Strived and strove both work as the past tense of strive. Both forms are many centuries old, and both appear regularly throughout the English-speaking world, so you’re safe using the one that sounds best to you.
The past participle is more complicated. Striven is the traditional form, but strived has gained ground and is now more common. So, for example, I have striven five years is the traditional construction, but I have strived five years is acceptable to modern English speakers. Constructions like I have strove five years are still rare, though.
Ceredigion council said the coastal town always strived to create an enjoyable and relaxing evening for residents as well as visitors. [BBC]
So there was a lot going on, yet, throughout, Springsteen strove to get as close as he could to the audience. [Telegraph]
Duma had always set the bar high, and strove to improve on a daily basis. [Independent Online]
She pleads for the audience to remember her good qualities, how hard she strived, and not how hard she suffered. [LA Times Show Tracker blog]
Disco as a fad was peaking, and Ms. Summer strove to outlast it. [New York Times]
Mr Addison said that the “quite low” balance was the reason the fund had strived for conservative investments. [The Australian]
China’s leaders have striven to project an aura of stability and unity amid global political upheaval. [New York Times]
The need to pay VAT on outsourced services has proved a major obstacle as social housing providers have strived to be more efficient. [Guardian]
[S]upply has risen sharply as companies have strived to produce more to meet demand. [Globe and Mail]
[T]he EU oligarchs have striven officiously, by every means at their disposal, to keep its life support system turned on. [Scotsman]