The Afrobarometer survey also found it was often the poorest people in each country who pressured to pay bribes at health clinics and hospitals.
The West African nation of Sierra Leone fared the worst overall, with 63 percent of respondents saying they'd paid up at least once in the previous year. Morocco and Guinea came next, each with 57 percent.
"Corruption is a cancer which has spread nationwide," said Onesimus Johnson, an analyst in Sierra Leone where bribes are known as a "put for me." More than 69 percent of citizens polled there said that most or all police were corrupt.
In Guinea, the culture of corruption dates back to the decades of dictatorship that enriched its ruler and his associates. The West African country held its first democratic presidential election in 2010.
"Corruption is a national sport every day at the direction of customs officials," said Cherif Mohamed Haidara, who heads a group of businessmen.
Medical treatment was the second most common reason cited after paying off officials to obtain a document or permit, said Richard Houessou, who headed the Afrobarometer project in French-speaking Africa.
The problem of medical bribes was the worst in Uganda—reported by 46 percent of respondents. Swaziland at 41 percent and Niger with 40 percent were close behind.
"Among the poorest—those who went without food at least once in the past year—18 percent had to pay a bribe at least once in the previous year to receive treatment, compared to a substantially lower 12 percent among those who were better off," the report found.
The survey also found that more than half of the people polled were dissatisfied with their governments' efforts to battle corruption.
Nigerians gave the worst ratings to their government on its efforts to battle graft, with 82 percent saying the government was doing fairly badly or very badly.
Pollsters with the Afrobarometer project conducted 51,000 face-to-face interviews across Africa between October 2011 and June 2013. Afrobarometer selected 34 countries of the more than 50 on the continent to survey, but did not include many in Central Africa, leaving out Congo, Chad, Central African Republic and Gabon.
Associated Press writers Boubacar Diallo in Conakry, Guinea and Clarence Roy-Macaulay in Freetown, Sierra Leone contributed to this report.
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