By Henri Darmon (auth.), Guillaume Hanrot, François Morain, Emmanuel Thomé (eds.)
This booklet constitutes the refereed court cases of the ninth overseas Algorithmic quantity thought Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised complete papers offered including five invited papers have been conscientiously reviewed and chosen for inclusion within the publication. The papers are dedicated to algorithmic points of quantity idea, together with user-friendly quantity concept, algebraic quantity thought, analytic quantity thought, geometry of numbers, algebraic geometry, finite fields, and cryptography.
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Additional info for Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010. Proceedings
The decryption algorithm lifts the ciphertext in Fp and enumerate all the reduced forms equivalent to ϕÔfc Õ, looking for the pattern. Of course, the knowledge of q is needed to compute ϕ. There are only OÔlogÔpÕÕ of them. It will necessarily ﬁnd it, because the (unknown) lift of fs fc is an equivalent form ϕÔfs Õ Ôa, ¦, ¦Õ, whose normalization Ôa, ¦, ¦Õ is reduced due to the small size of a, and it satisﬁes the pattern by construction. Due to the small number of reduced forms, it is likely the only one of the small reduced cycle to satisfy the pattern, and the plaintext m is eventually extracted from a.
In other words, hÔf Õ is the shortest normalization of f . T Ôνf Õ is classically-normalized, νf being one of the integers h¡ f , hf but not necessarily the one with the smallest absolute value. Our reduction algorithm, is a variant of the Gauss reduction which operates in GL2 ÔZÕ. It alternates exchange E and the shortest normalization T ÔhÔf ÕÕ at each loop, and terminates on a largely-reduced form. As we will see later, any kind 1 of normalization by h¡ f or hf would make a reduction algorithm terminate , but the choice of the shortest normalization hÔf Õ instead of the classical νf (especially during the last steps) is the key element to minimize the reduction matrix.
Extend φK to a Frobenius lift on A† by setting φ(x) = xp , φ(y) = y p 1 + = yp ∞ i=0 φK (f )(xp ) − f (x)p f (x)p 1/2 1/2 (φK (f )(xp ) − f (x)p )i , i y 2pi noting the series converges in A† because φK (f )(xp ) − f (x)p has positive valuation. (This choice of φ(y) ensures that φ(y)2 = φ(f (x)), so that the action on A† is well-deﬁned. 2. Use a Newton iteration to compute y/φ(y). Then for i = 0, . . 1 to write φ∗ (ωi ) = pxpi+p−1 y dx = dfi + φ(y) 2y 2g−1 Mij ωj (5) j=0 for some fi ∈ A† and some 2g × 2g matrix M over K.