Point-to-Point　Protocol　over　Ethernet　(PPPoE)　is　a　network　protocol　for　encapsulating　PPP　frames　inside　Ethernet　frames.　It　is　widely　used　by　commercial　Internet　service　providers　to　provide　Internet　surfing　for　customers　who　pay　bills.　In　this　paper,　we　analyze　the　security　of　PPPoE　network.　We　find　that　we　can　easily　collect　information　about　both　the　peers　and　the　PPPoE　authentication　servers.　We　can　use　such　information　to　recover　the　peer＇s　authentication　password　by　silently　impersonating　the　server,　which　is　undetectable　in　the　network.　We　impersonate　the　server　in　the　peers＇　LAN　and　get　their　passwords　by　hijacking　the　peers＇　PPPoE　connections　and　negotiating　for　using　the　Password　Authentication　Protocol　(PAP).　We　further　propose　an　efficient　password　recovery　attack　against　the　Challenge-Handshake　Authentication　Protocol　(CHAP).　We　first　recover　the　length　of　the　used　password　through　on-line　queries,　based　on　the　weakness　of　MD5　input　pre-processing.　Then,　we　crack　the　known-length　password　off-line,　using　the　probabilistic　context-free　grammars.　We　point　out　that　PPPoE　cannot　be　used　anymore　until　all　of　the　weak　authentication　protocols　including　PAP,　CHAP,　and　Microsoft　CHAP　are　abolished　right　now　and　replaced　with　more　secure　Extensible　Authentication　Protocols.　Copyright　(c)　2012　John　Wiley　&　Sons,　Ltd.