On the reduction of a random basis

For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra & Lov\'asz}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r\_j^{(n)}-1)\_j$ tends in distribution in some sense to an explicit process $({\mathcal R}\_j -1)\_j$; some properties of this latter are provided

Study of Fully Homomorphic Encryption over Integers

Fully homomorphic encryption has long been regarded as an open problem of cryptography. The method of constructing first fully homomorphic encryption scheme by Gentry is complicate so that it has been considered difficult to understand. This paper explains the idea of constructing fully homomorphic encryption and presents a general framework from various scheme of fully homomorphic encryption. Specially, this general framework can show some possible ways to construct fully homomorphic encryption. We then analyze the procedure how to obtaining fully homomorphic encryption over the integers. The analysis of recrypt procedure show the growth of noise, and the bound of noise in recrypt procedure is given. Finally, we describe the steps of implementation.

Finding a closest point in a lattice of Voronoi's first kind

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network

Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any $\gamma \leq 1 + O(\log n/n)$, we obtain an efficient dimension-preserving reduction from $\gamma^{O(n/\log n)}$-SVP to $\gamma$-GapSVP and an efficient dimension-preserving reduction from $\gamma^{O(n)}$-CVP to $\gamma$-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when $\gamma = 1$. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction---we reduce $\gamma^{O(n/\log n)}$-SVP to $\gamma$-unique SVP, a potentially easier problem than $\gamma$-GapSVP.Comment: Updated to acknowledge additional prior wor

ANALISIS VITAMIN C, SIFAT FISIK, DAN SIFAT ORGANOLEPTIK TEMPE BERBAHAN DASAR KEDELAI KUNING (Glycine max L), KEDELAI HIJAU (Glycin Max (L) Merrill), KEDELAI HITAM (Glycine soja (L) Merrit)

Tempe merupakan salah satu makanan sumber protein bagi masyarakat Indonesia. Hal ini disebabkan karena tempe dibuat dari sumber protein nabati yaitu kedelai. Secara umum, jenis kedelai yang sering digunakan sebagai bahan baku adalah kedelai kuning (Glycine max L.), namun karena adanya peningkatan kebutuhan bahan baku, diperlukan sumber kedelai lain sebagai alternatif. Oleh karena itu, penelitian ini bertujuan untuk mengetahui kadar vitamin C, sifat fisik dan sifat organoleptic tempe yang berbahan dasar kedelai hijau (edamame) (Glycine max (L) Merrill) dan kedelai hitam (Glycine soja (L) Merrit) yang dibandingkan dengan tempe berbahan dasar kedelai kuning (Glycine max L.). dengan menggunakan jamur Rhizopus sp sebagai inokulum. Jenis penelitian yang digunakan adalah eksperimen murni (true experiment) dengan menggunakan rancangan acak kelompok dengan 3 sampel. Kedelai yang dianalisis adalah kacang kedelai kuning (Glycine max L), kedelai hijau (edamame) (Glycine max (L) Merrill), dan kedelai hitam (Glycine soja (L) Merrit).      Berdasarkan uji vitamin C dengan menggunakan iodine, warna ketiga tempe berubah menjadi lebih terang atau jernih yang menunjukkan tempe tersebut mengandung vitamin C. Sifat fisik ketiga tempe memiliki pertumbuhan miselium yang sama atau seragam. Pada uji organoleptik, ketiga tempe memiliki warna yang hampir sama yaitu putih dan tekstur padat. Rasa pada tempe kedelai kuning yaitu khas tempe pada umumnya, kedelai hijau  (edamame) terasa manis, dan kedelai hitam sedikit pahit. Pada tempe kedelai kuning dan hijau (edamame) memiliki aroma khas tempe, sedangkan tempe kedelai hitam memiliki aroma khas tempe yang lebih menyengat

Interpolation and Approximation of Polynomials in Finite Fields over a Short Interval from Noisy Values

Motivated by a recently introduced HIMMO key distribution scheme, we consider a modification of the noisy polynomial interpolation problem of recovering an unknown polynomial $f(X) \in Z[X]$ from approximate values of the residues of $f(t)$ modulo a prime $p$ at polynomially many points $t$ taken from a short interval