XOR

Mathematical symbol

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zero

XOR, exclusive OR in English, abbreviation Xor

XOR is a mathematics operator 。 It applies to Logical operation 。 The mathematical symbol of XOR is "⊕", and the computer symbol is "xor". The algorithm is:

a⊕b = (¬a ∧ b) ∨ (a ∧¬b)

If the values of a and b are different, the XOR result is 1. If the values of a and b are the same, the XOR result is 0.

XOR is also called Banjia operation , its algorithm is equivalent to binary addition without carry: in binary, 1 represents true, 0 represents false, then the exclusive OR algorithm is: 0 ⊕ 0=0, 1 ⊕ 0=1, 0 ⊕ 1=1, 1 ⊕ 1=0 (the same is 0, the different is 1), these algorithms are the same as addition It is the same, but without carry, so XOR is often regarded as non carry addition.

XOR is abbreviated as XOR, EX-OR

There are two operators in the program: XOR and ⊕.

The use method is as follows

z = x ⊕ y

z = x xor y

Chinese name: XOR
Foreign name: exclusive OR

Mathematical symbol: ⊕
English abbreviation: xor
Program symbol: ^

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Algorithm

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one Law of return to zero ：

two Law of identity ：

three Commutative law ：

four Associative law ：

;

five introspect ：

6. d=a ⊕ b ⊕ c can push a=d ⊕ b ⊕ c

7. If x is a binary number 0101, y is Binary number 1011；

Then x ⊕ y=1110

The result is 1 only when the two compared bits are different, otherwise the result is 0

That is, "0 if two inputs are the same, 1 if they are different"

a	b	a⊕b
zero	zero	zero
zero	one	one
one	zero	one
one	one	zero

logic

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Logical expression ：

The truth table of XOR logic is as follows Figure 1 As shown in.

chart

Its logic symbols are as follows Figure 2 As shown in.

The relationship of XOR logic is: when AB is different, output P=1; When AB is the same, output P=0. "⊕" is an exclusive OR operation symbol, and exclusive OR logic is also a combination of AND or non logic, and its logical expression is:

P=A⊕B

From Figure 1 It can be seen that the rule of XOR operation is

0⊕0=0,0⊕1=1

1⊕0=1,1⊕1=0

Figure 2

Formula 1: Take 0 for the same and 1 for the different

Formula 2:

Input A takes 0, then output p=input B

Input A takes 1, then output p=inverse of input B

In fact, XOR is defined as either one (is one), but not both in English, that is, when only one is true (1), the truth (1) is taken.

effect

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Widely used in computers, xor Logical symbol Generally, xor is also used:

True False=True

False True=True

False=False

True True=False

Or:

True ⊕ False = True

False ⊕ True = True

False⊕ False = False

True ⊕ True = False

Some computer languages use 1 to indicate true and 0 to indicate false, so the bitwise XOR of the two bytes is as follows

-	00000000
xor	00000000
-------------
result	00000000

The following two binary values are used for XOR calculation:

-	eleven million one hundred and eleven thousand one hundred and eleven
xor	00000000
--------------
result	eleven million one hundred and eleven thousand one hundred and eleven

In reality, all decimal values are used. Let's take a look at how two decimal values perform XOR calculation:

5 ⊕ 3 = ？

1. The values will be converted to binary before XOR calculation:

5 and 3 are converted to binary: 0101 and 0011 respectively

-	0101
xor	0011
--------
result	0110

2. Then convert the result 0110 to decimal: 6

3. So 5 ⊕ 3=6

Skillful use

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Unlike other languages, C language and C++Language (C++supports xor, whose usage and effect are the same as' ^ '.) Instead of xor, "^" is used, and the typing method is Shift+6. ("^" in other languages generally means power)

If it is necessary to exchange the values of two variables, in addition to borrowing intermediate variables, you can also use XOR to exchange only two variables, such as:

void swap(int &a,int &b) { a=a^b; b=b^a; a=a^b; }

Detailed explanation:

a1=a^b b=b^a1=b^a^b=a //Now a1=a ^ b b=a a=a1^a=a^b^a=b

be careful:

a=a^b^(b=a);// This type of form is incorrect UB behavior, and different results can be obtained in different compilers. Do not use

This completes the exchange of a and b.

To sum up, the same variable and another variable and their exclusive OR value are exclusive or equal to another number, such as (a ^ b) ^ b=a.

Use case: It can be used in one or more links of the encryption algorithm, making the algorithm more complex, difficult to crack, and more secure.

Addition and subtraction relation

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Addition and subtraction operations on the Galois field are equivalent, that is, XOR operations. While multiplication and division are directly polynomial multiplication and division, and then Primitive polynomial Take the mold.

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