The existing law says that presidential elections are valid only if 50 percent of the electorate vote. Parliamentary elections require a turnout of 25 percent.
The new bill, which needs the approval of the upper house of parliament and the president before becoming law, also abolishes advance voting and erects some barriers intended to prevent alleged extremists from running for public office.
This is the latest in a series of changes to Russia's election laws that have ended the direct election of regional leaders, barred voters from rejecting candidates, and raised the threshold for parties to enter parliament.
In comments addressed to leaders from the pro-Kremlin Unified Russia party on November 17, President Vladimir Putin warned parties and politicians against using extremist statements and actions to further their goals.
"There is no place for extremism on the Russian political scene," he said, warning that "otherwise we will simply not preserve the unity of our country."