My boyfriend is too secretive and it makes me anxious. What should I do?

My boyfriend and I have only been together a couple months. I am well aware that we got together rather soon after he broke up with the girl he left back in his home city. I don't mind this as I am generally a rather secure person, and wasn't jealous or anxious about it. What did bother me was that he has lied about several things now.

1. One night he was at my apartment and he left his phone in the hall. As I was passing by it to join him, his phone rang with an unfamiliar name. When I handed it to him, he said that it was a coworker and did not answer it.

2. A couple weeks later he was at my flat again and I noticed that the jumper that he was wearing was embroidered, and I couldn't help but notice that the initial matched the initial of whoever had called him before.

3. A few weeks later I asked him about the name again, and he said that it was a friend from his home country. I had already done research already and found that it was only a common name in one country, which was not his. At that point I confronted him and he admitted that she was his most recent ex, and that he only hid it because he didn't want to hurt my feelings or ruin the mood. This was really annoying as he knows that I wouldn't be bothered by it in the slightest if he had just told me the truth from the start.

4. Finally, and most importantly, is since we started dating he has been very defensive over his phone. He doesn't let me hold it if he's showing me something, he doesn't like me looking at his phone if I am next to him (which he does to me frequently), he won't let me use his phone to check anything even if he is watching, and he has it constantly on DND and airplane mode, and has control centre disabled if the phone is locked. This all seems extreme to me as I am very open with my phone, and its not as if I want to deliberately snoop through. But the simple fact that he has taken so many precautions and is so careful with his phone really makes me anxious.

My boyfriend is too secretive and it makes me anxious. What should I do?
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