Fault-tolerant modular reconstruction of rational numbers

Abstract

In this paper we present two efficient methods for reconstructing a rational number from several residue-modulus pairs, some of which may be incorrect. One method is a natural generalization of that presented by Wang, Guy and Davenport in [WGD1982] (for reconstructing a rational number from correct modular images), and also of an algorithm presented by Abbott in [Abb1991] for reconstructing an integer value from several residue-modulus pairs, some of which may be incorrect. We compare our heuristic method with that of Böhm, Decker, Fieker and Pfister [BDFP2012].

Ideals Modulo a Prime

“…Remark 4. 1. We observe that "σ-precedes" is just the "σ-lexicographical" ordering on the σ-ordered tuples (T, x ∞ ) where T is a σ-ordered tuple of distinct powerproducts, and x ∞ is σ-greater than any power-product.…”

“…Remark 3. 1. In [6] the theory of minimal strong Gröbner bases is fully developed, in particular it is stated that every non-zero ideal in Z[x 1 , .…”

“…Two main interrelated obstacles to the success of this kind of approach are the existence of bad, good and lucky primes and the difficulty of reconstructing the correct rational coefficients possibly in the presence of undetected bad primes. We refer to [1] for a discussion of the second problem and to [4] and [3] for some new results in this direction and applications to the problem of the implicitization of hypersurfaces and of the computation of minimal polynomials.…”

Resolving Zero Divisors Using Hensel Lifting

“…Our technique handles both the expected zero-divisors (such as z 1 + 12z 2 (mod 13) in the above example) and the unexpected zero-divisors (such as z 1 + z 2 (mod 17)). A different approach that we tried is Abbott's fault tolerant rational reconstruction as described in [1]; although this is effective, we prefer Hensel lifting as it enables us to split the triangular set immediately thus saving work.…”

Gröbner bases for everyone with CoCoA-5 and CoCoALib

“…See Section 5 for an example of how to ask CoCoA to compute a Gröbner basis with a term-ordering given by a matrix. [3,7,7], [3,6,8], [6,7,14], [6,5,15], [0, 0, -1]]) [c+b^2+a^2-1, -b^6-3*a^2*b^4-3*a^4*b^2+b^5+3*b^4+6*a^2*b^2+3*a^4-3*b^2-3*a^2, a^5+b^4+2*a^2*b^2+a^4+b^3-2*b^2-2*a^2]…”