user90638: I've been reading about HD wallet derivation key mechanism. 
I understood that if an attacker gets somehow a child's private key plus the chaincode used to derive that child' key, then he could calculate the parent's private key.
How could that be possible? Through a sort of brute force attack?
Furthermore, how can the hardened derivation scheme neutralize this problem by using the parent's private key instead of his public key as input for the 
HMAC-SHA512 functions?

user90504: Here is a solid answer by Andrew Chow. Can we derive parent’s private key using child’s private key?
As for the specific questions: 

How could that be possible?

Once again, Andrew Chow: Derivation of parent private key from non-hardened child

Furthermore, how can the hardened derivation scheme neutralize this problem by using the parent's private key instead of his public key as input for the HMAC-SHA512 functions?

(From the linked post above) Hardened derivation protects against this because it generates the child private key by hashing the parent private key. However this also means that you cannot generate the child public key from the parent public key.

90638: <p>I've been reading about HD wallet derivation key mechanism. 
I understood that if an attacker gets somehow a <strong>child's private key</strong> plus the <strong>chaincode</strong> used to derive that child' key, then he could calculate the <strong>parent's private key</strong>.
How could that be possible? Through a sort of brute force attack?</p>

<p>Furthermore, how can the <strong>hardened derivation</strong> scheme neutralize this problem by using the parent's private key instead of his public key as input for the 
HMAC-SHA512 functions?</p>
 90504: <p>Here is a solid answer by Andrew Chow. <a href="https://bitcoin.stackexchange.com/questions/76296/can-we-derive-parent-s-private-key-using-child-s-private-key">Can we derive parent’s private key using child’s private key?</a></p>

<p>As for the specific questions: </p>

<blockquote>
  <p>How could that be possible?</p>
</blockquote>

<p>Once again, Andrew Chow: <a href="https://bitcoin.stackexchange.com/questions/56916/derivation-of-parent-private-key-from-non-hardened-child?noredirect=1&amp;lq=1">Derivation of parent private key from non-hardened child</a></p>

<blockquote>
  <p>Furthermore, how can the hardened derivation scheme neutralize this problem by using the parent's private key instead of his public key as input for the HMAC-SHA512 functions?</p>
</blockquote>

<p>(From the linked post above) Hardened derivation protects against this because it generates the child private key by hashing the parent private key. However this also means that you cannot generate the child public key from the parent public key.</p>
