The rupee strengthened by 31 paise to 81.51 against the US dollar in early trade on Tuesday, tracking weakness in the greenback in overseas markets and a rally in domestic equities.

However, rising crude price in the international market restricted the rupee, forex traders said.

At the interbank foreign exchange, the local currency opened sharply up at 81.66 against the dollar, then gained further ground to 81.51, showing a gain of 31 paise over its previous close.

The rupee settled 42 paise lower at 81.82 against the dollar on Monday.

In the domestic equity market, the 30-share BSE Sensex was trading 1,006.59 points or 1.77 per cent higher at 57,795.40, and the broader NSE Nifty climbed 306.40 points or 1.81 per cent to 17,193.75.

Meanwhile, the dollar index, which gauges the greenback's strength against a basket of six currencies, slipped 0.14 per cent to 111.58.

Brent crude futures, the global oil benchmark, rose 0.46 per cent to USD 89.27 per barrel.

Foreign institutional investors were net buyers in the capital market on Monday as they bought shares worth Rs 590.58 crore, as per exchange data.