The Ohio Republican Party spearheaded a lawsuit Friday over a directive from the office of Democratic Secretary of State Jennifer Brunner that would allow some early voters to register and vote on the same day.

The suit, filed by two Ohio voters in the Supreme Court of Ohio, in Columbus, ramps up the battle over voting procedures in a critical swing state with 20 electoral votes. But the parties' roles are reversed from the 2004 election. This time, a Democrat is setting the rules, and the state Republican Party is charging...