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I would like to know if the address of the smart contract is fixed.

I want some privacy for my application, so I would like to change the sending address and receiving address each time the smart contract receives a transaction.

Is it possible ? Something like Hierarchical Key Derivation ?

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A contract's address is fixed, for the formula see How is the address of an Ethereum contract computed?

There is currently no privacy on the blockchain, all transaction details are viewable by everyone.

Yes, HD wallets are possible and for a wiki of them see What Hierarchical Deterministic (HD) wallets are available for Ethereum?

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