. It is a XOR operation.
Solution 7 - Math
It's an exclusive or (XOR). If I remember correctly, when doing bitwise mathematics the dot (.) means AND and the plus (+) means OR. Putting a circle around the plus to mean XOR is consistent with the style used for OR.
Solution 8 - Math
I used the logic in the replies by rampion and schnaader. I will summarise how I confirmed the results. I changed the numbers to binary and then used the XOR-operation. Alternatively, you can use the Hexadecimal tables: http://www.garykessler.net/library/byte_logic_table.html">Click here!
Solution 9 - Math
It is XOR. Another name for the XOR function is addition without carry. I suppose that's how the symbol might make sense.