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  2. Montgomery modular multiplication - Wikipedia

    en.wikipedia.org/wiki/Montgomery_modular...

    When R is a power of a small positive integer b, N′ can be computed by Hensel's lemma: The inverse of N modulo b is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of b, stopping when the inverse modulo R is known; N′ is ...

  3. Modular multiplicative inverse - Wikipedia

    en.wikipedia.org/wiki/Modular_multiplicative_inverse

    A modular multiplicative inverse of a modulo m can be found by using the extended Euclidean algorithm. The Euclidean algorithm determines the greatest common divisor (gcd) of two integers, say a and m. If a has a multiplicative inverse modulo m, this gcd must be 1. The last of several equations produced by the algorithm may be solved for this gcd.

  4. Berlekamp–Massey algorithm - Wikipedia

    en.wikipedia.org/wiki/Berlekamp–Massey_algorithm

    The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse. [1] Reeds and Sloane offer an extension to handle a ring. [2] Elwyn Berlekamp invented an algorithm for decoding Bose–Chaudhuri–Hocquenghem (BCH) codes.

  5. Extended Euclidean algorithm - Wikipedia

    en.wikipedia.org/wiki/Extended_Euclidean_algorithm

    Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a ...

  6. Multiplicative inverse - Wikipedia

    en.wikipedia.org/wiki/Multiplicative_inverse

    The graph forms a rectangular hyperbola. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/ x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a / b is b / a. For the multiplicative inverse of a real number, divide 1 by the number.

  7. Quadratic reciprocity - Wikipedia

    en.wikipedia.org/wiki/Quadratic_reciprocity

    The observations about −3 and 5 continue to hold: −7 is a residue modulo p if and only if p is a residue modulo 7, −11 is a residue modulo p if and only if p is a residue modulo 11, 13 is a residue (mod p) if and only if p is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which ...

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  9. Eisenstein's criterion - Wikipedia

    en.wikipedia.org/wiki/Eisenstein's_criterion

    For example, in the case x 2 + x + 2 given above, the discriminant is −7 so that 7 is the only prime that has a chance of making it satisfy the criterion. Modulo 7, it becomes (x − 3) 2 — a repeated root is inevitable, since the discriminant is 0 mod 7. Therefore the variable shift is actually something predictable.

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