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The merchant / business person is always responsible for security of the Internet-connected PC where their customer details are handled. The minimum would be a Virus protection, a firewall and a commercial secure back-up where confidential information are stored.
Here are some the basic principles that needs to be a must for Customer Security.
1.) Privacy : Personal Details / Information should be kept confidential from unauthorized parties.
2.) Integrity: The information / Message which is provided should not be altered or tampered with any.
3.) Authentication: Here lies the Trust, the sender and recipient must prove their identities to each other to bring comfidence to the user & merchant.
4.) Proof of identity or Non-repudiation: An proof is needed that the message / information transmitted is indeed received by the merchant/ Customer.
The above specified principles can be done via encryption where credit cards are taken online and processed later,or credit cards are taken online and processed in real time. Its the merchant's responsibility to check the security and safety of transactions of the hosting company's webserver.