## Minimal redundant digit expansions in the Gaussian integers.(English)Zbl 1076.11005

Let $$q\geq 2$$ be an integer. There are infinitely many ways to represent a given integer $$n\in {\mathbb N}$$ in the form $n=\sum_{j=0}^l \varepsilon_j q^j,\quad l \in {\mathbb N},\;\varepsilon_j\in {\mathbb Z}.$ In cryptography, it is sometimes interesting to find a representation with a minimal ‘cost’ (that is, such that $$\sum_{j=0}^l| \varepsilon_j|$$ is minimal).
C. Heuberger and H. Prodinger [Computing 66, No. 4, 377–393 (2001; Zbl 1030.11003)] have shown that knowing the digits $$\eta_0,\eta_1$$ of the usual $$q$$-ary expansion $$n=\sum_{j=0}^L \eta_jq^j$$ ($$0\leq \eta_j<q$$) is enough to decide what digit $$\varepsilon_0$$ should be taken to obtain such a minimal representation of $$n$$. This provides an easy algorithm to produce a representation of minimal cost for a given integer $$n$$.
In the paper under review, the corresponding problem is investigated for so-called canonical number systems in the ring of Gaussian integers. The main result shows that the situation becomes very different. Indeed, given an arbitrary positive integer $$L$$, the author constructs two Gaussian integers $$\alpha,\alpha'$$ with the following properties: The ‘usual’ representations of $$\alpha$$ and $$\alpha'$$ coincide in the first $$L$$ digits, while for all pairs of strings $$(\varepsilon_0,\dots \varepsilon_s)$$ and $$(\varepsilon'_0,\dots \varepsilon'_{s'})$$ corresponding to minimal cost representations of $$\alpha$$ and $$\alpha'$$, respectively, we have $$\varepsilon_0 \not=\varepsilon'_0$$.
This proves that there is no algorithm, in the vein of the one mentioned above, for finding the minimal representation for canonical number systems in Gaussian integers.

### MSC:

 11A63 Radix representation; digital problems 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

### Keywords:

minimal redundant expansions; Gaussian integers

Zbl 1030.11003
Full Text:

### References:

  Heuberger, C., Prodinger, H., On minimal expansions in redundant number systems: Algorithms and quantitative analysis. Computing66 (2001), 377-393. · Zbl 1030.11003  Kátai, I., Szabó, J., Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975), 255-260. · Zbl 0309.12001  Knuth, D.E., Seminumerical algorithms, third ed. The Art of Computer Programming, vol. 2, Addison-Wesley, 1998. · Zbl 0895.65001  Kovács, B., Pethö, A., Number systems in integral domains, especially in orders of algebraic number fields. Acta Sci. Math.(Szeged) 55 (1991), 287-299. · Zbl 0760.11002
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