Question: Write a simple four-function calculator in (mathrm{GF}left(2^{4}ight)). You may use table lookups for the multiplicative inverses.
Write a simple four-function calculator in \(\mathrm{GF}\left(2^{4}ight)\). You may use table lookups for the multiplicative inverses.
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Here is a simple fourfunction calculator in mathrmGF24 with table lookups for multiplicative inverses Representation Elements in mathrmGF24 are represented as 4bit binary stringswhere the most significant bit represents the coefficient of x3and the least significant bit represents the coefficient of x0For examplethe string 1011 represents the element x3 x 1 Addition and Subtraction Addition and subtraction in mathrmGF24 are simply bitwise XOR operationsThis is because addition in modulo 2 arithmetic is just XOR Multiplication Multiplication in mathrmGF24 can be done using the standard polynomial multiplication algorithmbut modulo 2 arithmetic simplifies the processWe only need to keep track of the carry bits and XOR them with the appropriate terms based on the irreducible polynomial for mathrmGF24which is x4 x 1 Table for Multiplicative Inverses Since this is a small Galois fieldprecomputing the multiplicative inverses is efficient and allows for faster lookupsHere is a table showing the multiplicative inverses of all nonzero elements in mathrmGF24 Element Inverse 1001 1001 1010 1010 1011 1101 1100 1111 1101 1100 1110 0111 1111 1110 Calculator Implementation Input Take two elements in mathrmGF24 as input represented as 4bit binary strings Choose operation Select the desired operation addition subtraction multiplication or division Perform operation AdditionSubtraction Perform bitwise XOR on the corresponding bits of the two elements Multiplication Use the polynomial multiplication algorithm with modulo 2 and XOR operations considering the irreducible polynomial x4 x 1 Look up the multiplicative inverses from the table if needed Division Divide the first element by the second element multiplicative inverse ... View full answer
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