Binary Quadratic Forms: An Algorithmic Approach (Algorithms - download pdf or read online

By Johannes Buchmann

ISBN-10: 3540463674

ISBN-13: 9783540463672

The booklet offers with algorithmic difficulties with regards to binary quadratic varieties, akin to discovering the representations of an integer by way of a kind with integer coefficients, discovering the minimal of a kind with genuine coefficients and determining equivalence of 2 types. for you to clear up these difficulties, the ebook introduces the reader to special parts of quantity thought resembling diophantine equations, aid concept of quadratic kinds, geometry of numbers and algebraic quantity concept. The ebook explains purposes to cryptography. It calls for merely simple mathematical wisdom.

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Additional info for Binary Quadratic Forms: An Algorithmic Approach (Algorithms and Computation in Mathematics)

Example text

15. We explain how kronecker calculates 17 3 = 2 3 =− 1 3 =− 0 1 17 3 . = −1. 16. Algorithm kronecker(m, n) takes time O size(m) size (n) to compute m n . Proof. 3 in [BS96]. 16 we also obtain an estimate for the running time of Algorithm numberOfPrimeForms. 17. Algorithm O size(∆) size(p) . 4 kronecker (m, n) Input: Integers m and n Output: m n if 2 | n and 2 | m then return 0. if sign(m) = sign(n) = −1 then j ← −1 else j ← 1. m ← |m|, n ← |n|. 4 Computing square roots modulo p In order to determine prime forms, algorithm primeForm must find a square root of a discriminant modulo a prime number.

Determine the proper automorphism group Aut+ (f ) and the Aut+ (f )-orbit Aut+ (f )(x, y) for each proper representation found in step 2. If (a, b, c) is an integral form, then in the first step it suffices to find all Γorbits of forms (n, B, C) that have the same content as f . In particular, if f is primitive, it suffices to find the primitive forms (n, B, C) of discriminant ∆. 4. We determine all proper representations of 1 by f = X 2 + Y 2 . The discriminant of f is ∆(f ) = −4. So we compute all Γ-orbits of forms 32 2 Equivalence of Forms (1, B, C) of discriminant −4.

From this form we obtain the form (52 , 31, 18). From this form we obtain the form (53 , 81, 26). Hence there are two Γ-orbits of forms (53 , b, c) of discriminant 61, namely (53 , ±81, 26)Γ. Now we apply the Chinese remainder theorem. We solve b ≡ 13 (mod 32 ) , b ≡ 81 (mod 53 ) , b ≡ 1 (mod 2) . We obtain the form (32 · 53 , 1831, 745). Next, we solve b ≡ −13 (mod 32 ) , b ≡ 81 (mod 53 ) , b≡1 (mod 2) . We obtain the form (32 · 53 , 581, 75). Hence F(61, 32 · 53 ) = {(32 · 53 , ±581, 75)Γ, (32 · 53 , ±1831, 745)Γ}.

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Binary Quadratic Forms: An Algorithmic Approach (Algorithms and Computation in Mathematics) by Johannes Buchmann


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