A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=a-b$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the difference size of $G$ and is denoted by $Delta[G]$. We prove that for every $ninmathbb N$ the cyclic group $C_n$ of order $n$ has difference size $frac{1+sqrt{4|n|-3}}2le Delta[C_n]lefrac32sqrt{n}$. If $nge 9$ (and $nge 2cdot 10^{15}$), then $Delta[C_n]lefrac{12}{sqrt{73}}sqrt{n}$ (and $Delta[C_n]<frac2{sqrt{3}}sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $le 100$.
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