The five men "should not roam scot-free as they are dangerous to society", she told the BBC.
Police have arrested three of the five accused.
The 21-year-old student had been pursuing a case in court against the men, when she was attacked and left for dead in Haryana state last week.
She said she was forced inside a car and the men tried to strangle her. A passerby later took her to hospital.
The woman, who is from a poor low-caste Dalit (formerly known as untouchables) family, was first assaulted in 2013 in Bhiwani town.
Her family has alleged that they had moved to Rohtak after they were threatened by the accused who have been out on bail.
Speaking to the BBC from her hospital bed, where she is recovering from her injuries, the woman said the five were the same men who had gang raped her in 2013.
"These people gang-raped me in 2013. This time too, they're the accused," she said.
"I know them very well. I only want justice. They should be hanged.  It's not just only about me or anybody else's daughter. These people are dangerous," she said.
Though she could not speak much due to health complications, her cousin said the victim's family "do not have faith" in the local police.
"It took police seven days to arrest three of the five men identified by my sister. The accused were all roaming about here. We pleaded before the police. But their bias was obvious. We are being victimised because we are from a lower caste," he told the BBC.
He said his family embraced Buddhism last year after being fed up of "caste discrimination".
The victim's family has also alleged that the accused offered them money to withdraw the 2013 gang rape case. "When we refused the money, we were threatened," he said.
The arrested men have been remanded to custody by a local court.
The state government has also announced setting up of a Special Investigation Team (SIT) to probe the incident. The team is led by two senior police officials and includes forensic experts.
Police official Mohammed Akil said the accused have handed over some CCTV footage claiming that they were at different places when the incident took place.
"The matter is complicated and there are many aspects that have to be taken into consideration during the investigation. The SIT will do its job," Mr Akil told the BBC.
The DNA samples of the accused were being sent for examination, he said.
Family members of the arrested men, however, alleged that they were being "falsely implicated" in the case.
They accused the victim's family of "demanding money" to withdraw the case.
Garima, wife of one of the accused, said her husband Sandeep was in a different city on the day the incident was said to have taken place.
Harbans, brother of accused Jagmohan, told the BBC that they had handed over evidence to the police to prove that his brother was "several kilometres away from the place of the incident".
Police officials say that the matter has become complicated and only a thorough investigation will establish what actually happened.
Nearly four years ago, a 23-year-old woman was brutally raped by multiple men and murdered on a bus in Delhi, causing global outrage.
Tougher laws on sexual violence were introduced the year after, including a new death sentence if a person was convicted of rape a second time.
But correspondents in India say questions are being asked about how five men who were accused of a serious sexual crime could have been free to conduct another attack.
A woman in India who was allegedly gang raped for the second time in three years by the same men says she wants "stringent action" against the accused.