A nonce is a number that is used only once in a cryptographic operation. It provides protection against a replay attack. In order to guarantee that a message hasn't been replayed, the recipient generates the nonce and sends it to the sender. The sender includes that nonce in the message. Because the sender presumably has no way of knowing what nonce the recipient will choose, they can't replay a message that was prepared earlier. But of course, you must take care that a previously prepared message can't just be minimally modified. When sending commands to a TPM and receiving the results back, nonces provide proof to the user that the results of the command were sent by the TPM.
In a typical TPM use, the nonce is included in the calculation of the HMAC of a command message. After an operation using the message is complete, the TPM changes the nonce. If the caller attempts to replay the message which had an HMAC that used the previous nonce, the TPM will attempt to verify the HMAC of the replayed message using the new nonce, and this verification will fail.
For many applications, a nonce can simply be a number that increments at each use and is large enough to never wrap around. However, this would require the TPM to keep track of the last-used value. Instead, the TPM takes advantage of its random-number generator. It uses random numbers as its nonces, and it uses large enough values (for example, 20 bytes) that the odds of a repeat are nil.
Asymmetric keys are used by asymmetric algorithms for digital identities and key management. They are actually a key pair: a private key known only to one party and a public key known to everyone. Asymmetric keys make use of mathematical one-way functions. These functions have the property that calculating the public key from the private key is relatively easy computationally, but calculating the private key from the public key is computationally infeasible.
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If the owner of the private key uses it to encrypt some data (and the key is large enough), everyone can use the public key to decrypt the data, but everyone will know that only the holder of the private key could have encrypted that data. This is called signing the data. There are some complications to using this securely—the data that is signed should be in a particular format called a signing scheme, but the TPM ensures sure that the correct format is used.
If someone wants to share data (usually a symmetric key) with the owner of a private key in a secure way, they can provide the data to the owner by encrypting it with the owner's public key. Then they can be certain that only the owner will be able to recover the shared data by using the owner's private key. This is done in different ways depending on the type of asymmetric algorithm, but we skip these deep mathematical details here.