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After restoring configuration backup using Luci web interface, the ssh public key authentication stopped working:

$ ssh root@192.168.1.254 -i ~/.ssh/id_rsa

root@192.168.1.254: Permission denied (publickey)

Checked the dropbear:

root@OpenWrt:/etc/dropbear# ls -l

-rw-------    1 501      20             724 Apr 29 09:07 authorized_keys

/etc/dropbear/authorized_keys contains the correct id_rsa.pub key.

Luci confirms this too.
What can be wrong?

$ ssh root@192.168.1.254 -i ~/.ssh/id_rsa -vvvv

OpenSSH_7.6p1, LibreSSL 2.6.2

debug1: Reading configuration data /Users/drew/.ssh/config

debug1: /Users/drew/.ssh/config line 1: Applying options for *

debug1: Reading configuration data /etc/ssh/ssh_config

debug1: /etc/ssh/ssh_config line 48: Applying options for *

debug2: ssh_connect_direct: needpriv 0

debug1: Connecting to 192.168.1.254 port 22.

debug1: Connection established.

debug1: identity file /Users/drew/.ssh/id_rsa type 0

debug1: key_load_public: No such file or directory

debug1: identity file /Users/drew/.ssh/id_rsa-cert type -1

debug1: Local version string SSH-2.0-OpenSSH_7.6

debug1: Remote protocol version 2.0, remote software version dropbear_2015.67

debug1: no match: dropbear_2015.67

debug3: fd 5 is O_NONBLOCK

debug1: Authenticating to 192.168.1.254:22 as 'root'

debug3: hostkeys_foreach: reading file "/Users/drew/.ssh/known_hosts"

debug3: record_hostkey: found key type RSA in file /Users/drew/.ssh/known_hosts:57

debug3: load_hostkeys: loaded 1 keys from 192.168.1.254

debug3: order_hostkeyalgs: prefer hostkeyalgs: ssh-rsa-cert-v01@openssh.com,rsa-sha2-512,rsa-sha2-256,ssh-rsa

debug3: send packet: type 20

debug1: SSH2_MSG_KEXINIT sent

debug3: receive packet: type 20

debug1: SSH2_MSG_KEXINIT received

debug2: local client KEXINIT proposal

debug2: KEX algorithms: curve25519-sha256,curve25519-sha256@libssh.org,ecdh-sha2-nistp256,ecdh-sha2-nistp384,ecdh-sha2-nistp521,diffie-hellman-group-exchange-sha256,diffie-hellman-group16-sha512,diffie-hellman-group18-sha512,diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha256,diffie-hellman-group14-sha1,ext-info-c

debug2: host key algorithms: ssh-rsa-cert-v01@openssh.com,rsa-sha2-512,rsa-sha2-256,ssh-rsa,ecdsa-sha2-nistp256-cert-v01@openssh.com,ecdsa-sha2-nistp384-cert-v01@openssh.com,ecdsa-sha2-nistp521-cert-v01@openssh.com,ssh-ed25519-cert-v01@openssh.com,ecdsa-sha2-nistp256,ecdsa-sha2-nistp384,ecdsa-sha2-nistp521,ssh-ed25519

debug2: ciphers ctos: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com

debug2: ciphers stoc: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com

debug2: MACs ctos: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1

debug2: MACs stoc: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1

debug2: compression ctos: none,zlib@openssh.com,zlib

debug2: compression stoc: none,zlib@openssh.com,zlib

debug2: languages ctos:

debug2: languages stoc:

debug2: first_kex_follows 0

debug2: reserved 0

debug2: peer server KEXINIT proposal

debug2: KEX algorithms: diffie-hellman-group14-sha1,diffie-hellman-group1-sha1,kexguess2@matt.ucc.asn.au

debug2: host key algorithms: ssh-rsa,ssh-dss

debug2: ciphers ctos: aes128-ctr,aes256-ctr,aes128-cbc,aes256-cbc,3des-ctr,3des-cbc

debug2: ciphers stoc: aes128-ctr,aes256-ctr,aes128-cbc,aes256-cbc,3des-ctr,3des-cbc

debug2: MACs ctos: hmac-sha1,hmac-md5

debug2: MACs stoc: hmac-sha1,hmac-md5

debug2: compression ctos: none

debug2: compression stoc: none

debug2: languages ctos:

debug2: languages stoc:

debug2: first_kex_follows 0

debug2: reserved 0

debug1: kex: algorithm: diffie-hellman-group14-sha1

debug1: kex: host key algorithm: ssh-rsa

debug1: kex: server->client cipher: aes128-ctr MAC: hmac-sha1 compression: none

debug1: kex: client->server cipher: aes128-ctr MAC: hmac-sha1 compression: none

debug1: sending SSH2_MSG_KEXDH_INIT

debug2: bits set: 1001/2048

debug3: send packet: type 30

debug1: expecting SSH2_MSG_KEXDH_REPLY

debug3: receive packet: type 31

debug1: Server host key: ssh-rsa SHA256:MGw2pwNSJ41mmcBheN3x+BimgVePcLDr15ynkP49L9I

debug3: hostkeys_foreach: reading file "/Users/drew/.ssh/known_hosts"

debug3: record_hostkey: found key type RSA in file /Users/drew/.ssh/known_hosts:57

debug3: load_hostkeys: loaded 1 keys from 192.168.1.254

debug1: Host '192.168.1.254' is known and matches the RSA host key.

debug1: Found key in /Users/drew/.ssh/known_hosts:57

debug2: bits set: 1025/2048

debug3: send packet: type 21

debug2: set_newkeys: mode 1

debug1: rekey after 4294967296 blocks

debug1: SSH2_MSG_NEWKEYS sent

debug1: expecting SSH2_MSG_NEWKEYS

debug3: receive packet: type 21

debug1: SSH2_MSG_NEWKEYS received

debug2: set_newkeys: mode 0

debug1: rekey after 4294967296 blocks

debug2: key: /Users/drew/.ssh/id_rsa (0x7fc70eb01490), explicit, agent

debug2: key:  (0x7fc70e903480), agent

debug2: key:  (0x7fc70e9046e0), agent

debug3: send packet: type 5

debug3: receive packet: type 6

debug2: service_accept: ssh-userauth

debug1: SSH2_MSG_SERVICE_ACCEPT received

debug3: send packet: type 50

debug3: receive packet: type 51

debug1: Authentications that can continue: publickey

debug3: start over, passed a different list publickey

debug3: preferred gssapi-with-mic,publickey,keyboard-interactive,password

debug3: authmethod_lookup publickey

debug3: remaining preferred: keyboard-interactive,password

debug3: authmethod_is_enabled publickey

debug1: Next authentication method: publickey

debug1: Offering public key: RSA SHA256:NWHiMGnH1Sz8K/cLRV9x493V6B6P8+oat6xtwFViZl4 /Users/drew/.ssh/id_rsa

debug3: send_pubkey_test

debug3: send packet: type 50

debug2: we sent a publickey packet, wait for reply

debug3: receive packet: type 51

debug1: Authentications that can continue: publickey

debug1: Offering public key: RSA SHA256:OLw0tzPx2/CQeq3PzmwR3UdCK0RJuy2t7frV1WyX7fk

debug3: send_pubkey_test

debug3: send packet: type 50

debug2: we sent a publickey packet, wait for reply

debug3: receive packet: type 51

debug1: Authentications that can continue: publickey

debug1: Offering public key: RSA SHA256:8Jahyt/2O/nfBJYtKQuSkwPxg/Ol9ZJcH60hOneKKLQ

debug3: send_pubkey_test

debug3: send packet: type 50

debug2: we sent a publickey packet, wait for reply

debug3: receive packet: type 51

debug1: Authentications that can continue: publickey

debug2: we did not send a packet, disable method

debug1: No more authentication methods to try.

root@192.168.1.254: Permission denied (publickey).

Ok, how I figured this out:

ssh into OpenWRT with password authentication,
Run

logread |grep dropbear

...

Sun Apr 29 10:23:49 2018 authpriv.info dropbear[8716]: Child connection from 192.168.1.10:62572

Sun Apr 29 10:23:50 2018 authpriv.info dropbear[8716]: /etc/dropbear must be owned by user or root, and not writable by others

Sun Apr 29 10:23:50 2018 authpriv.info dropbear[8716]: Exit before auth (user 'root', 0 fails): Exited normally

...

For some reason the ownership of /etc/dropbear was broken.
So I did

chown -R root:root /etc/dropbear

Which fixed everything