Defending champion Paula Badosa solved the riddle of Veronika Kudermetova, cruising past the 21st seed 6-3, 6-2 to set up a semi-final clash with Maria Sakkari at the Indian Wells WTA and ATP Masters on Thursday.

Sakkari, coming off a run to the final in Saint Petersburg that helped propel her past Badosa into sixth in the world rankings, beat Kazakhstan's Elena Rybakina 7-5, 6-4.

Badosa had lost all three of her prior meetings with Kudermetova, but she was in control throughout on a sunsplashed Stadium Court.

"I played against her and I was always making the same mistakes so today I didn't want to do it for the fourth time in a row. I started to serve very well, I think that was the key, and I stayed very aggressive and didn't let her move a lot."

Badosa gained the only break she needed in the fourth game of the opening set and broke Kudermetova twice for a 4-1 lead in the second. She saved the only break point she faced in the match and wrapped it up on her first match point when Kudermetova sent a second straight service return long.

"I think it's the court, it does magic with me," Badosa said of the venue where she lifted the trophy in October, when the tournament was moved from its usual March slot because of the coronavirus pandemic.

She was the first of three Spaniards to take center stage on Thursday.

Next up was 21-time Grand Slam champion Rafael Nadal, who was taking on Australian Nick Kyrgios in a men's quarter-final.

Magic: Defending champion Paula Badosa celebrates victory over Veronika Kudermetova in the WTA quarter-finals of the BNP Paribas Open at Indian Wells
Magic: Defending champion Paula Badosa celebrates victory over Veronika Kudermetova in the WTA quarter-finals of the BNP Paribas Open at Indian Wells AFP / Frederic J. BROWN

Then 18-year-old Spanish sensation Carlos Alcaraz, who faced defending champion Cameron Norrie of Britain for a chance to take on his idol Nadal in the final four.

Nadal wasn't expecting an easy ride from Kyrgios, even though the Australian has slumped to 132nd in the world.

The two have endured a sometimes bitter relationship on court.

In 2019 Nadal accused the Aussie of "lacking respect" after Kyrgios won an ill-tempered encounter in Mexico.

Kyrgios branded Nadal "super-salty" and when they met again at Wimbledon that year Nadal was fuming after his win during which Kyrgios appeared to spear a ball directly at him.

In all they've met eight times, with Nadal coming out on top five times.

That's a good omen for the Spaniard as he tries to extend his perfect 18-0 start to the season, although he acknowledged that in his fourth-round win over Reilly Opelka on Wednesday he was feeling the left foot trouble that brought his 2021 season to a premature end.

Alcaraz, the youngest Indian Wells men's quarter-finalist since 17-year-old Michael Chang in 1979, is aiming to become the second-youngest semi-finalist in tournament history behind 17-year-old Andre Agassi in 1988.

But Norrie will be out for revenge after a straight-sets loss to Alcaraz in the 2021 US Open first round.