Voting is in two stages, with some governorates voting Saturday and the rest on Dec. 22.
The new constitution has divided the country and sparked weeks of deadly confrontations between opponents and supporters of President Mohamed Morsi.
Presidential spokesperson Yasser Ali told CNN Morsi voted Saturday, denying a claim by "some opponents" that he voted outside his district, violating election laws. Ali said Morsi changed his place of residence after he was elected president.
Ahram Online said electoral violations were being reported at several polling places.
Sami Yassin of the National Human Rights Council, a monitor at polling stations in the Shubra district in Cairo, said the Salafist Nour Party is campaigning for a "yes" too close to polling stations in violation of the law.
Yassin said voting cards were not stamped in several other polling stations and that judges had to open ballot boxes to restamp them.
An elections observer was detained by bearded men in a polling station in Belbes for videotaping confrontations outside of a polling station, an activist told Ahram Online.
Ahram Online also said very few judges were available -- many judges said they would protest against the draft constitution -- resulting in long lines.
The Supreme Judicial Elections Commission at different times said between 7,000 and 10,000 judges were supervising the vote, Egypt Independent reported. A number of judges said they would boycott the poll to protest the draft constitution, which they said restricted human and civil rights. They also were angered over Morsi's edict that granted his actions immunity from judicial review, which he later rescinded.
The National Salvation Front, formed recently amid mass protests against Morsi and the draft constitution, reported a judge was telling voters to vote "yes" on their ballots at a polling station in Cairo, the Independent said.
Egypt Independent said several voters indicated they would vote "yes" because the country needed stability.
"I'm voting 'yes' so that the country moves forward," Galal Fahmy said.
Women were turning out in large numbers at several polling places in Cairo, the Independent said.
Fatemah Moussalem told the newspaper she voted "no" because she thought the draft constitution and its authors didn't represent all Egyptians. Opponents said the draft was written by Islamists at the exclusion of other opinions and suppressed rights of different groups, including women.
"It only represents them," she said, adding that she thought the draft was meant to ensure Morsi's Muslim Brotherhood remained in power.
The BBC said about 250,000 security forces were deployed and Morsi granted the army the power to arrest civilians.
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