Reveling in their opportunity to vote in a post-Mubarak era, Egyptians headed to the polls Tuesday for the second day of the country’s parliamentary elections.

The voting this week marks the first time some Egyptians — young and old — have ever cast ballots. Citizens are picking members of the lower house of Parliament, which will be tasked with drafting a new constitution.

Egypt’s stock exchange opened considerably higher Tuesday and saw an unprecedented wave of buying amid the elections.

The head of Egypt’s election committee promised a smoother voting process Tuesday after logistical problems and illegal campaigning affected polling in some areas Monday. Voters decried the late opening of polling stations and a delay in the arrival of ballots.

State TV reported that 25 people were injured in election-related violence.

Election officials said they have received 964 complaints, 579 of which have been addressed, according to Egypt TV.

Activist Hafez Abu Saeeda with the Egyptian Organization for Human Rights said on Twitter he was in Al-Khaleefa “working on an election tour, and an attack took place on our supporters before I arrived.” The Army was able to control the situation immediately, he said. One person was injured.

But many voters expressed jubilation at their chance to help build a new Egypt after a popular revolt toppled President Hosni Mubarak’s 30-year regime in February.

“Before, there was always cheating. Now — I could be wrong — but I think my vote will count,” Mohamed Rida’a Mohamed Abdulla said as he left a Cairo polling station.

Some polling areas were segregated by gender. Lines at both men’s and women’s stations snaked around buildings for hours.

“It’s an awakening,” one woman said, beaming, at a Cairo polling station. “I’m very happy, and I feel that even when I see old ladies hardly walking, it makes me feel that really Egypt is reviving.”

The stakes are high for Egyptian women, who worry that if Islamists gain a majority in the lower house of Parliament, their hopes for a more liberal life will be quashed.

In Alexandria, the Al Noor Salafi Muslim party and the Freedom and Justice Party accused one another of breaking an “honorable agreement” aimed at cooperation. The Freedom and Justice Party is part of the once-banned Muslim Brotherhood, one of the nation’s largest organizations.

Yousri Hamad, a spokesman for the Al Noor Salafi party, said the Muslim Brotherhood spread false rumors and launched a “smear campaign” against the party.

“We were not as prepared for the elections as we should have been and did not spend enough money on the campaigning,” Hamad said.

But Esam El Erian, spokesman for the Freedom and Justice Party, said the Salafis had breached the agreement by making such accusations without evidence.

Ali Al Dali, an official monitor for the Egyptian Association of Human Rights, said eight cases of vote-buying had been documented in Alexandria, and police had been notified. About 45 percent of eligible voters in the city had cast ballots, he said.

Elections for the lower house are scheduled to take place in three stages, based on geography. The last of the three stages is set to take place in January.

Upper house elections will run between January and March.

Egyptians have dozens of political parties and thousands of independent candidates from which to choose. The Muslim Brotherhood is expected to perform well in the election.

Asked what he would say to those, be they secular or liberal, who might have concerns about the Muslim Brotherhood, Issam Al-Aryan, a senior leader of the organization, said, “They are our friends, our neighbors, our citizens, they are Egyptians as we, and they have the same rights and duties … if they oppose us, they are participating in building this country, and correcting our mistakes, if we commit a mistake … this is very important for a democratic system.”

The U.S. State Department said Monday it would welcome the Muslim Brotherhood’s participation in the political process. Asked for his reaction on the U.S. stance, Al-Aryan said, “We are going to have a mutual relationship with countries all over the world, including the [United] States, and we need also respect of our cultures, of our civilization, of our beliefs, respect of our will and choices, approval of the results and of course we need balance of relationships in our region.”

Presidential elections will be held by June, according to the Supreme Council of the Armed Forces, Egypt’s acting ruling body. Military leaders have said they will hand over power to a new government when one is elected, but many Egyptians say they don’t trust the council and fear the military will cling to power.

Over the past two weeks, at least 42 people have been killed in clashes as protesters called for an immediate end to military rule. An additional 3,250 have been wounded, according to the Health Ministry.

Some Egyptians expressed skepticism or even boycotted the voting on Monday.

“There is no inclination that the judiciary is independent, so there is no way to prove the election will be free and fair,” said Amr Hamzawy, a 32-year-old shopkeeper.

Despite the masses who flocked to polling stations Monday, some remained cautious about how much Egypt could evolve after one election cycle.

“We’re not changing in one month, or a year, or five years. It will take a long time to change from one system to the other,” one woman said on the streets of Cairo. “We’ve been going with this system for the past 30 years, and it’s not like a button we push to change everything.”

CNN’s Ivan Watson, Leone Lakhani and Jim Clancy and journalists Mohamed Fadel Fahmy and Ian Lee contributed to this report.