  The invariant mass spectra of $\phi \to K^{+}K^{-}$ are measured in 12 GeV
$p+A$ reactions in order to search for the in-medium modification of $\phi$
mesons. The observed $K^{+}K^{-}$ spectra are well reproduced by the
relativistic Breit-Wigner function with a combinatorial background shape in
three $\beta\gamma$ regions between 1.0 and 3.5. The nuclear mass-number
dependence of the yields of the $K^{+}K^{-}$ decay channel is compared to the
simultaneously measured $e^{+}e^{-}$ decay channel for carbon and copper
targets. We parameterize the production yields as $\sigma (A) = \sigma_0 \times
A^\alpha$ and obtain $\alpha_{\phi\to K^+K^-} - \alpha_{\phi\to e^+e^-}$ to be
0.14 $\pm$ 0.12. Limits are obtained for the partial decay widths of $\phi$
mesons in nuclear matter.
