"Let me extend to you my warmest congratulations on your decisive electoral victory even in these trying times," Mr Kumar, who met Mr Sharif during his visit to Pakistan in 2012, said in a letter dated May 13 and made available to the press on Monday.
"Your electoral performance augurs well for the furtherance of relations between the two countries and what we can together do to improve the economics and politics of both the countries and the region as a whole. Your bold and innovative approach can be decisive catalyst," he said.
Mr Kumar fondly remembered the meeting with Mr Sharif when he had visited Pakistan in November last year and renewed the invitation to him to visit Bihar whenever he comes to India.
"I greatly look to continuing our dialogue which commenced in Lahore and to build on the numerous avenues constructive cooperation between the two countries and with the people of Bihar," he said.
Mr Kumar wrote another letter to cricketer-turned politician Imran Khan and congratulated him for the 'spectacular performance' of his party Pakistan Terik-e-Insaf (PTI) in the elections.
"I was deeply touched that during your electoral campaign you referred to the success of the Bihar development model which encourages us to pursue our vision with great vigour," Mr Kumar said in the letter and prayed for his speedy recovery.