“Google is seeking to extend its dominance from the online world to the real, offline world, and the FTC really needs to look at that,” said Marc Rotenberg, the organization’s executive director.

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Google called its advertising approach "common" and said it had "invested in building a new, custom encryption technology that ensures users' data remains private, secure and anonymous."

The Washington Post detailed Google's program, Store Sales Measurement, in May. Executives have hailed it as a “revolutionary” breakthrough in advertisers’ abilities to track consumer behavior. The company said that, for the first time, it would be able to prove, with a high degree of confidence, that clicks on online ads led to purchases at the cash register of physical stores.

To do this, Google said it had obtained access to the credit and debit card records of 70 percent of U.S. consumers. It had then developed a mathematical formula that would anonymize and encrypt the transaction data, and then automatically match the transactions to the millions of U.S. users of Google and Google-owned services such as Gmail, search, YouTube and maps. This approach prevents Google from accessing the credit or debit card data for individuals.

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But the company did not disclose the mathematical formula it uses to protect consumers' data. In a statement, Google said it had taken pains to build custom encryption technology that ensures the data the company receives remains private and anonymous.

The privacy organization is asking the government not to take Google’s word for it and to review the algorithm itself. In its complaint, the organization said the mathematical technique that Store Sales Measurement is based on, CryptDB, has known security

flaws. Researchers hacked into a CryptDB-protected health-care database in 2015, accessing more than 50 percent of the stored records.

Google also would not disclose which companies were providing it with the transaction records. When asked if users had consented to having their credit and debit transactions shared, Google would not specifically say. The company replied it requires that its unnamed partners have “the rights necessary” to use this data.

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In its complaint, reviewed by The Washington Post, the privacy group alleges that if consumers don’t know how Google gets its purchase data, then they cannot make an informed decision about which cards not to use or where not to shop if they don’t want their purchases tracked. The organization points out that purchases can reveal medical conditions, religious beliefs and other intimate information.

Google also told The Post that it does not have access to the names or other personal information of the credit and debit card users, and that it does not share any information about individual Google users with partners.

Advertisers receive aggregate information. For example, for an ad campaign for sneakers that received 10,000 clicks, the advertiser learns that 12 percent of the clickers made a purchase.

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Users can opt out anytime, Google says. To do so, users of Google’s products can go to their My Activity Page, click on Activity Controls, and uncheck “Web and Web Activity,” Google says.

The privacy group says the opt-out settings and the descriptions of what users are opting out of are confusing and opaque. The group says the company continues to store server and click data even when Web and App Activity is turned off, and that to opt out of everything requires a labyrinthine process of going to a number of third-party sites. Meanwhile, opting out of location-tracking requires going to a separate button and interface. None of the opt out descriptions specifically describes credit card data.