* Euro rebounds from four-month low vs dollar * Cypriot banks open amidst tight capital control measures * Italy's funding costs rise as political worries persist * Yen slide loses downward momentum before BOJ meeting

By Daniel Bases and Julie Haviv

NEW YORK, March 28 (Reuters) - The euro rose against the dollar on Thursday, a day after hitting a four-month low, but analysts called the rebound tenuous because investors remained nervous about the Cyprus crisis and Italy's political gridlock.

Analysts say German businesses worried about recovery though the economy is still seen growing in Q1 and the full year.

Under a new draft bill discussed in Parliament's finance committee Tuesday, deposits below €20,000 ($25,900) would be exempt from any charge.

Reflects country's credit rating downgrade and a political stalemate that shows no signs of easing.

The good news overshadowed a credit downgrade slapped on Italy due to the country's inconclusive election.

Ratings agency cites political uncertainty after last week's election, deep recession and rising debt.

Leaves door open to Fiat buyout.

Draghi says markets have settled quickly after Italy's fractured election results, suggesting central bank is in no rush to take any action. | Bank of England holds fire on more stimulus | Shares gain on U.S. jobs data, euro reises after ECB

Libya's defence ministry sends security personnel to secure the Mellitah complex, a joint venture between Libya's National Oil Corporation and Italy's Eni.

Treasury sells the maximum planned amount of €4 billion of the new 10-year bond, but yield it had to offer jumped to 4.83%.