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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 ...
Note that is equivalent to zero in the above equation because addition of coefficients is performed modulo 2: = + = (+) = (). 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.
Any set of m integers, no two of which are congruent modulo m, is called a complete residue system modulo m. The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class modulo m. [4] For example, the least residue system modulo 4 is {0, 1, 2, 3}.
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.
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):
For example, given a bit pattern 0011 (decimal 3), to determine whether the second bit is set we use a bitwise AND with a bit pattern containing 1 only in the second bit: 0011 (decimal 3) AND 0010 (decimal 2) = 0010 (decimal 2) Because the result 0010 is non-zero, we know the second bit in the original pattern was set.
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 .
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} , and the question is to decide whether any subset of the integers sum to precisely T {\displaystyle T} . [ 1 ]