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Do you know what your company is doing?

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The operations of the 124 largest publicly traded companies are opaque, with relatively few fully revealing their corporate holdings, financial information or anti-corruption efforts, according to a new report from Transparency International.

"Acts of corruption are very often aided by the use of opaque company structures and secrecy jurisdictions," the report said. "Comprehensive public reporting is a key component of the measures companies must take to address corruption and provide the transparency that is the basis for robust and accountable governance."

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Transparency International studied three categories: reporting on anticorruption programs, organizational transparency and country-by-country reporting.

Just one company, Vodafone, managed to score at least 50 percent across all three categories, the report said.

Only three of the 124 companies studied don't commit to complying with anti-corruptions laws, but 68 don't disclose their political contributions and only 56 forbid "facilitation payments," or small bribes, the report said. No Chinese companies disclosed financial data in the 59 foreign companies they operate in, while a total of 90 don't disclose any tax-payment information in foreign countries, the report said.

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But while only 45 percent of the companies forbid facilitation payments, it's up from only 20 percent in 2012, the report noted.

Companies scored best on the anti-corruption category, with an average score of 70 percent, helped by more stringent foreign bribery laws, such as the 2010 U.K. Bribery Act and more aggressive enforcement, the report said, noting two British companies, BP and Vodafone, actually scored 100 percent. Bank of China was worst-performing in the category, scoring only 4 percent, as banks and financial services companies ranked considerably lower than other industries, averaging only 58 percent.

On organizational transparency, the average score was only 39 percent, with many companies failing to fully disclose information about subsidiaries, affiliates, joint ventures and other holdings. Only 34 companies of the 124 scored more than 50 percent and just one, Italian oil major Eni, scored 100 percent, it said. Technology and consumer services companies, mostly U.S.-based, performed worst, it said.

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Country-by-country reporting of basic financial data was the category where the 124 companies performed worst, the report said, with an average score of just 6 percent, more than 50 companies at zero percent and just three companies scoring over 50 percent.

"We need more transparency from multinational companies, whose power in the world economy closely rivals the biggest countries. With greater economic power comes greater responsibility," said Jose Ugaz, chair of Transparency International, in a statement.

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The organization recommends global companies prohibit facilitation payments, publicly disclose all political donations and exhaustive lists of subsidiaries, affiliates, joint ventures and other entities. It also recommends companies publish financial accounts for each country they operate in.

—By CNBC.Com's Leslie Shaffer; Follow her on Twitter @LeslieShaffer1