Mathematical Finite Fields

Mathematical Finite Fields 00404

Solve the following problems in the final field F(31 below) - [3/24] , [17(-3 degree)] , [4(-4degree) x 11] . This Python code illustrates the essential concepts of finite fields and demonstrates operations such as division, exponentiation, and multiplication within a finite field. It uses the finite field F(31) and finds multiplicative inverses to perform these operations, providing a valuable resource for those learning or working with finite fields in mathematics and cryptography.

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[Category] <-------------------------------------------> [Blockchain Creatione]
[Creator] <-----------------------------------------------------> [InNotSystem13]
[Project] <----------------------------------------------> [Mathematical Finite Fields]
[Identifier] <--------------------------------------------------------> [00404]
[Contract] <------------> [0x86E926fEB6e2Fe2562fe9C558347a4b785f60c8C]
[Chain] <---------------------------------------------------------> [BSC Chain]
[Type] <----------------------------------------------------> [Application]

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    # Function to find the multiplicative inverse in the finite field F(31)
    def find_inverse(a, prime):
        for x in range(1, prime):
            if (a * x) % prime == 1:
                return x

    # Prime modulus of the finite field
    prime = 31

    # Given numbers
    numerator = 3
    denominator = 24

    # Finding the multiplicative inverse for the denominator (24)
    inverse_denominator = find_inverse(denominator, prime)

    # Performing division
    result1 = (numerator * inverse_denominator) % prime

    # Given numbers
    base = 17
    exponent = -3

    # Finding the multiplicative inverse for the base (17)
    inverse_base = find_inverse(base, prime)

    # Calculating (-3) exponent as the reciprocal exponent
    result2 = (inverse_base ** abs(exponent)) % prime

    # Given numbers
    base = 4
    exponent = -4
    multiplier = 11

    # Finding the multiplicative inverse for the base (4)
    inverse_base = find_inverse(base, prime)

    # Calculating (-4) exponent as the reciprocal exponent
    result3 = (inverse_base ** abs(exponent)) % prime

    # Performing multiplication
    result3 = (result3 * multiplier) % prime

    print("Division result:", result1)
    print("Result of (-3) exponent:", result2)
    print("Result of 4^(-4) * 11:", result3)

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Author's message: Explore the world of finite fields with this Python code. It delves into the intricacies of finite field arithmetic, including finding multiplicative inverses and performing arithmetic operations within the context of F(31). Whether you're a math enthusiast or a cryptography enthusiast, this code provides insights into the mathematical foundations of finite fields.