"I thank you for congratulating me on the resounding victory in the Tamil Nadu Legislative Assembly election, 2016," she said.
"I value your good wishes and greetings," the 68-year-old AIADMK General Secretary told Shah in a letter.
Shah, during his election campaign in the state, had attacked the AIADMK government, describing it as the "most corrupt."
Senior BJP leader LK Advani also greeted Jayalalithaa.
Advani congratulated her over phone and the Chief Minister thanked the veteran leader, a release said.
".... Under your dynamic leadership and determination to serve the people above self, Tamil Nadu is set to lead the nation in all spheres once again," Rosaiah said.
(REOPEN MDS5)
In a separate statement, Jayalalithaa said the outcome of the elections had proved that "I am on the side of people and people are on my side."
"The victory is people's endorsement of the various welfare measures implemented by my government in the last five years," she said.
"DMK, through media and their campaign, unleashed lies. They didn't heed the repeated clarifications issued and maintained their lies like Goebbels," she said referring to the propaganda minister of the Nazi government.
DMK had made "imaginary allegations" against the state government but the "people proved they cannot be fooled," she said, adding they had realised the AIADMK was the protector of the poor and therefore handed it a good mandate.
Jayalalithaa thanked the people for electing her party again and reiterated her commitment to work with a new vigour for their welfare.
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