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Does a chip exist which will take input data and output encrypted data in a public private key scheme, where the public key is on the chip and the private key is located remotely?

The ultimate goal would be to design a circuit which stores routine log data in a form which is not readable locally. It is an express goal that hardware carry out this task.

Any advice?

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In general you don't need to protect the Public key on hardware storage. It's called the public key for a good reason.

If you encrypt something with a public key, then it is only possible to decrypt it with the private key (and vice versa).

So, in general it doesn't matter who has the public key. Your public key could be on a billboard and your server could encrypt with the public key as much as you like and nobody will be able to read the encrypted data except whoever has the private key.

All the PKI hardware (smartcards, HSMs, etc) I am aware of is divided into public sectors and private sectors so if you want to store the public key privately then the private key is loaded onto the public partition (i.e. not an ideal solution).

If you want something more like mutual authentication in the log distribution mechanism then it's much better to use two sets of keys (one set for the client and one set for the Server).

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Smart Cards have capabilities to process data, while the private key is stored on them and never leaves them. HSM (hardware security module) does similar things, with much better processing power than a smart card, but also for a much higher price.

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Simply put, YES

Smart cards already do this.

Many off the shelf embedded CPUs have this feature built in.

Or, you can cook your own in an FPGA (Field programmable gate array)

Free IP hardware IP in the form of HDL (hardware description language) is available from OpenCores.

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