ORLANDO, Fla. — Maria Rubin is one of the coveted independent voters in this swing state — so independent that she will not say whether she is voting for President Obama or Mitt Romney. She does share her age (63) and, more quickly, her opinion on Medicare: “I’m not in favor of changing it, or eliminating it.”

Her attitude speaks directly to one of the biggest challenges facing the Republican ticket this year: countering the Democrats’ longstanding advantage as the party more trusted to deal with Medicare.

In the 2010 Congressional races, successful Republicans believed that they had finally found a way to do that, by linking the program’s future to Mr. Obama’s unpopular health insurance overhaul and accusing Democrats of cutting Medicare to pay for it. This summer Mr. Romney resumed the offensive, eventually joined by his running mate, Representative Paul D. Ryan.

Initially, polls suggested that the Republican strategy was working. Democrats fretted that Mr. Romney would win the retiree-heavy Florida and increase his support nationwide among older voters, who lean Republican anyway. David Winston, a Republican pollster, wrote a month ago of “a structural shift in the issue” that left the parties in “a dead heat” and Mr. Obama unable to mount an effective response.