In this article, we formalize the definition of divisible Z-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible Z-modules are not finitely-generated. We introduce a divisible Z-module, equivalent to a vector space of a torsion-free Z-module with a coefficient ring Q. Z-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) b...

In this article, we formalize in Mizar [7] the definition of “torsion part” of Z-module and its properties. We show Z-module generated by the field of rational numbers as an example of torsion-free non free Z-modules. We also formalize the rank-nullity theorem over finite-rank free Z-modules (previously formalized in [1]). Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lo...

In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lovász) base reduction algorithm [20], cryptographic...

RSA cheating cryptosystems were first introduced by Anderson and developed by Young and Yung with their SETUP mechanism. Recently, Crépeau and Slakmon suggested very simple backdoors for this purpose. This paper sums up these methods. We also describe the LLL-reduction algorithm, which is useful to prove that Anderson’s trapdoor is insecure and to factor integers where the high bits of their pr...

In this article, we propose an adaptation of the algorithmic reduction theory lattices to binary codes. This includes celebrated LLL algorithm (Lenstra, Lenstra, Lovasz, 1982), as well adaptations associated algorithms such Nearest Plane Algorithm Babai (1986). Interestingly, codes can be interpreted version bound Griesmer (1960) on minimal distance a code. Using these algorithms, demonstrate—b...

The Lopsided Lovász Local Lemma (LLLL) is a probabilistic tool which cornerstone of the method combinatorics, shows that it possible to avoid collection "bad" events as long their probabilities and interdependencies are sufficiently small. strongest criterion can be stated in these terms due Shearer (1985), although technically difficult apply constructions combinatorics.
 
 original ...

For RSA, May showed a deterministic polynomial time equivalence of computing d to factoring N(= pq). On the other hand, Takagi showed a variant of RSA such that the decryption algorithm is faster than the standard RSA, where N = pq while ed = 1 mod (p−1)(q−1). In this paper, we show that a deterministic polynomial time equivalence also holds in this variant. The coefficient matrix T to which LL...

We present a system for the identification of syntax patterns describing interactions between genes and proteins in scientific text. The system uses sequence alignments applied to sentences annotated with interactions and syntactical information (part-of-speech), as well as finite state automata optimized with a genetic algorithm. Both methods identified syntactical patterns that are generaliza...

We show that the discrete set of pair amplitudes $A_m$ introduced by Haldane are an angular-momentum resolved generalization Tan two-body contact, which parametrizes universal short-range correlations in atomic quantum gases. The provide a complete description translation-invariant and rotation-invariant states lowest Landau level (LLL), both compressible incompressible. To leading nontrivial o...

The idea behind the SizeReduce(B) subroutine is, in the Gram-Schmidt decomposition B = B̃ ·U, to shift the entries in the upper triangle of U by integers (via unimodular transformations), so that they lie in [−12 , 1 2). Because changing an entry of U may affect the ones above it (but not below it) in the same column, we must make the changes upward in each column. Formally, the algorithm works ...