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In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the processor. Most bitwise operations are presented as two-operand ...
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition):
In cryptography, differential equations of addition (DEA) are one of the most basic equations related to differential cryptanalysis that mix additions over two different groups (e.g. addition modulo 2 32 and addition over GF(2)) and where input and output differences are expressed as XORs.
GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z. Notations Z 2 and Z 2 {\displaystyle \mathbb {Z} _{2}} may be encountered although they can be confused with the notation of 2 -adic integers .
1110 2 XOR 1001 2 = 0111 2 (this is equivalent to addition without carry) As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two n -bit strings is identical to the standard vector of addition in the vector space ( Z / 2 Z ) n {\\displaystyle (\\mathbb {Z} /2\\mathbb {Z} )^{n}} .
Polynomial addition modulo 2 is the same as bitwise XOR. Since XOR is the inverse of itself, polynominal subtraction modulo 2 is the same as bitwise XOR too. Multiplication is similar (a carry-less product): (+) (+) = + + + ().
This operation is also known as "bitwise xor" or "vector addition over GF" (bitwise addition modulo 2). Within combinatorial game theory it is usually called the nim-sum, as it will be called here. The nim-sum of x and y is written x ⊕ y to distinguish it from the ordinary sum, x + y. An example of the calculation with heaps of size 3, 4, and ...
For example, += and -= are often called plus equal(s) and minus equal(s), instead of the more verbose "assignment by addition" and "assignment by subtraction". The binding of operators in C and C++ is specified (in the corresponding Standards) by a factored language grammar, rather than a precedence table.