Story highlights China's Huawei is now the world's third biggest smartphone maker

The Shenzhen-based manufacturer has designs on markets outside China

U.S, lawmakers worry Huawei products could be hidden channel spies, hackers

The company produces smartphones priced much lower than rivals such as Apple

Quick question: Who is the world's third biggest smartphone maker?

Yes, it's a paltry 5% of the global market, but the Chinese company that made its name selling telecom equipment is making a big push into the already crowded global smartphone market.

I saw that ambition first hand at the Mobile World Congress in Barcelona earlier this year. Huawei ads were posted all over the airport to greet the throngs of international conference-goers. The company also occupied a huge booth at the venue positioned just across from Samsung.

But just how big does Huawei -- the smartphone maker -- want to get?

At company headquarters in Shenzhen in southern China, I talked to Richard Yu, CEO of Huawei's Consumer Business Group. He told me his division hopes to boost its mobile business revenue from $7.5 billion to $9 billion this year.

JUST WATCHED Chinese companies as Trojan horses? Replay More Videos ... MUST WATCH Chinese companies as Trojan horses? 02:45

JUST WATCHED Huawei on the hot seat Replay More Videos ... MUST WATCH Huawei on the hot seat 01:51

JUST WATCHED Cell phones a front for China spying? Replay More Videos ... MUST WATCH Cell phones a front for China spying? 02:43

"Last year, we shipped out 32 million smartphones in quantity and this year we hope we will ship out 50 to 60 million smartphones worldwide," he said. "We are growing."

The market priority for Huawei's Consumer Business Group is China, followed by Europe and Japan. But -- when it comes to smartphones -- it's not ruling out the U.S. market despite the recent back and forth about Huawei's commitment there.

In the U.S., the name Huawei is widely regarded with suspicion. Lawmakers are worried that Huawei products can be used as a hidden channel for Chinese spies and cyber hackers -- a charge that Huawei has denied.

Despite the trust issues and the fact that many Americans can't even pronounce the name of the company, Huawei has sold and will continue to sell its phones in America under the Huawei brand.

"Gradually, step by step, more and more people will trust Huawei," said Yu. "I think with a brand, the most important thing is trust."

Huawei prides itself on its investment in research and development -- 70,000 of its 150,000 employees are in R&D -- as well as its high-end products like the $500 Ascend P2, which is billed as the "world's fastest 4G LTE smartphone."

Yu told me his personal favorite is the Ascend D2, Huawei's $600 flagship smartphone that is water-resistant -- a point famously made by Yu when he posted a photo of himself swimming with the smartphone on his Sina Weibo account.

But Huawei is more widely known for its $100 (Y-300) smartphone -- a cut-price device that Josh Ong of The Next Web says is the company's competitive advantage and branding bane.

"Huawei is going to be able to execute on smartphones at lower costs than some of its international competitors," he said.

"It doesn't have the cachet or consumer loyalty that Apple and Samsung do."

My colleague Dayu Zhang in Beijing offered this on-the-ground Chinese consumer perspective: "To me and my friends, Huawei is more like a brand of low-end, cheap smartphones.

"(Such) Chinese brands are popular among the 'ant tribe community,' which refers to young people who come to the city for a better job but got stuck with low-paid jobs and high costs to live in the city. In their eyes, these Chinese smartphones are a lot cheaper than big brands like the iPhone and almost as good."

Huawei has a reputation for making smartphones that are just good enough. But that's not good enough for Huawei.

To upgrade its image, the company says it will bypass expensive branding campaigns and splashy ads to focus on innovation to generate viral, word-of-mouth recommendations.

And that will take time.

"Rome was not built in a day," said Yu. "We have the ambition to be the best -- to have the best products and be the best solution provider."

Would you buy a Huawei smartphone? Perhaps, if that's what you're after. They're "good enough."

Would you want to buy one? Not yet.

And that's the question Huawei has its designs on.