France has seen its most extraordinary presidential election campaign in recent history. Beyond politics as usual, it points to a deep institutional crisis.

The French are going to the polls today to vote in the first round of the presidential elections. About a third still don't know who to vote for according to polls. And can you really blame them?

French presidential election campaigns normally produce two clear front runners – often from the main center-right and center-left parties. In 2012, the center-right candidate- Nicolas Sarkozy- was facing current Socialist President Francois Hollande. Admittedly, far-right National Front leader Jean-Marie Le Pen reached the second round of voting in 2002, but that had been undetected in the polls.

This time around though, four candidates could potentially reach the decisive run-off vote on May 7. The gaps between their projected tallies are so small that they lie within the margin of error. That's unheard of and hasn't occurred since the beginning of France's Fifth Republic in 1958.

"This is the first time that the media's projections published on the first day of voting will probably not give us the names of the two candidates that'll get into the second round – the vote will just be too close," said Nicolas Lebourg, political historian at Montpellier University.

"It's extraordinary - never has a presidential election been so chaotic," he added.

Full of surprises

Thursday night's terror attack just added to the confusion focusing the campaign on terrorism in its last stretch. Unemployment had been the main talking point up until then. During the attack, one policeman was killed and three other people wounded after a 39-year old French man opened fire on a police van on Paris' iconic Champs-Elysées boulevard. However, it doesn't seem to have given any of the candidates a huge edge according to the latest polls.

But the whole campaign has been full of surprises. To start off with, none of the winners of the Republican and Socialist parties' primaries were expected to come first.

Then came scandal for the conservatives. The Republican candidate, former Prime Minister Francois Fillon, had for months been the favorite to become France's next President. He's a social conservative and intends to get the country back on track with Thatcher-like radical economic reforms. He is also planning on repositioning the country internationally – by seeking closer ties with Russia and Syria.

But in January, scandals around alleged fake jobs for his family saw his poll numbers drop from 28 percent to about 18 percent. He is now competing with far-left candidate Jean-Luc Melenchon for third position.

Melenchon – recently and surprisingly so – zoomed upwards from ten percent, with Socialist candidate Benoit Hamon plummeting into single-figures.

The left-extremist Melenchon still thinks in terms of class warfare. A gifted orator, he wants to strengthen the French welfare state by increasing the minimum wage and bringing down weekly working hours – currently at 35. He also intends to renegotiate EU treaties or, if that fails, push for France to leave the EU altogether.

Another extremist is among the front runners – again a first in French history. Marine Le Pen from the far-right National Front has a very good chance of getting through to the decisive run-off vote – and this is reflected in the polls. She has proven popular with her recipe of anti-immigrant, economic protectionism and nationalistic rhetoric. But she has managed to smoothen the party's image by no longer making controversial statements like her father Jean-Marie. He was tarnished by charges of xenophobia and anti-Semitism.

Traditional parties in crisis

Le Pen's closest rival is independent centrist Emmanuel Macron. The former Economics Minister is pro-European and pro-business but also intends to maintain and strengthen France's welfare state. His movement "En Marche!" (On The Move!), founded only a year ago, skyrocketed in the polls. He's now in first place.

"The fact that a lightning party like 'En Marche!' can attract members from the Communists and the Republicans just shows to what extent the traditional parties are in a crisis," said Florence Faucher, Professor for Political Science at Paris University Sciences Po.

But historian Lebourg says it's not just the Socialist and the Republican Parties that are in a dire state – but the whole Fifth Republic: "This is a deep institutional crisis."

"Our Republican Monarchy was made for the agrarian France of 1958, when less then ten percent of the people under 50 had a degree – now that figure is at 38 percent."

"People want to be included in political decisions – and no longer be dictated to. Our institutions and very authoritarian and centralized political system are just not suitable any more."

Perhaps for this reason, many in this election campaign were trying to depict themselves as anti-system candidates – including Melenchon, Le Pen, and Macron.

No clear favorite

Lebourg says the current political system, based on two rounds of voting, only works with two frontrunners and two strong political poles. But with four strong candidates, none of them is likely to get enough votes to appear legitimate. "The two run-off candidates will not have been able to gather much more than 20 percent in the first round," he stated.

And more problems could lie ahead. Parliamentary elections will take place in June and the resulting majority will form the new government.

But only if Francois Fillon became president, would he have a chance of getting such a majority. He could fall back on a large base of traditional voters of his party.

The other candidates, if elected, would not have that base and would hardly be able to get the necessary number of MPs. Those who are voting for the winner in the Presidential elections would not necessarily support his or her candidates in parliamentary elections, Lebourg explained.

The result would then be a coalition – a so-called "cohabitation."

But coalitions have never worked very well in French history. "It would be total chaos - the French are just not good at making compromises," Lebourg said adding that it wasn't for nothing that the French had come up with the term "Franco-French war”.

"In any case, the system is at breaking point – it's almost impossible not to reform it as things stand."

Lebourg thinks the electoral rules need to be changed towards a proportional system and that more space needs to be given to citizen initiatives.

Political scientist Faucher says it's no wonder people are confused given all the ups and downs of the election campaign. "The French are just not happy with the status quo. This campaign is the expression of a resentment against the established system, just like the Brexit vote and the outcome of the US presidential elections."

"Many just don't know who to vote for now – especially as they are more worried than ever to get things right."