They also said the Muslim Brotherhood is paying gangs to beat up men who are taking part in the latest round of protests, which followed a decree by President Morsi to give himself sweeping new powers.
It comes as the Muslim Brotherhood co-ordinated a demonstration today in support of President Mohamed Morsi, who is rushing through a constitution to try to defuse opposition fury over his newly expanded powers.
Just 24 hours earlier around 200,000 people gathered in Tahrir Square, the heart of last year's revolution which toppled President Hosni Mubarak, yesterday to protest against a new draft constitution.
Large marches from around Cairo flowed into the square, chanting 'Constitution: Void!' and The people want to bring down the regime.'
But amid the calls for democracy a sinister threat has emerged.
Magda Adly, the director of the Nadeem Centre for Human Rights, said that under Mubarak, the Government paid thugs to beat male protestors and sexually assault women.
'This is still happening now,' she told The Times. 'I believe thugs are being paid money to do this ... the Muslim Brotherhood have the same political approaches as Mubarak,' she said.
One protestor, Yasmine, told the newspaper how she had been in the square filming the demonstrations for a few hours when the crowd suddenly turned.
Before she knew what was happening, about 50 men had surrounded her and began grabbing her breasts. She said they ripped off her clothes, starting with her headscarf and for nearly an hour, indecently assaulted her with their hands.
A few men tried to help her but they were beaten away. Eventually some residents who had seen the attack from their windows came to her aid and an elderly couple pulled her into their home. She suffered internal injuries and was unable to walk for a week.
Four of Yasmine's friends were also sexually assaulted in the square that day, in the summer.
Afaf el-Sayed, a journalist and activist, told the newspaper she was assaulted by a group of men while protesting in Tahrir Square just over a month ago and she was sure her attackers were 'thugs from the Muslim Brotherhood'.
In February 2011 the correspondent for the American network CBS, Lara Logan, endured a half-hour sexual assault in Tahrir Square by a group of men. She said after the ordeal that she had been 'raped with their hands'.
While the exact frequency of these attacks is unknown, activists have reported nearly 20 attacks in the last ten days and say there has been a dramatic increase in mob sex attacks on protestors in the last year.
Most attacks take place in one particular corner of the square, at roughly the same time every evening, and usually starts with a group of men forming a human chain around women as if to protect them.
Yasmine said she was almost sure the assault was planned. She managed to throw her camera to a friend and was able to watch the footage later. She told The Times: 'Just before the attack it looks like men are getting into position. They look like they're up to something, they don't look like random protestors.'
The newspaper spoke to two men who admitted they were paid to target female protestors. Victor and Tutu, both in their thirties, said they operate in a group of around 65 local men and got paid between £10 and £20 a time. But they would not reveal who pays them.
'We're told to go out and sexually harass girls so they leave the demonstration,' Victor told The Times. He said the aim was to cause disruption and instil fear in protesters. He said members of the public sometimes joined in.
Protestors in Tahrir Square yesterday angrily vowed to bring down a draft constitution approved by allies of President Morsi.
The protests have highlighted an increasingly united opposition leadership of prominent liberal and secular politicians trying to direct public anger against Morsi and the Islamists - a contrast to the leaderless youth uprising last year which toppled autocrat Hosni Mubarak.
Figures from a new leadership coalition took the stage to address the crowds. The coalition, known as the National Salvation Front, includes prominent democracy advocate Nobel Peace laureate Mohamed ElBaradei, leftist Hamdeen Sabbahi and former Arab League chief Amr Moussa.
'We are determined to continue with all peaceful means, whatever it takes to defend our legitimate rights,' ElBaradei told the crowd. He later posted on Twitter that Morsi and his allies are "staging a coup against democracy" and that the regime's legitimacy 'is eroding'.
Sabbahi vowed protests would go on until 'we topple the constitution'.
The opposition announced plans for an intensified street campaign of protests and civil disobedience and even a possible march on Morsi's presidential palace to prevent him from calling a nationwide referendum on the draft, which it must pass to come into effect. Top judges announced Friday they may refuse to monitor any referendum, rendering it invalid.
If a referendum is called, 'we will go to him at the palace and topple him,' insisted one protester, Yasser Said, a businessman who said he voted for Morsi in last summer's presidential election.
Islamists, however, are gearing up as well. The Muslim Brotherhood drummed up supporters for its own mass rally today and boasted the turnout would show that the public supports Morsi's efforts to push through a constitution.
Brotherhood activists in several cities handed out fliers calling for people to come out and "support Islamic law". A number of Muslim clerics in Friday sermons in the southern city of Assiut called the president's opponents "enemies of God and Islam".
The week-long unrest has already seen clashes between Islamists and the opposition that left two dead and hundreds injured. On Friday, Morsi opponents and supporters rained stones and firebombs on each other in the cities of Alexandria and Luxor.
The Islamist-led assembly that worked on the draft for months passed it in a rushed, 16-hour session that lasted until sunrise on Friday.
The vote was abruptly moved up to pass the draft before Egypt's Constitutional Court rules on Sunday whether to dissolve the assembly. Liberal, secular and Christian members and secular members had already quit the council to protest what they call Islamists' hijacking of the process.
The draft was to be sent to Morsi today to decide on a date for a referendum, possibly in mid-December.
The draft has a distinctive Islamic bent - enough to worry many that civil liberties could be restricted, though its provisions for enforcing Sharia, or Islamic law, are not as firm as ultra-conservatives wished.
Protests were first sparked when Morsi last week issued decrees granting himself sweeping powers that neutralized the judiciary. Morsi said the move was needed to stop the courts - where anti-Islamist or Mubarak-era judges hold many powerful posts - from dissolving the assembly and further delaying Egypt's transition.
Opponents, however, accused Morsi of grabbing near-dictatorial powers by sidelining the one branch of government he doesn't control.