Fast Triangularization of Ideal Latttice Basis
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摘要:
为了提高理想格上格基的三角化算法的效率，该文通过研究理想格上的多项式结构提出了一个理想格上格基的快速三角化算法，其时间复杂度为O(n3log2B)，其中n是格基的维数，B是格基的无穷范数。基于该算法，可以得到一个计算理想格上格基Smith标准型的确定算法，且其时间复杂度也比现有的算法要快。更进一步，对于密码学中经常所使用的一类特殊的理想格，可以用更快的算法将三角化矩阵转化为格基的Hermite标准型。
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关键词:
- 理想格 /
- Hermite标准型 /
- Smith标准型 /
- 三角化
Abstract:To improve the efficiency of the triangularization of ideal lattice basis, a fast algorithm for triangularizing an ideal lattice basis is proposed by studying the polynomial structure, which runs in time O(n3log2B), where n is the dimension of the lattice, B is the infinity norm of lattice basis. Based on the algorithm, a deterministic algorithm for computing the Smith Normal Form (SNF) of ideal lattice is given, which has the same time complexity and thus is faster than any previously known algorithms. Moreover, for a special class of ideal lattices, a method to transform such triangular bases into Hermite Normal Form (HNF) faster than previous algorithms will be present.
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Key words:
- Ideal lattice /
- Hermite Normal Form (HNF) /
- Smith Normal Form (SNF) /
- Triangularization
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表 1 本原格向量序列
输入：$ {\mathbb{Z}}\left[ {{x}} \right]$中$ f\left( x \right)$和$ g\left( x \right)$，次数分别为$ n$和$ m$，且$ n>m$； (1) 利用扩展欧几里得算法计算$ {\mathbb{Q}}[x]$上$ {r_i}^{'}\left( x \right)$, $ {s_i}^{'}\left( x \right)$, $ {t_i}^{'}\left( x \right)$，使得$ {r_i}^{'}\left( x \right) = {r_{i - 2}}^{'}\left( x \right) + {q_i}^{'}\left( x \right){r_{i - 1}}^{'}\left( x \right)$ 和${r_i}^{'}\left( x \right) = {s_i}^{'}\left( x \right)f\left( x \right) + $$ {t_i}^{'}\left( x \right)g\left( x \right)$成立，这里 $ i = 1,2, ··· ,l$； (2) 计算每一个$ {t_i}^{'}\left( x \right)$系数分母的最小公倍数$ {C_i}$, $ i = 1,2, ··· ,l$； (3) 令$ {r_i}\left( x \right) = {r_i}^{'}\left( x \right) \cdot {C_i}$为余式序列中第$ i$个余式， $ i = 1,2, ··· ,l$； 输出：$ {r_1}\left( x \right)$, $ {r_2}\left( x \right)$, ···, $ {r_l}\left( x \right)$ 
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表 2 理想格上格基的三角化
输入：本原格向量序列，$ {r_0}\left( x \right)$, $ {r_1}\left( x \right)$, ···, $ {r_l}\left( x \right)$(向量形式为$ {{{r}}_{\bf 0}}$,
$ {{{r}}_{\bf 1}}$, ···, $ {{{r}}_{ l}}$)(1) 令$ {{T}} \leftarrow {0^{n \times n}}$ (2) 如果$ k \in {I_l},{{ T}_k}\left( x \right) = {r_l}\left( x \right){x^{n - k}},i \leftarrow l - 1$ (3) 如果$ k \in {I_i}$， (a) 计算$ \phi $和ψ使得
$ \phi {\rm{lc}}\left( {{r_i}} \right) + \psi {\rm{lc}}\left( {{{{T}}_{n - {n_{i + 1}}}}} \right) = {\rm{gcd}}\left( {\rm{lc}}\left( {{{{T}}_{n - {n_{i + 1}}}}} \right),\right.$
$\left.{\rm{lc}}\left( {{r_i}} \right) \right) $(b) 令$ {{{T}}_{n - {n_i}}}\left( {{x}} \right) = \phi {r_i}\left( x \right) + \psi {{{T}}_{n - {n_{i + 1}}}}\left( x \right){x^{{\delta _i}}}$ (c) 如果$ {\rm{lc}}\left( {{{{T}}_{n - {n_i}}}} \right) = 1$，则令
$ {{{T}}_j}\left( x \right) = {{{T}}_{n - {n_i}}}\left( x \right){x^{n - {n_i} - j}}$,
$ j = 1,2, ··· ,n - {n_i}$，并结束循环(d) 否则$ {{{T}}_k}\left( x \right) = {{{T}}_{n - {n_i}}}\left( x \right){x^{n - {n_i} - k}}$, $ i \leftarrow i - 1$ (e) 如果$ i>0$，到(3)开始循环，否则结束循环 输出：$ {{T}}$
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