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When bruteforcing a captured packet containing the encrypted password, can the owner/admin of the router see each auth attempt? I'm assuming you just encrypt each passphrase in the dictionary file and compare it to the hash of the password, but this can't be done (Aircrack-ng) without a WiFi connection and the AP in range. Why?

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No they cannot, certainly not after you've captured the handshake.

N.b. I have assumed you are talking about WPA/WPA2 because your description ('a captured packet containing the encrypted password') seems to match the PSK brute-force attack vector for those schemes.

Once you have captured a WPA/WPA2 handshake (with a tool like airodump-ng), the brute-forcing itself is all offline. You can use aircrack-ng (or another tool like oclHashcat or Pyrit) without needing to have your wireless on or be in the vicinity of the access point. The fact is that an attacker can capture the required handshake and go away to crack it at their leisure.

WEP cracking, incidentally, is similar in that the actual cracking phase is offline, although there's rarely any need to bother going offsite once you've spent a few minutes capturing the necessary packets.

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