A Cryptographic Countermeasure Against Connection ... - NDSS

1 downloads 0 Views 220KB Size Report
Let h denote a non invertible hash function whose output is of length l. (Since we won t require that h be collision resistant, h might be a fast hash function.
&OLHQW 3 X]]OHV $ &U \SWRJU DSKLF &RXQWHU PHDVXU H $JDLQVW &RQQHFWLRQ 'HSOHWLRQ $WWDFNV $ U L -X H OV 5 6   % H G ID U LME

-R K Q % U D LQ D U G $ / D E R U D W R U LH V &U R V E \ ' U LYH IR U G  0$      U D LQ D U G J#U V D F R P

$EVWU DFW :H LQWURGXFH D FU\SWRJUDSKLFDOO\ EDVHG FRXQWHUPHD VXUH DJDLQVW FRQQHFWLRQ GHSOHWLRQ DWWDFNV &RQQHFWLRQ GHSOHWLRQ LV D GHQLDORIVHUYLFH DWWDFN LQ ZKLFK DQ DW WDFNHU VHHNV WR LQLWLDWH DQG OHDYH XQUHVROYHG D ODUJH QXPEHU RI FRQQHFWLRQ UHTXHVWV WR D VHUYHU H[KDXVWLQJ LWV UHVRXUFHV DQG UHQGHULQJ LW LQFDSDEOH RI VHUYLFLQJ OH JLWLPDWH UHTXHVWV 7&3 6@ FUHDWLQJ GLJLWDO WLPH FDSVXOHV >@ DQG PHWHULQJ :HE VLWH XVDJH >@



7 &3

6@ VXJJHVWV WKDW KDOIRSHQ 7&3 6@ 7KXV D EDFNRI WKHHQYHORSH FDOFXODWLRQ UHYHDOV WKDW WKH URXJKO\  LQVWUXFWLRQ RYHUKHDG DVVRFLDWHG ZLWK FRQVWUXFWLRQ RI D SX]]OH VHH VHFWLRQ  WUDQVODWHV LQWR D SHUIRUPDQFH SHQDOW\ RI OHVV WKDQ  UHODWLYH WR 7&3 FRQQHFWLRQ FRPSXWDWLRQ WLPH 9HULpFDWLRQ RI DQ LQFRUUHFW SX]]OH WDNHV DW PRVW DQ DYHUDJH RI  LQVWUXFWLRQV HTXLY DOHQW WR DW PRVW  RI WKH WLPH UHTXLUHG WR HVWDEOLVK D 7&3 FRQQHFWLRQ $V QRWHG LQ 6HFWLRQ  SX]]OHV FDQ EH FRQVWUXFWHG WR EH DERXW  E\WHV LQ VL]H SOXV VHUYLFH LQIRUPDWLRQ ZKLOH VROXWLRQV DUH DERXW  E\WHV LQ VL]H 7KHVH SHQDOWLHV DUH VPDOO DQG RI FRXUVH SDLG RQO\ ZKHQ WKH VHUYHU LV XQGHU DWWDFN 1RZ E\ 7KHRUHP  ZH UHTXLUH DW OHDVW F y   DQG E y  6LQFH E F7 J PNb  WKLV PHDQV WKDW ZH UHTXLUH E y  dd d }   +HQFH ZH QHHG % WR FRQWDLQ DERXW    PD[FRQ EXoHU VORWV +HXULVWLF  KRZHYHU VSHFLpHV WKDW F y   DQG E y  7KLV LQVWHDG JLYHV XV E y   d  d   d  }   +HQFH LQ SUDFWLFH ZH ZRXOG ZDQW % WR FRQWDLQ RQ WKH RUGHU RI    PD[FRQ EXoHU VORWV $JDLQ E\ DSSHDOLQJ WR &RUROODU\  DV JLYHQ LQ WKH $SSHQGL[ PRUH FDUHIXO DQDO\VLV ZRXOG \LHOG D VPDOOHU YDOXH RI E ,Q FRQYHQWLRQDO HQYLURQPHQWV LW LV VWDQGDUG WR DGRSW D PLQLPXP VHUYHU 7&3 EXoHU VL]H RI  VORWV >@ 7KLV LQFOXGHV EXoHU VORWV GHYRWHG WR GHIHQVH DJDLQVW 7&3 6 @ ' $ 53$  7 UDQV PLV  V LRQ FRQWURO SURWRFRO 7 & 3 '$ 53 $ ,QWHUQHW SURJUDP SURWRFRO V SHFLpFDWLRQ 6H SWH PEH U      $ Y D LOD EOH D W KWWS  GVLQWH UQLF QH W UIF  UIF    W[ W > @ &  ' D Y LGVR Q 7 KH ? 6< 1 qR R G J D WH V R SH Q IR U :H E & R P L:RUOG :HHNO\   ' H F H PEH U      > @ ' LJ LWD O (T XLSPH QW & R USR UD WLR Q 3 HUIRUPDQFH 7 X Q LQJ 7 LSV IRU 'LJLWDO 8 QL[  -XQH      $ Y D LOD EOH D W Z Z Z VH UY LF H GLJ LWD OF R P PD QXD O PLVF  SH UIGH F KWPO > @ +  ' R EEH UWLQ & U\ SWD QD O\ VLV R I 0 '   ,Q '  *UR OOPD Q H GLWR U ) DV W 6RIWZDUH ( QFU\SWLRQ 7 KLUG ,QWHUQDWLRQDO :RUNV KRS SD J H V   ^    6SULQJ H U9 H UOD J       /H F  WXUH 1 R WH V LQ & R PSXWH U 6F LH QF H 1 R       >  @ & F 3 

 ' Z R PED URF    

R UN D QG 0  1 D R U 3ULF LQJ WWLQJ MXQN PD LO ,Q (UQH VW & 5< 3 7 2  SD J H V    ^   /H F WXUH 1 R WH V LQ & R PSXWH U

Y LD SUR F H VVLQJ R U )  %ULF N H OO H GLWR U   6SULQJ H U9 H UOD J  6F LH QF H 1 R     

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

WKH PH WKR G R I ER XQGH G GLoH U & RPELQDWRULFV  SD J H V    ^     3UH VV      /R QGR Q 0 D WK 6R F    

>  @ 5 5LY H VW 0  5R EVKD Z  5 6LGQH \  D QG / < LQ 7 KH 5&  EOR F N F LSKH U      3UH VH QWH G D W WKH 1 ,67 ) LUVW $ (6 & D QGLGD WH & R QIH UH QF H  9 H QWXUD  & $ 

> @ 7 KH 1 HZ < RUN 7 LPHV    6H SWH PEH U     

>  @ 5/ 5LY H ,Q $ - 0 & 5< 3 7 2 /H F WXUH 1

> @ 5 $ J XLOD U D QG - . R UQEOXP 1 HZ < RUN 7 LPHV VLWH KD F N H G & 1 ( 7 1 ( :6& 2 0   1 R Y H PEH U     

>  @ 5/ 5LY H VW 7 KH 0 '  0 H VVD J H ' LJ H VW $ OJ R ULWKP $ SULO      5) &      ,QWH UQH W $ F WLY LWLH V %R D UG

> @ 5 %UD GH Q 7  7 & 3 7 & 3 H [ WH QVLR QV IR U WUD QVD F WLR QV IXQF WLR QD O VSH F LpF D WLR Q -XO\      5) &      ,QWH U QH W $ F WLY LWLH V %R D UG

>  @ 5/ 5LY H VW $  6KD PLU D QG '  :D J QH U 7 LPH OR F N SX] ] OH V D QG WLPH GUH OH D VH F U\ SWR  0 D QXVF ULSW 7 R D S SH D U   0 D UF K     

> @ & (57  $ GYLV RU\ & $  7 & 3 6< 1 q RRGLQJ DQG ,3 V SRRpQJ DWWDFNV    6H SWH PEH U     

>  @ 7 KH 6WD QGD UG 3H UIR UPD QF H (Y D OXD WLR Q & R USR UD WLR Q 6SHF:HE %HQFKPDUN 5HV X OWV       $ Y D LOD EOH D W KWWS  Z Z Z VSH F EH QF KR UJ  R VJ  Z H E   UH VXOWV

5HIHU HQFHV

> @ ' D H PR Q  3UR MH F W 1 H SWXQH  3 KUDFN 0 DJD] LQH    ) LOH   R I     1 R Y H PEH U      $ Y D LOD EOH D W Z Z Z IF QH W SKUD F N  pOH V S   S    KWPO

VW H QH ]  R WH V

7 KH H V D SD J H LQ &

0 '  QG 6$ V    ^  R PSXWH

PH VVD J H GLJ H VW D OJ R ULWKP  9 D QVWR QH  H GLWR UV 3 URF    6SULQJ H U9 H UOD J       U 6F LH QF H 1 R     

>  @ % =LH J OH U + D F N H U WD QJ OH V 3D QL[ :H E VLWH  :DOO 6WUHHW -RX UQDO   6H SWH PEH U     

$

$SSHQGL[ 3 U RRIV

,Q WKLV DSSHQGL[ ZH DQDO\]H DQG SURYH UHVXOWV DERXW WKH OHYHO RI GHIHQVH SURYLGHG E\ WKH FOLHQW SX]]OH SUR WRFRO $W pUVW JODQFH WKLV ZRXOG VHHP VLPSOH DV ZH FDQ HDVLO\ FRPSXWH WKH DYHUDJH WLPH UHTXLUHG E\ WKH DGYHUVDU\ WR VROYH D SX]]OH DQG ZH NQRZ WKH QXPEHU RI SX]]OHV UHTXLUHG WR RYHUORDG WKH VHUYHU 7ZR IDFWV KRZHYHU PDNH WKH DQDO\VLV WULFN\ 7KH pUVW LV WKDW WKH WLPH UHTXLUHG WR VROYH D FOLHQW SX]]OH LV D UDQGRP YDULDEOH 6LQFH VROYLQJ D FOLHQW SX]]OH LQYROYHV D VHDUFK RI WKH VSDFH RI SRWHQWLDO VROXWLRQV LW LV SRVVLEOH WKDW DQ DGYHUVDU\ PD\ ZLWK OXFN KDSSHQ XSRQ WKH VROX WLRQV WR D ODUJH QXPEHU RI SX]]OHV LQ D VKRUW VSDFH RI WLPH 7KH VHFRQG IDFW LV WKDW LW LV SRVVLEOH IRU DQ DGYHUVDU\ WR JXHVV DW WKH VROXWLRQ WR D FOLHQW SX]]OH :LWK VRPH VPDOO SUREDELOLW\ DQ DGYHUVDU\ V JXHVV ZLOO EH FRUUHFW DQG WKH VHUYHU ZLOO DOORFDWH EXo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

$

3 U HOLPLQDU LHV VXPSWLRQV

QRWDWLRQ DQG DV

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oHFWLYH VWUDWHJ\ IRU VROYLQJ D VXESX]]OH 3 >M@ LV EUXWH IRUFH LH DQ DUELWUDULO\ RUGHUHG EXW QRQUHGXQGDQW VHDUFK RI WKH VSDFH 6>M@ $V DERYH ZH GHpQH D WLPH VWHS WR EH WKH H[DPLQDWLRQ RI D VLQJOH HOHPHQW RI 6>M@ 5HFDOO WKDW WKH H[SHFWHG QXPEHU RI WLPH VWHSV IRU D EUXWH IRUFH VHDUFK WR \LHOG D VROXWLRQ WR 3 >M@ LV Nb  DQG WKH H[SHFWHG QXPEHU RI WLPH VWHSV IRU D EUXWH IRUFH VHDUFK WR \LHOG D VROXWLRQ WR 3 LV PNb  :H DVVXPH WKDW WKH WLPH IRU WKH DGYHUVDU\ WR JXHVV DW DQG VXEPLW D VROXWLRQ WR D SX]]OH LV RQH WLPH VWHS $V WKH DSSOLFD

WLRQ RI D KDVK IXQFWLRQ VXFK DV 0' UHTXLUHV DERXW  LQVWUXFWLRQV WKLV DVVXPSWLRQ LV DFFXUDWH DQG SHUKDSV HYHQ FRQVHUYDWLYH IRU UHDOZRUOG VHWWLQJV /HW XV GHQRWH E\ 6 WKH VHW RI SRVVLEOH VROXWLRQV WR WKH SX]]OH 3  /HW XV XVH d WR GHQRWH WKH SURGXFW RI VHWV 7KH VHW 6 RI SRVVLEOH VROXWLRQV WR 3 FRQ VLVWV RI VROXWLRQV WR DOO RI WKH VXESX]]OHV LQ 3  LH 6 6>@ d 6>@ d    d 6>P@ +HQFH WKH VHW 6 LV LVR PRUSKLF WR WKH VHW RI DOO ELWVWULQJV RI OHQJWK NP /HW 6 e GHQRWH WKH VXEVHW RI 6 FRQWDLQLQJ DOO VROXWLRQV WR 3 ZKLFK WKH DGYHUVDU\ KDV QRW \HW HOLPLQDWHG IURP LWV VHDUFK 6XSSRVH IRU H[DPSOH WKDW WKH DGYHU VDU\ KDV VROYHG VXESX]]OH 3 >@ DQG KDV XQVXFFHVV IXOO\ VHDUFKHG D VHW FRQVLVWLQJ RI KDOI RI WKH SRVVL [  EOH VROXWLRQV WR VXESX]]OH 3 >@ 7KHQ 6 e     N ! d 6>@ b 6 e >@ d 6>@ d    d 6>P@ VR 6 e HoHFWLYHO\ FRQWDLQV NPbNb ELWVWULQJV /HW XV GHpQH 3 e V 3 P b M M6 >M@M M6>M@M  +HQFH LQ RXU H[DPSOH KHUH V 3   7KH TXDQWLW\ V 3  ZKLFK ZH UHIHU WR DV WKH VROXWLRQ OHYHO RI 3  PD\ EH ORRVHO\ UHJDUGHG DV WKH QXPEHU RI VXESX]]OHV ZKLFK WKH DGYHUVDU\ KDV VROYHG LQ 3  :H PD\ GHpQH V 3 >M@ DQDORJRXVO\ WR EH WKH SRUWLRQ RI VXESX]]OH 3 >M@ ZKLFK WKH DGYHUVDU\ KDV VROYHG ,Q SDUWLFXODU V 3 >M@  b M6 e >M@M M6>M@M  ,Q WKH FOLHQW SX]]OH SURWRFRO DV GHVFULEHG DERYH WKHUH DUH WKUHH H[SLUDWLRQ WLPHV H[SLUDWLRQ WLPH 7 RQ FOLHQW SX]]OHV 7 RQ SHUPLVVLRQ WR LQLWLDWH SUR WRFRO H[HFXWLRQ 0L  DQG 7 RQ WKH EXoHU DVVRFLDWHG ZLWK 0L  $Q DWWDFNHU VHHNV WR VROYH DV PDQ\ FOLHQW SX]]OHV DV SRVVLEOH VXEMHFW WR WKHVH FRQVWUDLQWV /HW 7 7  7  7  ,W LV HDV\ WR VHH WKDW DW D JLYHQ SRLQW LQ WLPH €  WKHUH ZLOO FRUUHVSRQG WR HDFK DOOR FDWHG EXoHU VORW LQ % D VROYHG FOLHQW SX]]OH ZKLFK ZDV UHTXHVWHG DW WKH ODWHVW DW WLPH € b 7  7KHUHIRUH WKH PD[LPXP QXPEHU RI EXoHU VORWV DQ DGYHUVDU\ FDQ KDYH DOORFDWHG LQ % LV ERXQGHG DERYH E\ WKH QXPEHU RI FOLHQW SX]]OHV VKH FDQ VROYH LQ WLPH 7  ,I E EXo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

6eM

RWH WK DW Z T X HVWLRQ S X U S

Q

RWDWLRQ G

R OLN HZ

K HQ

M

6 e M  WK H DG Y HU VDU 6 e M  DQ G M6 e M

 VR WK DW M

RVHV HT X LY DOHQ W FRQ G

 G DQ

HQ RWH D p G

LVH IRU

P

Q LVK HG

DWK HP

S X

LWLRQ V : ]]OH  Z

DQ \

VX E

S X

K DV VROY HG  DU H IRU

H OHW HLWK HU

K LFK HY HU

DWLFDO FRQ VLVWHQ F\

6 e >M @ IRU

\

]]OH

LQ

WK H S

X ]

DOO LQ WHQ WV

M

6eM

 RU

LV FRQ Y HQ LHQ W IRU

D JLY HQ

FRQ WH[ W :

H

pQH DQ RSWLPDO DOJRULWKP RU VWUDWHJ\ IRU DQ SHU IRUPLQJ FRPSXWDWLRQ & LQ 8 WLPH VWHSV DV IRO ORZV /HW FW EH WKH VWDWH RI WKH FRPSXWDWLRQ LQ WLPH VWHS W DQG OHW SU> ; 8 & @ EH WKH SUREDELOLW\ WKDW DOJRULWKP ; FRPSOHWHV FRPSXWDWLRQ & E\ WLPH 8  $OJRULWKP ; LV RSWLPDO LI IRU DOO DOJRULWKPV < DQG WLPHV W x 8  SU> ; 8 & M FW  FWb      F @ y SU> M@  V 3 >M  @   ,Q RWKHU ZRUGV VXSSRVH WKDW WKH DGYHUVDU\ KDV QRW VROYHG VXESX]]OHV 3 >M@ DQG 3 >M  @ FRPSOHWHO\ /HW T EH WKH SUREDELOLW\ WKDW LI WKH DGYHUVDU\ JXHVVHV DW D VR OXWLRQ WR 3  LW ZLOO JXHVV FRUUHFW VROXWLRQV WR 3 >M@ DQG 3 >M  @ ,W LV VWUDLJKWIRUZDUG WR VHH WKDW T  M6 e 3 >M@ MM6 e 3 >M  @ M  ,I V 3 >M@  V 3 >M  @   WKHQ M6 e 3 >M@ M  M6 e 3 >M  @ M ! N  ,W LV HDV\ WR N DQG VKRZ WKDW T LV PD[LPL]HG ZKHQ M6 e 3 >M  @ M e M6 3 >M@ M LV PLQLPL]HG LH ZKHQ V 3 >M@ V 3 >M@   ,I V 3 >M@  V 3 >M  @ y  V 3 >M  @  DQG V 3 >M  @  WKHQ T LV VLPLODUO\ PD[LPL]HG ZKHQ M6 e 3 >M@ M LH ZKHQ V 3 >M@  $OO RI WKLV LV WR VD\ LQ ORRVH WHUPV WKDW T LV PD[LPL]HG ZKHQ WKH VHDUFK RI WKH DG YHUVDU\ KDV EHHQ FRQFHQWUDWHG DV PXFK DV SRVVLEOH RQ RQH VXESX]]OH %\ D VWUDLJKWIRUZDUG LQGXFWLYH DUJX PHQW LW IROORZV WKDW WKH SUREDELOLW\ WKDW DQ DGYHUVDU\ PDNHV D FRUUHFW JXHVV RQ D SX]]OH 3 LV PD[LPL]HG ZKHQ WKH VHDUFK RI WKH DGYHUVDU\ KDV EHHQ FRQFHQ WUDWHG RQ DV IHZ VXESX]]OHV DV SRVVLEOH &DOO WKLV

SUREDELOLW\ U 7KH SUREDELOLW\ U LV PD[LPL]HG ZKHQ V 3 >@ V 3 >@  V 3 >EV 3 F@  DQG V 3 >GV 3 H@ V 3 b EV 3 F  ,W LV HDV\ WR VKRZ QRZ WKDW IRU DQ\ VKRUW VROXWLRQ U x bN PbV 3  7KH /HPPD IROORZV W X 2EVHUYDWLRQ  /HW 63 DQG 63  EH WZR VXESX]]OHV /HW S EH WKH SUREDELOLW\ WKDW WKH DGYHUVDU\ VROYHV 63 RQ WKH QH[W VWHS RI ZRUN RQ WKDW VXESX]]OH DQG S EH WKH SUREDELOLW\ WKDW WKH DGYHUVDU\ VROYHV 63  RQ WKH QH[W VWHS RI ZRUN RQ 63   ,I V 63 x V 63   WKHQ S x S  3URRI 7KH SUREDELOLW\ WKDW DQ DGYHUVDU\ VROYHV D XQVROYHG VXESX]]OH 63 E\ ZRUNLQJ RQ LW IRU RQH WLPH VWHS LV  M6 e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y ] LQ Q WLPH

D VHTXHQFH RI SDU y  7KHQ IRU DQ\ IRU DQ DGYHUVDU\ WR VWHSV

3URRI /HW Q EH DQ DUELWUDU\ QXPEHU RI3WLPH VWHSV DQG OHW & GHQRWH FRPSXWDWLRQ DFKLHYLQJ L V 63L y ] /HW + EH WKH KLVWRU\ RI FRPSXWDWLRQ RI DOO FRPSXWDWLRQ SHUIRUPHG E\ WKH DGYHUVDU\ RQ SX]]OHV 63  63     /HW F EH WKH FXUUHQW VWDWH RI FRPSXWD WLRQ :H VKDOO SURYH LQGXFWLYHO\ RQ M WKDW IRU DOO <  LW LV WKH FDVH WKDW SU> %) M & M +@ y SU>

Suggest Documents