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2021, Volume 15, Issue 3: 539-556. Doi: 10.3934/amc.2020081
This issue Previous Article The differential spectrum of a class of power functions over finite fields Next Article

Internal state recovery of Espresso stream cipher using conditional sampling resistance and TMDTO attack

  • Bosch India (RBEI/ESY), Bangalore, India

Received:  November 2019
Revised:  January 2020
Early access:  April 2020
Published:  August 2021
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    Abstract

  • Espresso is a stream cipher proposed for the 5G wireless communication system. Since the design of this cipher is based on the Galois configuration of NLFSR, the cipher has a short propagation delay, and it is the fastest among the ciphers below 1500 GE, including Grain-128 and Trivium. The time-memory-data tradeoff (TMDTO) attack on this cipher and finding the conditional BSW sampling resistance are difficult due to its Galois configuration. This paper demonstrates the calculation of conditional BSW-sampling resistance of Espresso stream cipher, which is based on Galois configuration, and also mounts the TMDTO attack on the cipher by employing the calculated sampling resistance. It is also shown that the attack complexities of TMDTO attack are lower than those claimed by the designers of the ciphers.

    Mathematics Subject Classification: Primary: 68P25, 94A60; Secondary: 06E30.

    Citation:
    shu

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  • Table 1.  State Bits required to calculate feedback bits

    Row Feedback bit calculaton because of (5) Column 0 Feedback bit calculaton because of (6) Column 1 Feedback bit calculaton because of (7) Column 2 Feedback bit calculaton because of (8) Column 3 Feedback bit calculaton because of (9) Column 4
    Feedback bits State bits appeared on RHS of (5) Feedback bits State bits appeared on RHS of (6) Feedback bits State bits appeared on RHS of (7) Feedback bits State bits appeared on RHS of (8) Feedback bits State bits appeared on RHS of (2)
    0 $ x_{256}^0 $ $ x_{0}, \underline{x_{41}}, \overline{x_{70}} $ $ x_{252}^1 $ $ x_{252}, x_{42}, $ $ x_{83}, x_{8} $ $ x_{248}^2 $ $ x_{248}, x_{44}, $ $ x_{102}, x_{40} $ $ x_{244}^3 $ $ x_{244}, x_{43}, $ $ x_{118}, x_{103} $ $ x_{240}^4 $ $ x_{240}, \overline{x_{46}}, $ $ \underline{x_{141}}, x_{117} $
    1 $ x_{257}^0 $ $ x_{1}, \underline{x_{42}}, \overline{x_{71}} $ $ x_{253}^1 $ $ x_{253}, x_{43}, $ $ x_{84}, x_{9} $ $ x_{249}^2 $ $ x_{249}, x_{45}, $ $ x_{103}, x_{41} $ $ x_{245}^3 $ $ x_{245}, x_{44}, $ $ x_{119}, x_{104} $ $ x_{241}^4 $ $ x_{241}, \overline{x_{47}}, $ $ \underline{x_{142}}, x_{118} $
    2 $ x_{258}^0 $ $ x_{2}, \underline{x_{43}}, \overline{x_{72}} $ $ x_{254}^1 $ $ x_{254}, x_{44}, $ $ x_{85}, x_{10} $ $ x_{250}^2 $ $ x_{250}, \overline{x_{46}}, $ $ \underline{x_{104}}, x_{42} $ $ x_{246}^3 $ $ x_{246}, x_{45}, $ $ x_{120}, x_{105} $ $ x_{242}^4 $ $ x_{242}, \overline{x_{48}}, $ $ \underline{x_{143}}, x_{119} $
    3 $ x_{259}^0 $ $ x_{3}, \underline{x_{44}}, \overline{x_{73}} $ $ x_{255}^1 $ $ x_{255}, x_{45}, $ $ x_{86}, x_{11} $ $ x_{251}^2 $ $ x_{251}, \overline{x_{47}}, $ $ \underline{x_{105}}, x_{43} $ $ x_{247}^3 $ $ x_{247}, \overline{x_{46}}, $ $ \underline{x_{121}}, x_{106} $ $ x_{243}^4 $ $ x_{243}, \overline{x_{49}}, $ $ \underline{x_{144}}, x_{120} $
    4 $ x_{260}^0 $ $ x_{4}, \underline{x_{45}}, \overline{x_{74}} $ $ x_{256}^1 $ $ x_{256}^0, \overline{x_{46}}, $ $ \underline{x_{87}}, x_{12} $ $ x_{252}^2 $ $ x_{252}^1, \overline{x_{48}}, $ $ \underline{x_{106}}, x_{44} $ $ x_{248}^3 $ $ x_{248}^2, \overline{x_{47}}, $ $ \underline{x_{122}}, x_{107} $ $ x_{244}^4 $ $ x_{244}^3, \overline{x_{50}}, $ $ \underline{x_{145}}, x_{121} $
    5 $ x_{261}^0 $ $ x_{5}, \overline{x_{46}}, \overline{x_{75}} $ $ x_{257}^1 $ $ x_{257}^0, \overline{x_{47}}, $ $ \underline{x_{88}}, x_{13} $ $ x_{253}^2 $ $ x_{253}^1, \overline{x_{49}}, $ $ \underline{x_{107}}, x_{45} $ $ x_{249}^3 $ $ x_{249}^2, \overline{x_{48}}, $ $ \underline{x_{123}}, x_{108} $ $ x_{245}^4 $ $ x_{245}^3, \overline{x_{51}}, $ $ \underline{x_{146}}, x_{122} $
    6 $ x_{262}^0 $ $ x_{6}, \overline{x_{47}}, \overline{x_{76}} $ $ x_{258}^1 $ $ x_{258}^0, \overline{x_{48}}, $ $ \underline{x_{89}}, x_{14} $ $ x_{254}^2 $ $ x_{254}^1, \overline{x_{50}}, $ $ \underline{x_{108}}, \overline{x_{46}} $ $ x_{250}^3 $ $ x_{250}^2, \overline{x_{49}}, $ $ \underline{x_{124}}, x_{109} $ $ x_{246}^4 $ $ x_{246}^3, \overline{x_{52}}, $ $ \underline{x_{147}}, x_{123} $
    7 $ x_{263}^0 $ $ x_{7}, \overline{x_{48}}, \underline{x_{77}} $ $ x_{259}^1 $ $ x_{259}^0, \overline{x_{49}}, $ $ \underline{x_{90}}, x_{15} $ $ x_{255}^2 $ $ x_{255}^1, \overline{x_{51}}, $ $ \underline{x_{109}}, \overline{x_{47}} $ $ x_{251}^3 $ $ x_{251}^2, \overline{x_{50}}, $ $ \underline{x_{125}}, x_{110} $ $ x_{247}^4 $ $ x_{247}^3, \overline{x_{53}}, $ $ \underline{x_{148}}, x_{124} $
    8 $ x_{264}^0 $ $ x_{8}, \overline{x_{49}}, \underline{x_{78}} $ $ x_{260}^1 $ $ x_{260}^0, \overline{x_{50}}, $ $ \underline{x_{91}}, x_{16} $ $ x_{256}^2 $ $ x_{256}^1, \overline{x_{52}}, $ $ \underline{x_{110}}, \overline{x_{48}} $ $ x_{252}^3 $ $ x_{252}^2, \overline{x_{51}}, $ $ \underline{x_{126}}, x_{111} $ $ x_{248}^4 $ $ x_{248}^3, \overline{x_{54}}, $ $ \underline{x_{149}}, x_{125} $
    9 $ x_{265}^0 $ $ x_{9}, \overline{x_{50}}, \underline{x_{79}} $ $ x_{261}^1 $ $ x_{261}^0, \overline{x_{51}}, $ $ \underline{x_{92}}, x_{17} $ $ x_{257}^2 $ $ x_{257}^1, \overline{x_{53}}, $ $ \underline{x_{111}}, \overline{x_{49}} $ $ x_{253}^3 $ $ x_{253}^2, \overline{x_{52}}, $ $ \underline{x_{127}}, x_{112} $ $ x_{249}^4 $ $ x_{249}^3, \overline{x_{55}}, $ $ \underline{x_{150}}, x_{126} $
    10 $ x_{266}^0 $ $ x_{10}, \overline{x_{51}}, \underline{x_{80}} $ $ x_{262}^1 $ $ x_{262}^0, \overline{x_{52}}, $ $ \underline{x_{93}}, x_{18} $ $ x_{258}^2 $ $ x_{258}^1, \overline{x_{54}}, $ $ \underline{x_{112}}, \overline{x_{50}} $ $ x_{254}^3 $ $ x_{254}^2, \overline{x_{53}}, $ $ \underline{x_{128}}, x_{113} $ $ x_{250}^4 $ $ x_{250}^3, \overline{x_{56}}, $ $ \underline{x_{151}}, x_{127} $
    11 $ x_{267}^0 $ $ x_{11}, \overline{x_{52}}, \underline{x_{81}} $ $ x_{263}^1 $ $ x_{263}^0, \overline{x_{53}}, $ $ \underline{x_{94}}, x_{19} $ $ x_{259}^2 $ $ x_{259}^1, \overline{x_{55}}, $ $ \underline{x_{113}}, \overline{x_{51}} $ $ x_{255}^3 $ $ x_{255}^2, \overline{x_{54}}, $ $ \underline{x_{129}}, x_{114} $ $ x_{251}^4 $ $ x_{251}^3, \overline{x_{57}}, $ $ \underline{x_{152}}, x_{128} $
    12 $ x_{268}^0 $ $ x_{12}, \overline{x_{53}}, \underline{x_{82}} $ $ x_{264}^1 $ $ x_{264}^0, \overline{x_{54}}, $ $ \underline{x_{95}}, x_{20} $ $ x_{260}^2 $ $ x_{260}^1, \overline{x_{56}}, $ $ \underline{x_{114}}, \overline{x_{52}} $ $ x_{256}^3 $ $ x_{256}^2, \overline{x_{55}}, $ $ \underline{x_{130}}, x_{115} $ $ x_{252}^4 $ $ x_{252}^3, \overline{x_{58}}, $ $ \underline{x_{153}}, x_{129} $
    13 $ x_{269}^0 $ $ x_{13}, \overline{x_{54}}, \underline{x_{83}} $ $ x_{265}^1 $ $ x_{265}^0, \overline{x_{55}}, $ $ \underline{x_{96}}, x_{21} $ $ x_{261}^2 $ $ x_{261}^1, \overline{x_{57}}, $ $ \underline{x_{115}}, \overline{x_{53}} $ $ x_{257}^3 $ $ x_{257}^2, \overline{x_{56}}, $ $ \underline{x_{131}}, x_{116} $ $ x_{253}^4 $ $ x_{253}^3, \overline{x_{59}}, $ $ \underline{x_{154}}, x_{130} $
    14 $ x_{270}^0 $ $ x_{14}, \overline{x_{55}}, \underline{x_{84}} $ $ x_{266}^1 $ $ x_{266}^0, \overline{x_{56}}, $ $ \underline{x_{97}}, x_{22} $ $ x_{262}^2 $ $ x_{262}^1, \overline{x_{58}}, $ $ \underline{x_{116}}, \overline{x_{54}} $ $ x_{258}^3 $ $ x_{258}^2, \overline{x_{57}}, $ $ \underline{x_{132}}, x_{117} $ $ x_{254}^4 $ $ x_{254}^3, \overline{x_{60}}, $ $ \underline{x_{155}}, x_{131} $
    15 $ x_{271}^0 $ $ x_{15}, \overline{x_{56}}, \underline{x_{85}} $ $ x_{267}^1 $ $ x_{267}^0, \overline{x_{57}}, $ $ \underline{x_{98}}, x_{23} $ $ x_{263}^2 $ $ x_{263}^1, \overline{x_{59}}, $ $ \underline{x_{117}}, \overline{x_{55}} $ $ x_{259}^3 $ $ x_{259}^2, \overline{x_{58}}, $ $ \underline{x_{133}}, x_{118} $ $ x_{255}^4 $ $ x_{255}^3, \overline{x_{61}}, $ $ \underline{x_{156}}, x_{132} $
    16 $ x_{272}^0 $ $ x_{16}, \overline{x_{57}}, \underline{x_{86}} $ $ x_{268}^1 $ $ x_{268}^0, \overline{x_{58}}, $ $ \underline{x_{99}}, x_{24} $ $ x_{264}^2 $ $ x_{264}^1, \overline{x_{60}}, $ $ \underline{x_{118}}, \overline{x_{56}} $ $ x_{260}^3 $ $ x_{260}^2, \overline{x_{59}}, $ $ \underline{x_{134}}, x_{119} $ $ x_{256}^4 $ $ x_{256}^3, \overline{x_{62}}, $ $ \underline{x_{157}}, x_{133} $
    17 $ x_{273}^0 $ $ x_{17}, \overline{x_{58}}, \underline{x_{87}} $ $ x_{269}^1 $ $ x_{269}^0, \overline{x_{59}}, $ $ \underline{x_{100}}, x_{25} $ $ x_{265}^2 $ $ x_{265}^1, \overline{x_{61}}, $ $ \underline{x_{119}}, \overline{x_{57}} $ $ x_{261}^3 $ $ x_{261}^2, \overline{x_{60}}, $ $ \underline{x_{135}}, x_{120} $ $ x_{257}^4 $ $ x_{257}^3, \overline{x_{63}}, $ $ \underline{x_{158}}, x_{134} $
    18 $ x_{274}^0 $ $ x_{18}, \overline{x_{59}}, \underline{x_{88}} $ $ x_{270}^1 $ $ x_{270}^0, \overline{x_{60}}, $ $ \underline{x_{101}}, x_{26} $ $ x_{266}^2 $ $ x_{266}^1, \overline{x_{62}}, $ $ \underline{x_{120}}, \overline{x_{58}} $ $ x_{262}^3 $ $ x_{262}^2, \overline{x_{61}}, $ $ \underline{x_{136}}, x_{121} $ $ x_{258}^4 $ $ x_{258}^3, \overline{x_{64}}, $ $ \underline{x_{159}}, x_{135} $
    19 $ x_{275}^0 $ $ x_{19}, \overline{x_{60}}, \underline{x_{89}} $ $ x_{271}^1 $ $ x_{271}^0, \overline{x_{61}}, $ $ \underline{x_{102}}, x_{27} $ $ x_{267}^2 $ $ x_{267}^1, \overline{x_{63}}, $ $ \underline{x_{121}}, \overline{x_{59}} $ $ x_{263}^3 $ $ x_{263}^2, \overline{x_{62}}, $ $ \underline{x_{137}}, x_{122} $ $ x_{259}^4 $ $ x_{259}^3, \overline{x_{65}}, $ $ \underline{x_{160}}, x_{136} $
    20 $ x_{276}^0 $ $ x_{20}, \overline{x_{61}}, \underline{x_{90}} $ $ x_{272}^1 $ $ x_{272}^0, \overline{x_{62}}, $ $ \underline{x_{103}}, x_{28} $ $ x_{268}^2 $ $ x_{268}^1, \overline{x_{64}}, $ $ \underline{x_{122}}, \overline{x_{60}} $ $ x_{264}^3 $ $ x_{264}^2, \overline{x_{63}}, $ $ \underline{x_{138}}, x_{123} $ $ x_{260}^4 $ $ x_{260}^3, \overline{x_{66}}, $ $ \underline{x_{161}}, x_{137} $
    21 $ x_{277}^0 $ $ x_{21}, \overline{x_{62}}, \underline{x_{91}} $ $ x_{273}^1 $ $ x_{273}^0, \overline{x_{63}}, $ $ \underline{x_{104}}, x_{29} $ $ x_{269}^2 $ $ x_{269}^1, \overline{x_{65}}, $ $ \underline{x_{123}}, \overline{x_{61}} $ $ x_{265}^3 $ $ x_{265}^2, \overline{x_{64}}, $ $ \underline{x_{139}}, x_{124} $ $ x_{261}^4 $ $ x_{261}^3, \overline{x_{67}}, $ $ \underline{x_{162}}, x_{138} $
    22 $ x_{278}^0 $ $ x_{22}, \overline{x_{63}}, \underline{x_{92}} $ $ x_{274}^1 $ $ x_{274}^0, \overline{x_{64}}, $ $ \underline{x_{105}}, x_{30} $ $ x_{270}^2 $ $ x_{270}^1, \overline{x_{66}}, $ $ \underline{x_{124}}, \overline{x_{62}} $ $ x_{266}^3 $ $ x_{266}^2, \overline{x_{65}}, $ $ \underline{x_{140}}, x_{125} $ $ x_{262}^4 $ $ x_{262}^3, \overline{x_{68}}, $ $ \underline{x_{163}}, x_{139} $
    23 $ x_{279}^0 $ $ x_{23}, \overline{x_{64}}, \underline{x_{93}} $ $ x_{275}^1 $ $ x_{275}^0, \overline{x_{65}}, $ $ \underline{x_{106}}, x_{31} $ $ x_{271}^2 $ $ x_{271}^1, \overline{x_{67}}, $ $ \underline{x_{125}}, \overline{x_{63}} $ $ x_{267}^3 $ $ x_{267}^2, \overline{x_{66}}, $ $ \underline{x_{141}}, x_{126} $ $ x_{263}^4 $ $ x_{263}^3, \overline{x_{69}}, $ $ \underline{x_{164}}, x_{140} $
    24 $ x_{280}^0 $ $ x_{24}, \overline{x_{65}}, \underline{x_{94}} $ $ x_{276}^1 $ $ x_{276}^0, \overline{x_{66}}, $ $ \underline{x_{107}}, x_{32} $ $ x_{272}^2 $ $ x_{272}^1, \overline{x_{68}}, $ $ \underline{x_{126}}, \overline{x_{64}} $ $ x_{268}^3 $ $ x_{268}^2, \overline{x_{67}}, $ $ \underline{x_{142}}, x_{127} $ $ x_{264}^4 $ $ x_{264}^3, \overline{x_{70}}, $ $ \underline{x_{165}}, x_{141} $
    25 $ x_{281}^0 $ $ x_{25}, \overline{x_{66}}, \underline{x_{95}} $ $ x_{277}^1 $ $ x_{277}^0, \overline{x_{67}}, $ $ \underline{x_{108}}, x_{33} $ $ x_{273}^2 $ $ x_{273}^1, \overline{x_{69}}, $ $ \underline{x_{127}}, \overline{x_{65}} $ $ x_{269}^3 $ $ x_{269}^2, \overline{x_{68}}, $ $ \underline{x_{143}}, x_{128} $ $ x_{265}^4 $ $ x_{265}^3, \overline{x_{71}}, $ $ \underline{x_{166}}, x_{142} $
    26 $ x_{282}^0 $ $ x_{26}, \overline{x_{67}}, \underline{x_{96}} $ $ x_{278}^1 $ $ x_{278}^0, \overline{x_{68}}, $ $ \underline{x_{109}}, x_{34} $ $ x_{274}^2 $ $ x_{274}^1, \overline{x_{70}}, $ $ \underline{x_{128}}, \overline{x_{66}} $ $ x_{270}^3 $ $ x_{270}^2, \overline{x_{69}}, $ $ \underline{x_{144}}, x_{129} $ $ x_{266}^4 $ $ x_{266}^3, \overline{x_{72}}, $ $ \underline{x_{167}}, x_{143} $
    27 $ x_{283}^0 $ $ x_{27}, \overline{x_{68}}, \underline{x_{97}} $ $ x_{279}^1 $ $ x_{279}^0, \overline{x_{69}}, $ $ \underline{x_{110}}, x_{35} $ $ x_{275}^2 $ $ x_{275}^1, \overline{x_{71}}, $ $ \underline{x_{129}}, \overline{x_{67}} $ $ x_{271}^3 $ $ x_{271}^2, \overline{x_{70}}, $ $ \underline{x_{145}}, x_{130} $ $ x_{267}^4 $ $ x_{267}^3, \overline{x_{73}}, $ $ \underline{x_{168}}, x_{144} $
    28 $ x_{284}^0 $ $ x_{28}, \overline{x_{69}}, \underline{x_{98}} $ $ x_{280}^1 $ $ x_{280}^0, \overline{x_{70}}, $ $ \underline{x_{111}}, x_{36} $ $ x_{276}^2 $ $ x_{276}^1, \overline{x_{72}}, $ $ \underline{x_{130}}, \overline{x_{68}} $ $ x_{272}^3 $ $ x_{272}^2, \overline{x_{71}}, $ $ \underline{x_{146}}, x_{131} $ $ x_{268}^4 $ $ x_{268}^3, \overline{x_{74}}, $ $ \underline{x_{169}}, x_{145} $
    29 $ x_{285}^0 $ $ x_{29}, \overline{x_{70}}, \underline{x_{99}} $ $ x_{281}^1 $ $ x_{281}^0, \overline{x_{71}}, $ $ \underline{x_{112}}, x_{37} $ $ x_{277}^2 $ $ x_{277}^1, \overline{x_{73}}, $ $ \underline{x_{131}}, \overline{x_{69}} $ $ x_{273}^3 $ $ x_{273}^2, \overline{x_{72}}, $ $ \underline{x_{147}}, x_{132} $ $ x_{269}^4 $ $ x_{269}^3, \overline{x_{75}}, $ $ \underline{x_{170}}, x_{146} $
    30 $ x_{286}^0 $ $ x_{30}, \overline{x_{71}}, \underline{x_{100}} $ $ x_{282}^1 $ $ x_{282}^0, \overline{x_{72}}, $ $ \underline{x_{113}}, x_{38} $ $ x_{278}^2 $ $ x_{278}^1, \overline{x_{74}}, $ $ \underline{x_{132}}, \overline{x_{70}} $ $ x_{274}^3 $ $ x_{274}^2, \overline{x_{73}}, $ $ \underline{x_{148}}, x_{133} $ $ x_{270}^4 $ $ x_{270}^3, \overline{x_{76}}, $ $ \underline{x_{171}}, x_{147} $
    31 $ x_{287}^0 $ $ x_{31}, \overline{x_{72}}, \underline{x_{101}} $ $ x_{283}^1 $ $ x_{283}^0, \overline{x_{73}}, $ $ \underline{x_{114}}, x_{39} $ $ x_{279}^2 $ $ x_{279}^1, \overline{x_{75}}, $ $ \underline{x_{133}}, \overline{x_{71}} $ $ x_{275}^3 $ $ x_{275}^2, \overline{x_{74}}, $ $ \underline{x_{149}}, x_{134} $ $ x_{271}^4 $ $ x_{271}^3, x_{77}, $ $ x_{172}, x_{148} $
    32 $ x_{288}^0 $ $ x_{32}, \overline{x_{73}}, \underline{x_{102}} $ $ x_{284}^1 $ $ x_{284}^0, \overline{x_{74}}, $ $ \underline{x_{115}}, x_{40} $ $ x_{280}^2 $ $ x_{280}^1, \overline{x_{76}}, $ $ \underline{x_{134}}, \overline{x_{72}} $ $ x_{276}^3 $ $ x_{276}^2, \overline{x_{75}}, $ $ \underline{x_{150}}, x_{135} $ $ x_{272}^4 $ $ x_{272}^3, x_{78}, $ $ x_{173}, x_{149} $
    33 $ x_{289}^0 $ $ x_{33}, \overline{x_{74}}, \underline{x_{103}} $ $ x_{285}^1 $ $ x_{285}^0, \overline{x_{75}}, $ $ \underline{x_{116}}, x_{41} $ $ x_{281}^2 $ $ x_{281}^1, x_{77}, $ $ x_{135}, \overline{x_{73}} $ $ x_{277}^3 $ $ x_{277}^2, \overline{x_{76}}, $ $ \underline{x_{151}}, x_{136} $ $ x_{273}^4 $ $ x_{273}^3, x_{79}, $ $ x_{174}, x_{150} $
    34 $ x_{290}^0 $ $ x_{34}, \overline{x_{75}}, \underline{x_{104}} $ $ x_{286}^1 $ $ \underline{x_{286}^0}, \overline{x_{76}}, $ $ \underline{x_{117}}, x_{42} $ $ x_{282}^2 $ $ x_{282}^1, x_{78}, $ $ x_{136}, \overline{x_{74}} $ $ x_{278}^3 $ $ x_{278}^2, x_{77}, $ $ x_{152}, x_{137} $ $ x_{274}^4 $ $ x_{274}^3, x_{80}, $ $ x_{175}, x_{151} $
     | Show Table
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    Table 2.  State Bits required to calculate feedback bits

    Row Feedback bit calculaton because of (10) Column 5 Feedback bit calculaton because of (11) Column 6 Feedback bit calculaton because of (12) Column 7 Feedback bit calculaton because of (13) Column 8 Feedback bit calculaton because of (14) Column 9
    Feedback bits State bits appeared on RHS of (10) Feedback bits State bits appeared on RHS of (11) Feedback bits State bits appeared on RHS of (12) Feedback bits State bits appeared on RHS of (13) Feedback bits State bits appeared on RHS of (14)
    0 $ x_{236}^5 $ $ x_{236}, \overline{x_{67}}, \underline{x_{90}, x_{110}, x_{137}} $ $ x_{232}^6 $ $ x_{232}, \overline{x_{50}}, $ $ \underline{x_{159}}, x_{189} $ $ x_{218}^7 $ $ x_{218}, \underline{x_{3}}, \overline{x_{32}} $ $ x_{214}^8 $ $ x_{214}, x_{4}, x_{45} $ $ x_{210}^9 $ $ x_{210}, \underline{x_{6}}, \overline{x_{64}} $
    1 $ x_{237}^5 $ $ x_{237}, \overline{x_{68}}, \underline{x_{91}, x_{111}, x_{138}} $ $ x_{233}^6 $ $ x_{233}, \overline{x_{51}}, $ $ \underline{x_{160}}, x_{190} $ $ x_{219}^7 $ $ x_{219}, \underline{x_{4}}, \overline{x_{33}} $ $ x_{215}^8 $ $ x_{215}, \underline{x_{5}}, \overline{x_{46}} $ $ x_{211}^9 $ $ x_{211}, \underline{x_{7}}, \overline{x_{65}} $
    2 $ x_{238}^5 $ $ x_{238}, \overline{x_{69}}, \underline{x_{92}, x_{112}, x_{139}} $ $ x_{234}^6 $ $ x_{234}, \overline{x_{52}}, $ $ \underline{x_{161}}, x_{191} $ $ x_{220}^7 $ $ x_{220}, \underline{x_{5}}, \overline{x_{34}} $ $ x_{216}^8 $ $ x_{216}, \underline{x_{6}}, \overline{x_{47}} $ $ x_{212}^9 $ $ x_{212}, \underline{x_{8}}, \overline{x_{66}} $
    3 $ x_{239}^5 $ $ x_{239}, \overline{x_{70}}, \underline{x_{93}, x_{113}, x_{140}} $ $ x_{235}^6 $ $ x_{235}, \overline{x_{53}}, $ $ \underline{x_{162}}, x_{192} $ $ x_{221}^7 $ $ x_{221}, \underline{x_{6}}, \overline{x_{35}} $ $ x_{217}^8 $ $ x_{217}, \underline{x_{7}}, \overline{x_{48}} $ $ x_{213}^9 $ $ x_{213}, \underline{x_{9}}, \overline{x_{67}} $
    4 $ x_{240}^5 $ $ x_{240}^4, \overline{x_{71}}, \underline{x_{94}, x_{114}, x_{141}} $ $ x_{236}^6 $ $ x_{236}^5, \overline{x_{54}}, $ $ \underline{x_{163}}, x_{193} $ $ x_{222}^7 $ $ x_{222}, \underline{x_{7}}, \overline{x_{36}} $ $ x_{218}^8 $ $ x_{218}^7, \underline{x_{8}}, \overline{x_{49}} $ $ x_{214}^9 $ $ x_{214}^8, \underline{x_{10}}, \overline{x_{68}} $
    5 $ x_{241}^5 $ $ x_{241}^4, \overline{x_{72}}, \underline{x_{95}, x_{115}, x_{142}} $ $ x_{237}^6 $ $ x_{237}^5, \overline{x_{55}}, $ $ \underline{x_{164}}, x_{194}^{13} $ $ x_{223}^7 $ $ x_{223}, x_{8}, x_{37} $ $ x_{219}^8 $ $ x_{219}^7, \underline{x_{9}}, \overline{x_{50}} $ $ x_{215}^9 $ $ x_{215}^8, \underline{x_{11}}, \overline{x_{69}} $
    6 $ x_{242}^5 $ $ x_{242}^4, \overline{x_{73}}, \underline{x_{96}, x_{116}, x_{143}} $ $ x_{238}^6 $ $ x_{238}^5, \overline{x_{56}}, $ $ \underline{x_{165}}, x_{195}^{13} $ $ x_{224}^7 $ $ x_{224}, x_{9}, x_{38} $ $ x_{220}^8 $ $ x_{220}^7, \underline{x_{10}}, \overline{x_{51}} $ $ x_{216}^9 $ $ x_{216}^8, \underline{x_{12}}, \overline{x_{70}} $
    7 $ x_{243}^5 $ $ x_{243}^4, \overline{x_{74}}, \underline{x_{97}, x_{117}, x_{144}} $ $ x_{239}^6 $ $ x_{239}^5, \overline{x_{57}}, $ $ \underline{x_{166}}, x_{196}^{13} $ $ x_{225}^7 $ $ x_{225}, x_{10}, x_{39} $ $ x_{221}^8 $ $ x_{221}^7, \underline{x_{11}}, \overline{x_{52}} $ $ x_{217}^9 $ $ x_{217}^8, \underline{x_{13}}, \overline{x_{71}} $
    8 $ x_{244}^5 $ $ x_{244}^4, \overline{x_{75}}, \underline{x_{98}, x_{118}, x_{145}} $ $ x_{240}^6 $ $ x_{240}^5, \overline{x_{58}}, $ $ \underline{x_{167}}, x_{197}^{13} $ $ x_{226}^7 $ $ x_{226}, x_{11}, x_{40} $ $ x_{222}^8 $ $ x_{222}^7, \underline{x_{12}}, \overline{x_{53}} $ $ x_{218}^9 $ $ x_{218}^8, \underline{x_{14}}, \overline{x_{72}} $
    9 $ x_{245}^5 $ $ x_{245}^4, \overline{x_{76}}, \underline{x_{99}, x_{119}, x_{146}} $ $ x_{241}^6 $ $ x_{241}^5, \overline{x_{59}}, $ $ \underline{x_{168}}, x_{198}^{13} $ $ x_{227}^7 $ $ x_{227}, x_{12}, x_{41} $ $ x_{223}^8 $ $ x_{223}^7, \underline{x_{13}}, \overline{x_{54}} $ $ x_{219}^9 $ $ x_{219}^8, \underline{x_{15}}, \overline{x_{73}} $
    10 $ x_{246}^5 $ $ x_{246}^4, x_{77}, x_{100}, x_{120}, x_{147} $ $ x_{242}^6 $ $ x_{242}^5, \overline{x_{60}}, $ $ \underline{x_{169}}, x_{199}^{13} $ $ x_{228}^7 $ $ x_{228}, x_{13}, x_{42} $ $ x_{224}^8 $ $ x_{224}^7, \underline{x_{14}}, \overline{x_{55}} $ $ x_{220}^9 $ $ x_{220}^8, \underline{x_{16}}, \overline{x_{74}} $
    11 $ x_{247}^5 $ $ x_{247}^4, x_{78}, x_{101}, x_{121}, x_{148} $ $ x_{243}^6 $ $ x_{243}^5, \overline{x_{61}}, $ $ \underline{x_{170}}, x_{200}^{13} $ $ x_{229}^7 $ $ x_{229}, x_{14}, x_{43} $ $ x_{225}^8 $ $ x_{225}^7, \underline{x_{15}}, \overline{x_{56}} $ $ x_{221}^9 $ $ x_{221}^8, \underline{x_{17}}, \overline{x_{75}} $
    12 $ x_{248}^5 $ $ x_{248}^4, x_{79}, x_{102}, x_{122}, x_{149} $ $ x_{244}^6 $ $ x_{244}^5, \overline{x_{62}}, $ $ \underline{x_{171}}, x_{201}^{13} $ $ x_{230}^7 $ $ x_{230}, x_{15}, x_{44} $ $ x_{226}^8 $ $ x_{226}^7, \underline{x_{16}}, \overline{x_{57}} $ $ x_{222}^9 $ $ x_{222}^8, \underline{x_{18}}, \overline{x_{76}} $
    13 $ x_{249}^5 $ $ x_{249}^4, x_{80}, x_{103}, x_{123}, x_{150} $ $ x_{245}^6 $ $ x_{245}^5, \overline{x_{63}}, $ $ \underline{x_{172}}, x_{202}^{13} $ $ x_{231}^7 $ $ x_{231}, x_{16}, x_{45} $ $ x_{227}^8 $ $ x_{227}^7, \underline{x_{17}}, \overline{x_{58}} $ $ x_{223}^9 $ $ x_{223}^8, x_{19}, x_{77} $
    14 $ x_{250}^5 $ $ x_{250}^4, x_{81}, x_{104}, x_{124}, x_{151} $ $ x_{246}^6 $ $ x_{246}^5, \overline{x_{64}}, $ $ \underline{x_{173}}, x_{203}^{13} $ $ x_{232}^7 $ $ x_{232}^6, \underline{x_{17}}, \overline{x_{46}} $ $ x_{228}^8 $ $ x_{228}^7, \underline{x_{18}}, \overline{x_{59}} $ $ x_{224}^9 $ $ x_{224}^8, x_{20}, x_{78} $
    15 $ x_{251}^5 $ $ x_{251}^4, x_{82}, x_{105}, x_{125}, x_{152} $ $ x_{247}^6 $ $ x_{247}^5, \overline{x_{65}}, $ $ \underline{x_{174}}, x_{204}^{13} $ $ x_{233}^7 $ $ x_{233}^6, \underline{x_{18}}, \overline{x_{47}} $ $ x_{229}^8 $ $ x_{229}^7, \underline{x_{19}}, \overline{x_{60}} $ $ x_{225}^9 $ $ x_{225}^8, x_{21}, x_{79} $
    16 $ x_{252}^5 $ $ x_{252}^4, x_{83}, x_{106}, x_{126}, x_{153} $ $ x_{248}^6 $ $ x_{248}^5, \overline{x_{66}}, $ $ \underline{x_{175}}, x_{205}^{13} $ $ x_{234}^7 $ $ x_{234}^6, \underline{x_{19}}, \overline{x_{48}} $ $ x_{230}^8 $ $ x_{230}^7, \underline{x_{20}}, \overline{x_{61}} $ $ x_{226}^9 $ $ x_{226}^8, x_{22}, x_{80} $
    17 $ x_{253}^5 $ $ x_{253}^4, x_{84}, x_{107}, x_{127}, x_{154} $ $ x_{249}^6 $ $ x_{249}^5, \overline{x_{67}}, $ $ \underline{x_{176}}, x_{206}^{13} $ $ x_{235}^7 $ $ x_{235}^6, \underline{x_{20}}, \overline{x_{49}} $ $ x_{231}^8 $ $ x_{231}^7, \underline{x_{21}}, \overline{x_{62}} $ $ x_{227}^9 $ $ x_{227}^8, x_{23}, x_{81} $
    18 $ x_{254}^5 $ $ x_{254}^4, x_{85}, x_{108}, x_{128}, x_{155} $ $ x_{250}^6 $ $ x_{250}^5, \overline{x_{68}}, $ $ \underline{x_{177}}, x_{207}^{13} $ $ x_{236}^7 $ $ x_{236}^6, \underline{x_{21}}, \overline{x_{50}} $ $ x_{232}^8 $ $ x_{232}^7, \underline{x_{22}}, \overline{x_{63}} $ $ x_{228}^9 $ $ x_{228}^8, x_{24}, x_{82} $
    19 $ x_{255}^5 $ $ x_{255}^4, x_{86}, x_{109}, x_{129}, x_{156} $ $ x_{251}^6 $ $ x_{251}^5, \overline{x_{69}}, $ $ \underline{x_{178}}, x_{208}^{13} $ $ x_{237}^7 $ $ x_{237}^6, \underline{x_{22}}, \overline{x_{51}} $ $ x_{233}^8 $ $ x_{233}^7, \underline{x_{23}}, \overline{x_{64}} $ $ x_{229}^9 $ $ x_{229}^8, x_{25}, x_{83} $
    20 $ x_{256}^5 $ $ x_{256}^4, x_{87}, x_{110}, x_{130}, x_{157} $ $ x_{252}^6 $ $ x_{252}^5, \overline{x_{70}}, $ $ \underline{x_{179}}, x_{209}^{13} $ $ x_{238}^7 $ $ x_{238}^6, \underline{x_{23}}, \overline{x_{52}} $ $ x_{234}^8 $ $ x_{234}^7, \underline{x_{24}}, \overline{x_{65}} $ $ x_{230}^9 $ $ x_{230}^8, x_{26}, x_{84} $
    21 $ x_{257}^5 $ $ x_{257}^4, x_{88}, x_{111}, x_{131}, x_{158} $ $ x_{253}^6 $ $ x_{253}^5, \overline{x_{71}}, $ $ \underline{x_{180}}, x_{210}^{13} $ $ x_{239}^7 $ $ x_{239}^6, \underline{x_{24}}, \overline{x_{53}} $ $ x_{235}^8 $ $ x_{235}^7, \underline{x_{25}}, \overline{x_{66}} $ $ x_{231}^9 $ $ x_{231}^8, x_{27}, x_{85} $
    22 $ x_{258}^5 $ $ x_{258}^4, x_{89}, x_{112}, x_{132}, x_{159} $ $ x_{254}^6 $ $ x_{254}^5, \overline{x_{72}}, $ $ \underline{x_{181}}, x_{211}^{13} $ $ x_{240}^7 $ $ x_{240}^6, \underline{x_{25}}, \overline{x_{54}} $ $ x_{236}^8 $ $ x_{236}^7, \underline{x_{26}}, \overline{x_{67}} $ $ x_{232}^9 $ $ x_{232}^8, x_{28}, x_{86} $
    23 $ x_{259}^5 $ $ x_{259}^4, x_{90}, x_{113}, x_{133}, x_{160} $ $ x_{255}^6 $ $ x_{255}^5, \overline{x_{73}}, $ $ \underline{x_{182}}, x_{212}^{13} $ $ x_{241}^7 $ $ x_{241}^6, \underline{x_{26}}, \overline{x_{55}} $ $ x_{237}^8 $ $ x_{237}^7, \underline{x_{27}}, \overline{x_{68}} $ $ x_{233}^9 $ $ x_{233}^8, \overline{x_{29}}, \underline{x_{87}} $
    24 $ x_{260}^5 $ $ x_{260}^4, x_{91}, x_{114}, x_{134}, x_{161} $ $ x_{256}^6 $ $ x_{256}^5, \overline{x_{74}}, $ $ \underline{x_{183}}, x_{213}^{13} $ $ x_{242}^7 $ $ x_{242}^6, \underline{x_{27}}, \overline{x_{56}} $ $ x_{238}^8 $ $ x_{238}^7, \underline{x_{28}}, \overline{x_{69}} $ $ x_{234}^9 $ $ x_{234}^8, \overline{x_{30}}, \underline{x_{88}} $
    25 $ x_{261}^5 $ $ x_{261}^4, x_{92}, x_{115}, x_{135}, x_{162} $ $ x_{257}^6 $ $ x_{257}^5, \overline{x_{75}}, $ $ \underline{x_{184}}, x_{214}^{13} $ $ x_{243}^7 $ $ x_{243}^6, \underline{x_{28}}, \overline{x_{57}} $ $ x_{239}^8 $ $ x_{239}^7, \overline{x_{29}}, \overline{x_{70}} $ $ x_{235}^9 $ $ x_{235}^8, \overline{x_{31}}, \underline{x_{89}} $
    26 $ x_{262}^5 $ $ x_{262}^4, x_{93}, x_{116}, x_{136}, x_{163} $ $ x_{258}^6 $ $ x_{258}^5, \overline{x_{76}}, $ $ \underline{x_{185}}, x_{215}^{13} $ $ x_{244}^7 $ $ x_{244}^6, \overline{x_{29}}, \overline{x_{58}} $ $ x_{240}^8 $ $ x_{240}^7, \overline{x_{30}}, \overline{x_{71}} $ $ x_{236}^9 $ $ x_{236}^8, \overline{x_{32}}, \underline{x_{90}} $
    27 $ x_{263}^5 $ $ x_{263}^4, x_{94}, x_{117}, x_{137}, x_{164} $ $ x_{259}^6 $ $ x_{259}^5, x_{77}, $ $ x_{186}, x_{216}^{13} $ $ x_{245}^7 $ $ x_{245}^6, \overline{x_{30}}, \overline{x_{59}} $ $ x_{241}^8 $ $ x_{241}^7, \overline{x_{31}}, \overline{x_{72}} $ $ x_{237}^9 $ $ x_{237}^8, \overline{x_{33}}, \underline{x_{91}} $
    28 $ x_{264}^5 $ $ x_{264}^4, x_{95}, x_{118}, x_{138}, x_{165} $ $ x_{260}^6 $ $ x_{260}^5, x_{78}, $ $ x_{187}, x_{217}^{13} $ $ x_{246}^7 $ $ x_{246}^6, \overline{x_{31}}, \overline{x_{60}} $ $ x_{242}^8 $ $ x_{242}^7, \overline{x_{32}}, \overline{x_{73}} $ $ x_{238}^9 $ $ x_{238}^8, \overline{x_{34}}, \underline{x_{92}} $
    29 $ x_{265}^5 $ $ x_{265}^4, x_{96}, x_{119}, x_{139}, x_{166} $ $ x_{261}^6 $ $ x_{261}^5, x_{79}, $ $ x_{188}, x_{218}^{13} $ $ x_{247}^7 $ $ x_{247}^6, \overline{x_{32}}, \overline{x_{61}} $ $ x_{243}^8 $ $ x_{243}^7, \overline{x_{33}}, \overline{x_{74}} $ $ x_{239}^9 $ $ x_{239}^8, \overline{x_{35}}, \underline{x_{93}} $
    30 $ x_{266}^5 $ $ x_{266}^4, x_{97}, x_{120}, x_{140}, x_{167} $ $ x_{262}^6 $ $ x_{262}^5, x_{80}, $ $ x_{189}, x_{219}^{13} $ $ x_{248}^7 $ $ x_{248}^6, \overline{x_{33}}, \overline{x_{62}} $ $ x_{244}^8 $ $ x_{244}^7, \overline{x_{34}}, \overline{x_{75}} $ $ x_{240}^9 $ $ x_{240}^8, \overline{x_{36}}, \underline{x_{94}} $
    31 $ x_{267}^5 $ $ x_{267}^4, x_{98}, x_{121}, x_{141}, x_{168} $ $ x_{263}^6 $ $ x_{263}^5, x_{81}, $ $ x_{190}, x_{220}^{13} $ $ x_{249}^7 $ $ x_{249}^6, \overline{x_{34}}, \overline{x_{63}} $ $ x_{245}^8 $ $ x_{245}^7, \overline{x_{35}}, \overline{x_{76}} $ $ x_{241}^9 $ $ x_{241}^8, x_{37}, x_{95} $
    32 $ x_{268}^5 $ $ x_{268}^4, x_{99}, x_{122}, x_{142}, x_{169} $ $ x_{264}^6 $ $ x_{264}^5, x_{82}, $ $ x_{191}, x_{221}^{13} $ $ x_{250}^7 $ $ x_{250}^6, \overline{x_{35}}, \overline{x_{64}} $ $ x_{246}^8 $ $ x_{246}^7, \overline{x_{36}},\underline{x_{77}} $ $ x_{242}^9 $ $ x_{242}^8, x_{38}, x_{96} $
    33 $ x_{269}^5 $ $ x_{269}^4, x_{100}, x_{123}, x_{143}, x_{170} $ $ x_{265}^6 $ $ x_{265}^5, x_{83}, $ $ x_{192}, x_{222}^{13} $ $ x_{251}^7 $ $ x_{251}^6, \overline{x_{36}}, \overline{x_{65}} $ $ x_{247}^8 $ $ x_{247}^7, x_{37}, x_{78} $ $ x_{243}^9 $ $ x_{243}^8, x_{39}, x_{97} $
    34 $ x_{270}^5 $ $ x_{270}^4, x_{101}, x_{124}, x_{144}, x_{171} $ $ x_{266}^6 $ $ x_{266}^5, x_{84}, $ $ x_{193}, x_{223}^{13} $ $ x_{252}^7 $ $ x_{252}^6, \underline{x_{37}}, \overline{x_{66}} $ $ x_{248}^8 $ $ x_{248}^7, x_{38}, x_{79} $ $ x_{244}^9 $ $ x_{244}^8, x_{40}, x_{98} $
     | Show Table
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    Table 3.  State Bits required to calculate feedback bits

    Row Feedback bit calculaton because of (15) Column 10 Feedback bit calculaton because of (16) Column11 Feedback bit calculaton because of (17) Column 12 Feedback bit calculaton because of (18) Column 13
    Feedback bits State bits appeared on RHS of (15) Feedback bits State bits appeared on RHS of (16) Feedback bits State bits appeared on RHS of (17) Feedback bits State bits appeared on RHS of (18)
    0 $ x_{206}^{10} $ $ x_{206}, x_{5}, x_{80} $ $ x_{202}^{11} $ $ x_{202}, x_{8}, $ $ x_{103} $ $ x_{198}^{12} $ $ x_{198}, \overline{x_{29}}, \overline{x_{52}}, \overline{x_{72}}, \underline{x_{99}} $ $ x_{194}^{13} $ $ x_{194}, x_{12}, x_{121} $
    1 $ x_{207}^{10} $ $ x_{207}, x_{6}, x_{81} $ $ x_{203}^{11} $ $ x_{203}, x_{9}, $ $ x_{104} $ $ x_{199}^{12} $ $ x_{199}, \overline{x_{30}}, \overline{x_{53}}, \overline{x_{73}}, \underline{x_{100}} $ $ x_{195}^{13} $ $ x_{195}, x_{13}, x_{122} $
    2 $ x_{208}^{10} $ $ x_{208}, x_{7}, x_{82} $ $ x_{204}^{11} $ $ x_{204}, x_{10}, $ $ x_{105} $ $ x_{200}^{12} $ $ x_{200}, \overline{x_{31}}, \overline{x_{54}}, \overline{x_{74}}, \underline{x_{101}} $ $ x_{196}^{13} $ $ x_{196}, x_{14}, x_{123} $
    3 $ x_{209}^{10} $ $ x_{209}, x_{8}, x_{83} $ $ x_{205}^{11} $ $ x_{205}, x_{11}, $ $ x_{106} $ $ x_{201}^{12} $ $ x_{201}, \overline{x_{32}}, \overline{x_{55}}, \overline{x_{75}}, \underline{x_{102}} $ $ x_{197}^{13} $ $ x_{197}, x_{15}, x_{124} $
    4 $ x_{210}^{10} $ $ x_{210}^9, x_{9}, x_{84} $ $ x_{206}^{11} $ $ x_{206}^{10}, x_{12}, $ $ x_{107} $ $ x_{202}^{12} $ $ x_{202}^{11}, \overline{x_{33}}, \overline{x_{56}}, \overline{x_{76}}, \underline{x_{103}} $ $ x_{198}^{13} $ $ x_{198}^{12}, x_{16}, x_{125} $
    5 $ x_{211}^{10} $ $ x_{211}^9, x_{10}, x_{85} $ $ x_{207}^{11} $ $ x_{207}^{10}, x_{13}, $ $ x_{108} $ $ x_{203}^{12} $ $ x_{203}^{11}, \overline{x_{34}}, \overline{x_{57}}, \underline{x_{77}}, \underline{x_{104}} $ $ x_{199}^{13} $ $ x_{199}^{12}, x_{17}, x_{126} $
    6 $ x_{212}^{10} $ $ x_{212}^9, x_{11}, x_{86} $ $ x_{208}^{11} $ $ x_{208}^{10}, x_{14}, $ $ x_{109} $ $ x_{204}^{12} $ $ x_{204}^{11}, \overline{x_{35}}, \overline{x_{58}}, \underline{x_{78}}, \underline{x_{105}} $ $ x_{200}^{13} $ $ x_{200}^{12}, x_{18}, x_{127} $
    7 $ x_{213}^{10} $ $ x_{213}^9, x_{12}, x_{87} $ $ x_{209}^{11} $ $ x_{209}^{10}, x_{15}, $ $ x_{110} $ $ x_{205}^{12} $ $ x_{205}^{11}, \overline{x_{36}}, \overline{x_{59}}, \underline{x_{79}}, \underline{x_{106}} $ $ x_{201}^{13} $ $ x_{201}^{12}, x_{19}, x_{128} $
    8 $ x_{214}^{10} $ $ x_{214}^9, x_{13}, x_{88} $ $ x_{210}^{11} $ $ x_{210}^{10}, x_{16}, $ $ x_{111} $ $ x_{206}^{12} $ $ x_{206}^{11}, \underline{x_{37}}, \overline{x_{60}}, \underline{x_{80}}, \underline{x_{107}} $ $ x_{202}^{13} $ $ x_{202}^{12}, x_{20}, x_{129} $
    9 $ x_{215}^{10} $ $ x_{215}^9, x_{14}, x_{89} $ $ x_{211}^{11} $ $ x_{211}^{10}, x_{17}, $ $ x_{112} $ $ x_{207}^{12} $ $ x_{207}^{11}, \underline{x_{38}}, \overline{x_{61}}, \underline{x_{81}}, \underline{x_{108}} $ $ x_{203}^{13} $ $ x_{203}^{12}, x_{21}, x_{130} $
    10 $ x_{216}^{10} $ $ x_{216}^9, x_{15}, x_{90} $ $ x_{212}^{11} $ $ x_{212}^{10}, x_{18}, $ $ x_{113} $ $ x_{208}^{12} $ $ x_{208}^{11}, \underline{x_{39}}, \overline{x_{62}}, \underline{x_{82}}, \underline{x_{109}} $ $ x_{204}^{13} $ $ x_{204}^{12}, x_{22}, x_{131} $
    11 $ x_{217}^{10} $ $ x_{217}^9, x_{16}, x_{91} $ $ x_{213}^{11} $ $ x_{213}^{10}, x_{19}, $ $ x_{114} $ $ x_{209}^{12} $ $ x_{209}^{11}, \underline{x_{40}}, \overline{x_{63}}, \underline{x_{83}}, \underline{x_{110}} $ $ x_{205}^{13} $ $ x_{205}^{12}, x_{23}, x_{132} $
    12 $ x_{218}^{10} $ $ x_{218}^9, x_{17}, x_{92} $ $ x_{214}^{11} $ $ x_{214}^{10}, x_{20}, $ $ x_{115} $ $ x_{210}^{12} $ $ x_{210}^{11}, \underline{x_{41}}, \overline{x_{64}}, \underline{x_{84}}, \underline{x_{111}} $ $ x_{206}^{13} $ $ x_{206}^{12}, x_{24}, x_{133} $
    13 $ x_{219}^{10} $ $ x_{219}^9, x_{18}, x_{93} $ $ x_{215}^{11} $ $ x_{215}^{10}, x_{21}, $ $ x_{116} $ $ x_{211}^{12} $ $ x_{211}^{11}, \underline{x_{42}}, \overline{x_{65}}, \underline{x_{85}}, \underline{x_{112}} $ $ x_{207}^{13} $ $ x_{207}^{12}, x_{25}, x_{134} $
    14 $ x_{220}^{10} $ $ x_{220}^9, x_{19}, x_{94} $ $ x_{216}^{11} $ $ x_{216}^{10}, x_{22}, $ $ x_{117} $ $ x_{212}^{12} $ $ x_{212}^{11}, \underline{x_{43}}, \overline{x_{66}}, \underline{x_{86}}, \underline{x_{113}} $ $ x_{208}^{13} $ $ x_{208}^{12}, x_{26}, x_{135} $
    15 $ x_{221}^{10} $ $ x_{221}^9, x_{20}, x_{95} $ $ x_{217}^{11} $ $ x_{217}^{10}, x_{23}, $ $ x_{118} $ $ x_{213}^{12} $ $ x_{213}^{11}, \underline{x_{44}}, \overline{x_{67}}, \underline{x_{87}}, \underline{x_{114}} $ $ x_{209}^{13} $ $ x_{209}^{12}, x_{27}, x_{136} $
    16 $ x_{222}^{10} $ $ x_{222}^9, x_{21}, x_{96} $ $ x_{218}^{11} $ $ x_{218}^{10}, x_{24}, $ $ x_{119} $ $ x_{214}^{12} $ $ x_{214}^{11}, \underline{x_{45}}, \overline{x_{68}}, \underline{x_{88}}, \underline{x_{115}} $ $ x_{210}^{13} $ $ x_{210}^{12}, x_{28}, x_{137} $
    17 $ x_{223}^{10} $ $ x_{223}^9, x_{22}, x_{97} $ $ x_{219}^{11} $ $ x_{219}^{10}, x_{25}, $ $ x_{120} $ $ x_{215}^{12} $ $ x_{215}^{11}, \overline{x_{46}}, \overline{x_{69}}, \underline{x_{89}}, \underline{x_{116}} $ $ x_{211}^{13} $ $ x_{211}^{12}, \overline{x_{29}}, \underline{x_{138}} $
    18 $ x_{224}^{10} $ $ x_{224}^9, x_{23}, x_{98} $ $ x_{220}^{11} $ $ x_{220}^{10}, x_{26}, $ $ x_{121} $ $ x_{216}^{12} $ $ x_{216}^{11}, \overline{x_{47}}, \overline{x_{70}}, \underline{x_{90}}, \underline{x_{117}} $ $ x_{212}^{13} $ $ x_{212}^{12}, \overline{x_{30}}, \underline{x_{139}} $
    19 $ x_{225}^{10} $ $ x_{225}^9, x_{24}, x_{99} $ $ x_{221}^{11} $ $ x_{221}^{10}, x_{27}, $ $ x_{122} $ $ x_{217}^{12} $ $ x_{217}^{11}, \overline{x_{48}}, \overline{x_{71}}, \underline{x_{91}}, \underline{x_{118}} $ $ x_{213}^{13} $ $ x_{213}^{12}, \overline{x_{31}}, \underline{x_{140}} $
    20 $ x_{226}^{10} $ $ x_{226}^9, x_{25}, x_{100} $ $ x_{222}^{11} $ $ x_{222}^{10}, x_{28}, $ $ x_{123} $ $ x_{218}^{12} $ $ x_{218}^{11}, \overline{x_{49}}, \overline{x_{72}}, \underline{x_{92}}, \underline{x_{119}} $ $ x_{214}^{13} $ $ x_{214}^{12}, \overline{x_{32}}, \underline{x_{141}} $
    21 $ x_{227}^{10} $ $ x_{227}^9, x_{26}, x_{101} $ $ x_{223}^{11} $ $ x_{223}^{10}, \overline{x_{29}}, $ $ \underline{x_{124}} $ $ x_{219}^{12} $ $ x_{219}^{11}, \overline{x_{50}}, \overline{x_{73}}, \underline{x_{93}}, \underline{x_{120}} $ $ x_{215}^{13} $ $ x_{215}^{12}, \overline{x_{33}}, \underline{x_{142}} $
    22 $ x_{228}^{10} $ $ x_{228}^9, x_{27}, x_{102} $ $ x_{224}^{11} $ $ x_{224}^{10}, \overline{x_{30}}, $ $ \underline{x_{125}} $ $ x_{220}^{12} $ $ x_{220}^{11}, \overline{x_{51}}, \overline{x_{74}}, \underline{x_{94}}, \underline{x_{121}} $ $ x_{216}^{13} $ $ x_{216}^{12}, \overline{x_{34}}, \underline{x_{143}} $
    23 $ x_{229}^{10} $ $ x_{229}^9, x_{28}, x_{103} $ $ x_{225}^{11} $ $ x_{225}^{10}, \overline{x_{31}}, $ $ \underline{x_{126}} $ $ x_{221}^{12} $ $ x_{221}^{11}, \overline{x_{52}}, \overline{x_{75}}, \underline{x_{95}}, \underline{x_{122}} $ $ x_{217}^{13} $ $ x_{217}^{12}, \overline{x_{35}}, \underline{x_{144}} $
    24 $ x_{230}^{10} $ $ x_{230}^9, \overline{x_{29}}, \underline{x_{104}} $ $ x_{226}^{11} $ $ x_{226}^{10}, \overline{x_{32}}, $ $ \underline{x_{127}} $ $ x_{222}^{12} $ $ x_{222}^{11}, \overline{x_{53}}, \overline{x_{76}}, \underline{x_{96}}, \underline{x_{123}} $ $ x_{218}^{13} $ $ x_{218}^{12}, \overline{x_{36}}, \underline{x_{145}} $
    25 $ x_{231}^{10} $ $ x_{231}^9, \overline{x_{30}}, \underline{x_{105}} $ $ x_{227}^{11} $ $ x_{227}^{10}, \overline{x_{33}}, $ $ \underline{x_{128}} $ $ x_{223}^{12} $ $ x_{223}^{11}, \overline{x_{54}}, \underline{x_{77}}, \underline{x_{97}}, \underline{x_{124}} $ $ x_{219}^{13} $ $ x_{219}^{12}, x_{37}, x_{146} $
    26 $ x_{232}^{10} $ $ x_{232}^9, \overline{x_{31}}, \underline{x_{106}} $ $ x_{228}^{11} $ $ x_{228}^{10}, \overline{x_{34}}, $ $ \underline{x_{129}} $ $ x_{224}^{12} $ $ x_{224}^{11}, \overline{x_{55}}, \underline{x_{78}}, \underline{x_{98}}, \underline{x_{125}} $ $ x_{220}^{13} $ $ x_{220}^{12}, x_{38}, x_{147} $
    27 $ x_{233}^{10} $ $ x_{233}^9, \overline{x_{32}}, \underline{x_{107}} $ $ x_{229}^{11} $ $ x_{229}^{10}, \overline{x_{35}}, $ $ \underline{x_{130}} $ $ x_{225}^{12} $ $ x_{225}^{11}, \overline{x_{56}}, \underline{x_{79}}, \underline{x_{99}}, \underline{x_{126}} $ $ x_{221}^{13} $ $ x_{221}^{12}, x_{39}, x_{148} $
    28 $ x_{234}^{10} $ $ x_{234}^9, \overline{x_{33}}, \underline{x_{108}} $ $ x_{230}^{11} $ $ x_{230}^{10}, \overline{x_{36}}, $ $ \underline{x_{131}} $ $ x_{226}^{12} $ $ x_{226}^{11}, \overline{x_{57}}, \underline{x_{80}}, \underline{x_{100}}, \underline{x_{127}} $ $ x_{222}^{13} $ $ x_{222}^{12}, x_{40}, x_{149} $
    29 $ x_{235}^{10} $ $ x_{235}^9, \overline{x_{34}}, \underline{x_{109}} $ $ x_{231}^{11} $ $ x_{231}^{10}, x_{37}, $ $ x_{132} $ $ x_{227}^{12} $ $ x_{227}^{11}, \overline{x_{58}}, \underline{x_{81}}, \underline{x_{101}}, \underline{x_{128}} $ $ x_{223}^{13} $ $ x_{223}^{12}, x_{41}, x_{150} $
    30 $ x_{236}^{10} $ $ x_{236}^9, \overline{x_{35}}, \underline{x_{110}} $ $ x_{232}^{11} $ $ x_{232}^{10}, x_{38}, $ $ x_{133} $ $ x_{228}^{12} $ $ x_{228}^{11}, \overline{x_{59}}, \underline{x_{82}}, \underline{x_{102}}, \underline{x_{129}} $ $ x_{224}^{13} $ $ x_{224}^{12}, x_{42}, x_{151} $
    31 $ x_{237}^{10} $ $ x_{237}^9, \overline{x_{36}}, \underline{x_{111}} $ $ x_{233}^{11} $ $ x_{233}^{10}, x_{39}, $ $ x_{134} $ $ x_{229}^{12} $ $ x_{229}^{11}, \overline{x_{60}}, \underline{x_{83}}, \underline{x_{103}}, \underline{x_{130}} $ $ x_{225}^{13} $ $ x_{225}^{12}, x_{43}, x_{152} $
    32 $ x_{238}^{10} $ $ x_{238}^9, x_{37}, x_{112} $ $ x_{234}^{11} $ $ x_{234}^{10}, x_{40}, $ $ x_{135} $ $ x_{230}^{12} $ $ x_{230}^{11}, \overline{x_{61}}, \underline{x_{84}}, \underline{x_{104}}, \underline{x_{131}} $ $ x_{226}^{13} $ $ x_{226}^{12}, x_{44}, x_{153} $
    33 $ x_{239}^{10} $ $ x_{239}^9, x_{38}, x_{113} $ $ x_{235}^{11} $ $ x_{235}^{10}, x_{41}, $ $ x_{136} $ $ x_{231}^{12} $ $ x_{231}^{11}, \overline{x_{62}}, \underline{x_{85}}, \underline{x_{105}}, \underline{x_{132}} $ $ x_{227}^{13} $ $ x_{227}^{12}, x_{45}, x_{154} $
    34 $ x_{240}^{10} $ $ x_{240}^9, x_{39}, x_{114} $ $ x_{236}^{11} $ $ x_{236}^{10}, x_{42}, $ $ x_{137} $ $ x_{232}^{12} $ $ x_{232}^{11}, \overline{x_{63}}, \underline{x_{86}}, \underline{x_{106}}, \underline{x_{133}} $ $ x_{228}^{13} $ $ x_{228}^{12}, \overline{x_{46}}, \underline{x_{155}} $
     | Show Table
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    Table 4.  Equations used for recovery of 35 bits of the internal state

    Step/Row Equations used for recovery
    0 $\begin{aligned}x_{137}& = z_ 0 \oplus x_{ 80} \oplus x_{99} \oplus x_{227} \oplus x_{222} \oplus x_{187} \oplus x_{243}x_{217} \oplus x_{247}x_{231} \oplus x_{213}x_{235} \\ & \quad \oplus x_{255}x_{251} \oplus x_{181}x_{239} \oplus x_{174}x_{44}\oplus x_{164} \overline{x_{29}} \oplus x_{255}x_{247}x_{243}x_{213}x_{181}x_{174}\end{aligned}$
    1 $\begin{aligned}x_{ 138}& = z_ 1 \oplus x_{ 81} \oplus x_{ 100} \oplus x_{ 228} \oplus x_{ 223} \oplus x_{188} \oplus x_{ 244}^3x_{218}^7 \oplus x_{ 248}^2x_{ 232}^6 \oplus x_{214}^8x_{236}^5 \\ & \quad\oplus x_{ 256}^0x_{252}^1 \oplus x_{182}x_{240}^4 \oplus x_{175}x_{ 45}\oplus x_{165} \overline{x_{30}} \oplus x_{256}^0x_{248}^2x_{244}^3x_{214}^8x_{182}x_{175}\end{aligned}$
    2 $\begin{aligned}x_{ 139}& = z_ 2 \oplus x_{ 82} \oplus x_{ 101} \oplus x_{ 229} \oplus x_{ 224} \oplus x_{189} \oplus x_{ 245}^3x_{219}^7 \oplus x_{ 249}^2x_{ 233}^6 \oplus x_{215}^8x_{237}^5\\ & \quad\oplus x_{ 257}^0x_{253}^1 \oplus x_{183}x_{241}^4 \oplus x_{176}\overline{x_{ 46}}\oplus x_{166} \overline{x_{31}} \oplus x_{257}^0x_{249}^2x_{245}^3x_{215}^8x_{183}x_{176}\end{aligned}$
    3 $\begin{aligned}x_{ 140}& = z_ 3 \oplus x_{ 83} \oplus x_{ 102} \oplus x_{ 230} \oplus x_{ 225} \oplus x_{190} \oplus x_{ 246}^3x_{220}^7 \oplus x_{ 250}^2x_{ 234}^6 \oplus x_{216}^8x_{238}^5\\ & \quad\oplus x_{ 258}^0x_{254}^1 \oplus x_{184}x_{242}^4 \oplus x_{177}\overline{x_{ 47}}\oplus x_{167} \overline{x_{32}} \oplus x_{258}^0x_{250}^2x_{246}^3x_{216}^8x_{184}x_{177}\end{aligned}$
    4 $\begin{aligned}x_{ 141}& = z_ 4 \oplus x_{ 84} \oplus x_{ 103} \oplus x_{ 231} \oplus x_{ 226} \oplus x_{191} \oplus x_{ 247}^3x_{221}^7 \oplus x_{ 251}^2x_{ 235}^6 \oplus x_{217}^8x_{239}^5\\ & \quad\oplus x_{ 259}^0x_{255}^1 \oplus x_{185}x_{243}^4 \oplus x_{178}\overline{x_{ 48}}\oplus x_{168} \overline{x_{33}} \oplus x_{259}^0x_{251}^2x_{247}^3x_{217}^8x_{185}x_{178}\end{aligned}$
    5 $\begin{aligned}x_{ 142}& = z_ 5 \oplus x_{ 85} \oplus x_{ 104} \oplus x_{ 232}^6 \oplus x_{ 227} \oplus x_{192} \oplus x_{ 248}^3x_{222}^7 \oplus x_{ 252}^2x_{ 236}^6 \oplus x_{218}^8x_{240}^5\\ & \quad \oplus x_{ 260}^0x_{256}^1 \oplus x_{186}x_{244}^4 \oplus x_{179}\overline{x_{ 49}}\oplus x_{169}\overline{x_{34}} \oplus x_{260}^0x_{252}^2x_{248}^3x_{218}^8x_{186}x_{179}\end{aligned}$
    6 $\begin{aligned}x_{ 143}& = z_ 6 \oplus x_{ 86} \oplus x_{ 105} \oplus x_{ 233}^6 \oplus x_{ 228} \oplus x_{193} \oplus x_{ 249}^3x_{223}^7 \oplus x_{ 253}^2x_{ 237}^6 \oplus x_{219}^8x_{241}^5\\ & \quad \oplus x_{ 261}^0x_{257}^1 \oplus x_{187}x_{245}^4 \oplus x_{180}\overline{x_{ 50}}\oplus x_{170}\overline{x_{35}} \oplus x_{261}^0x_{253}^2x_{249}^3x_{219}^8x_{187}x_{180}\end{aligned}$
    7 $\begin{aligned}x_{ 144}& = z_ 7 \oplus x_{ 87} \oplus x_{ 106} \oplus x_{ 234}^6 \oplus x_{ 229} \oplus x_{194}^{13} \oplus x_{ 250}^3x_{224}^7 \oplus x_{ 254}^2x_{ 238}^6 \oplus x_{220}^8x_{242}^5\\ & \quad \oplus x_{ 262}^0x_{258}^1 \oplus x_{188}x_{246}^4 \oplus x_{181}\overline{x_{ 51}}\oplus x_{171}\overline{x_{36}} \oplus x_{262}^0x_{254}^2x_{250}^3x_{220}^8x_{188}x_{181}\end{aligned}$
    8 $\begin{aligned}x_{ 145}& = z_ 8 \oplus x_{ 88} \oplus x_{ 107} \oplus x_{ 235}^6 \oplus x_{ 230} \oplus x_{195}^{13} \oplus x_{ 251}^3x_{225}^7 \oplus x_{ 255}^2x_{ 239}^6 \oplus x_{221}^8x_{243}^5\\ & \quad\oplus x_{ 263}^0x_{259}^1 \oplus x_{189}x_{247}^4 \oplus x_{182}\overline{x_{ 52}}\oplus x_{172}x_{37} \oplus x_{263}^0x_{255}^2x_{251}^3x_{221}^8x_{189}x_{182}\end{aligned}$
    9 $\begin{aligned}x_{ 146}& = z_ 9 \oplus x_{ 89} \oplus x_{ 108} \oplus x_{ 236}^6 \oplus x_{ 231} \oplus x_{196}^{13} \oplus x_{ 252}^3x_{226}^7 \oplus x_{ 256}^2x_{ 240}^6 \oplus x_{222}^8x_{244}^5\\ & \quad \oplus x_{ 264}^0x_{260}^1 \oplus x_{190}x_{248}^4 \oplus x_{183}\overline{x_{ 53}}\oplus x_{173}x_{38} \oplus x_{264}^0x_{256}^2x_{252}^3x_{222}^8x_{190}x_{183}\end{aligned}$
    10 $\begin{aligned}x_{ 147}& = z_ {10} \oplus x_{ 90} \oplus x_{ 109} \oplus x_{ 237}^6 \oplus x_{ 232}^6 \oplus x_{197}^{13} \oplus x_{ 253}^3x_{227}^7 \oplus x_{ 257}^2x_{ 241}^6 \oplus x_{223}^8x_{245}^5\\ & \quad\oplus x_{ 265}^0x_{261}^1 \oplus x_{191}x_{249}^4 \oplus x_{184}\overline{x_{ 54}}\oplus x_{174}x_{39} \oplus x_{265}^0x_{257}^2x_{253}^3x_{223}^8x_{191}x_{184}\end{aligned}$
    11 $\begin{aligned}x_{ 148}& = z_ {11} \oplus x_{ 91} \oplus x_{ 110} \oplus x_{ 238}^6 \oplus x_{ 233}^6 \oplus x_{198}^{13} \oplus x_{ 254}^3x_{228}^7 \oplus x_{ 258}^2x_{ 242}^6 \oplus x_{224}^8x_{246}^5\\ & \quad \oplus x_{ 266}^0x_{262}^1 \oplus x_{192}x_{250}^4 \oplus x_{185}\overline{x_{ 55}}\oplus x_{175}x_{40} \oplus x_{266}^0x_{258}^2x_{254}^3x_{224}^8x_{192}x_{185}\end{aligned}$
    12 $\begin{aligned}x_{ 149}& = z_ {12} \oplus x_{ 92} \oplus x_{ 111} \oplus x_{ 239}^6 \oplus x_{ 234}^6 \oplus x_{199}^{13} \oplus x_{ 255}^3x_{229}^7 \oplus x_{ 259}^2x_{ 243}^6 \oplus x_{225}^8x_{247}^5\\ & \quad\oplus x_{ 267}^0x_{263}^1 \oplus x_{193}x_{251}^4 \oplus x_{186}\overline{x_{ 56}}\oplus x_{176}x_{41} \oplus x_{267}^0x_{259}^2x_{255}^3x_{225}^8x_{193}x_{186}\end{aligned}$
    13 $\begin{aligned}x_{ 150}& = z_ {13} \oplus x_{ 93} \oplus x_{ 112} \oplus x_{ 240}^6 \oplus x_{ 235}^6 \oplus x_{200}^{13} \oplus x_{ 256}^3x_{230}^7 \oplus x_{ 260}^2x_{ 244}^6 \oplus x_{226}^8x_{248}^5\\ & \quad\oplus x_{ 268}^0x_{264}^1 \oplus x_{194}^{13}x_{252}^4 \oplus x_{187}\overline{x_{ 57}}\oplus x_{177}x_{42} \oplus x_{268}^0x_{260}^2x_{256}^3x_{226}^8x_{194}^{13}x_{187}\end{aligned}$
    14 $\begin{aligned}x_{ 151}& = z_ {14} \oplus x_{ 94} \oplus x_{ 113} \oplus x_{ 241}^6 \oplus x_{ 236}^6 \oplus x_{201}^{13} \oplus x_{ 257}^3x_{231}^7 \oplus x_{ 261}^2x_{ 245}^6 \oplus x_{227}^8x_{249}^5\\ & \quad\oplus x_{ 269}^0x_{265}^1 \oplus x_{195}^{13}x_{253}^4 \oplus x_{188}\overline{x_{ 58}}\oplus x_{178}x_{43} \oplus x_{269}^0x_{261}^2x_{257}^3x_{227}^8x_{195}^{13}x_{188}\end{aligned}$
    15 $\begin{aligned}x_{ 152}& = z_ {15} \oplus x_{ 95} \oplus x_{ 114} \oplus x_{ 242}^6 \oplus x_{ 237}^6 \oplus x_{202}^{13} \oplus x_{ 258}^3x_{232}^7 \oplus x_{ 262}^2x_{ 246}^6 \oplus x_{228}^8x_{250}^5\\ & \quad\oplus x_{ 270}^0x_{266}^1 \oplus x_{196}^{13}x_{254}^4 \oplus x_{189}\overline{x_{59}}\oplus x_{179}x_{44} \oplus x_{270}^0x_{262}^2x_{258}^3x_{228}^8x_{196}^{13}x_{189}\end{aligned}$
    16 $\begin{aligned}x_{ 153}& = z_ {16} \oplus x_{ 96} \oplus x_{ 115} \oplus x_{ 243}^6 \oplus x_{ 238}^6 \oplus x_{203}^{13} \oplus x_{ 259}^3x_{233}^7 \oplus x_{ 263}^2x_{ 247}^6 \oplus x_{229}^8x_{251}^5\\ & \quad \oplus x_{ 271}^0x_{267}^1 \oplus x_{197}^{13}x_{255}^4 \oplus x_{190}\overline{x_{60}}\oplus x_{180}x_{45} \oplus x_{271}^0x_{263}^2x_{259}^3x_{229}^8x_{197}^{13}x_{190}\end{aligned}$
    17 $\begin{aligned}x_{ 154}& = z_ {17} \oplus x_{ 97} \oplus x_{ 116} \oplus x_{ 244}^6 \oplus x_{ 239}^6 \oplus x_{204}^{13} \oplus x_{ 260}^3x_{234}^7 \oplus x_{ 264}^2x_{ 248}^6 \oplus x_{230}^8x_{252}^5\\ & \quad \oplus x_{ 272}^0x_{268}^1 \oplus x_{198}^{13}x_{256}^4 \oplus x_{191}\overline{x_{61}}\oplus x_{181}\overline{x_{46}} \oplus x_{272}^0x_{264}^2x_{260}^3x_{230}^8x_{198}^{13}x_{191}\end{aligned}$
    18 $\begin{aligned}x_{ 155}& = z_ {18} \oplus x_{ 98} \oplus x_{ 117} \oplus x_{ 245}^6 \oplus x_{ 240}^6 \oplus x_{205}^{13} \oplus x_{ 261}^3x_{235}^7 \oplus x_{ 265}^2x_{ 249}^6 \oplus x_{231}^8x_{253}^5\\ & \quad\oplus x_{ 273}^0x_{269}^1 \oplus x_{199}^{13}x_{257}^4 \oplus x_{192}\overline{x_{62}}\oplus x_{182}\overline{x_{47}} \oplus x_{273}^0x_{265}^2x_{261}^3x_{231}^8x_{199}^{13}x_{192}\end{aligned}$
    19 $\begin{aligned}x_{ 156}& = z_ {19} \oplus x_{ 99} \oplus x_{ 118} \oplus x_{ 246}^6 \oplus x_{ 241}^6 \oplus x_{206}^{13} \oplus x_{ 262}^3x_{236}^7 \oplus x_{ 266}^2x_{ 250}^6 \oplus x_{232}^8x_{254}^5\\ & \quad \oplus x_{ 274}^0x_{270}^1 \oplus x_{200}^{13}x_{258}^4 \oplus x_{193}\overline{x_{63}}\oplus x_{183}\overline{x_{48}} \oplus x_{274}^0x_{266}^2x_{262}^3x_{232}^8x_{200}^{13}x_{193}\end{aligned}$
    20 $\begin{aligned}x_{ 157}& = z_ {20} \oplus x_{100} \oplus x_{ 119} \oplus x_{ 247}^6 \oplus x_{ 242}^6 \oplus x_{207}^{13} \oplus x_{ 263}^3x_{237}^7 \oplus x_{ 267}^2x_{ 251}^6 \oplus x_{233}^8x_{255}^5\\ & \quad \oplus x_{ 275}^0x_{271}^1 \oplus x_{201}^{13}x_{259}^4 \oplus x_{194}^{13}\overline{x_{64}}\oplus x_{184}\overline{x_{49}} \oplus x_{275}^0x_{267}^2x_{263}^3x_{233}^8x_{201}^{13}x_{194}^{13}\end{aligned}$
    21 $\begin{aligned}x_{ 158}& = z_ {21} \oplus x_{101} \oplus x_{ 120} \oplus x_{ 248}^6 \oplus x_{ 243}^6 \oplus x_{208}^{13} \oplus x_{ 264}^3x_{238}^7 \oplus x_{ 268}^2x_{ 252}^6 \oplus x_{234}^8x_{256}^5\\ & \quad \oplus x_{ 276}^0x_{272}^1 \oplus x_{202}^{13}x_{260}^4 \oplus x_{195}^{13}\overline{x_{65}}\oplus x_{185}\overline{x_{50}} \oplus x_{276}^0x_{268}^2x_{264}^3x_{234}^8x_{202}^{13}x_{195}^{13}\end{aligned}$
    22 $\begin{aligned}x_{ 159}& = z_ {22} \oplus x_{102} \oplus x_{ 121} \oplus x_{ 249}^6 \oplus x_{ 244}^6 \oplus x_{209}^{13} \oplus x_{ 265}^3x_{239}^7 \oplus x_{ 269}^2x_{ 253}^6 \oplus x_{235}^8x_{257}^5\\ & \quad \oplus x_{ 277}^0x_{273}^1 \oplus x_{203}^{13}x_{261}^4 \oplus x_{196}^{13}\overline{x_{66}}\oplus x_{186}\overline{x_{51}} \oplus x_{277}^0x_{269}^2x_{265}^3x_{235}^8x_{203}^{13}x_{196}^{13}\end{aligned}$
    23 $\begin{aligned}x_{ 160}& = z_ {23} \oplus x_{103} \oplus x_{ 122} \oplus x_{ 250}^6 \oplus x_{ 245}^6 \oplus x_{210}^{13} \oplus x_{ 266}^3x_{240}^7 \oplus x_{ 270}^2x_{ 254}^6 \oplus x_{236}^8x_{258}^5\\ & \quad \oplus x_{ 278}^0x_{274}^1 \oplus x_{204}^{13}x_{262}^4 \oplus x_{197}^{13}\overline{x_{67}}\oplus x_{187}\overline{x_{52}} \oplus x_{278}^0x_{270}^2x_{266}^3x_{236}^8x_{204}^{13}x_{197}^{13}\end{aligned}$
    24 $\begin{aligned}x_{ 161}& = z_ {24} \oplus x_{104} \oplus x_{ 123} \oplus x_{ 251}^6 \oplus x_{ 246}^6 \oplus x_{211}^{13} \oplus x_{ 267}^3x_{241}^7 \oplus x_{ 271}^2x_{ 255}^6 \oplus x_{237}^8x_{259}^5\\ & \quad \oplus x_{ 279}^0x_{275}^1 \oplus x_{205}^{13}x_{263}^4 \oplus x_{198}^{13}\overline{x_{68}}\oplus x_{188}\overline{x_{53}} \oplus x_{279}^0x_{271}^2x_{267}^3x_{237}^8x_{205}^{13}x_{198}^{13} \end{aligned}$
    25 $\begin{aligned}x_{ 162}& = z_ {25} \oplus x_{105} \oplus x_{ 124} \oplus x_{ 252}^6 \oplus x_{ 247}^6 \oplus x_{212}^{13} \oplus x_{ 268}^3x_{242}^7 \oplus x_{ 272}^2x_{ 256}^6 \oplus x_{238}^8x_{260}^5\\ & \quad\oplus x_{ 280}^0x_{276}^1 \oplus x_{206}^{13}x_{264}^4 \oplus x_{199}^{13}\overline{x_{69}}\oplus x_{189}\overline{x_{54}} \oplus x_{280}^0x_{272}^2x_{268}^3x_{238}^8x_{206}^{13}x_{199}^{13} \end{aligned}$
    26 $\begin{aligned}x_{ 163}& = z_ {26} \oplus x_{106} \oplus x_{ 125} \oplus x_{ 253}^6 \oplus x_{ 248}^6 \oplus x_{213}^{13} \oplus x_{ 269}^3x_{243}^7 \oplus x_{ 273}^2x_{ 257}^6 \oplus x_{239}^8x_{261}^5\\ & \quad\oplus x_{ 281}^0x_{277}^1 \oplus x_{207}^{13}x_{265}^4 \oplus x_{200}^{13}\overline{x_{70}}\oplus x_{190}\overline{x_{55}} \oplus x_{281}^0x_{273}^2x_{269}^3x_{239}^8x_{207}^{13}x_{200}^{13}\end{aligned}$
    27 $\begin{aligned}x_{ 164}& = z_ {27} \oplus x_{107} \oplus x_{ 126} \oplus x_{ 254}^6 \oplus x_{ 249}^6 \oplus x_{214}^{13} \oplus x_{ 270}^3x_{244}^7 \oplus x_{ 274}^2x_{ 258}^6 \oplus x_{240}^8x_{262}^5\\ & \quad\oplus x_{ 282}^0x_{278}^1 \oplus x_{208}^{13}x_{266}^4 \oplus x_{201}^{13}\overline{x_{71}}\oplus x_{191}\overline{x_{56}} \oplus x_{282}^0x_{274}^2x_{270}^3x_{240}^8x_{208}^{13}x_{201}^{13}\end{aligned}$
    28 $\begin{aligned}x_{ 165}& = z_ {28} \oplus x_{108} \oplus x_{ 127} \oplus x_{ 255}^6 \oplus x_{ 250}^6 \oplus x_{215}^{13} \oplus x_{ 271}^3x_{245}^7 \oplus x_{ 275}^2x_{ 259}^6 \oplus x_{241}^8x_{263}^5\\ & \quad\oplus x_{ 283}^0x_{279}^1 \oplus x_{209}^{13}x_{267}^4 \oplus x_{202}^{13}\overline{x_{72}}\oplus x_{192}\overline{x_{57}} \oplus x_{283}^0x_{275}^2x_{271}^3x_{241}^8x_{209}^{13}x_{202}^{13}\end{aligned}$
    29 $\begin{aligned}x_{ 166}& = z_ {29} \oplus x_{109} \oplus x_{ 128} \oplus x_{ 256}^6 \oplus x_{ 251}^6 \oplus x_{216}^{13} \oplus x_{ 272}^3x_{246}^7 \oplus x_{ 276}^2x_{ 260}^6 \oplus x_{242}^8x_{264}^5\\ & \quad\oplus x_{ 284}^0x_{280}^1 \oplus x_{210}^{13}x_{268}^4 \oplus x_{203}^{13}\overline{x_{73}}\oplus x_{193}\overline{x_{58}} \oplus x_{284}^0x_{276}^2x_{272}^3x_{242}^8x_{210}^{13}x_{203}^{13}\end{aligned}$
    30 $\begin{aligned}x_{ 167}& = z_ {30} \oplus x_{110} \oplus x_{ 129} \oplus x_{ 257}^6 \oplus x_{ 252}^6 \oplus x_{217}^{13} \oplus x_{ 273}^3x_{247}^7 \oplus x_{ 277}^2x_{ 261}^6 \oplus x_{243}^8x_{265}^5\\ & \quad\oplus x_{ 285}^0x_{281}^1 \oplus x_{211}^{13}x_{269}^4 \oplus x_{204}^{13}\overline{x_{74}}\oplus x_{194}^{13}\overline{x_{59}} \oplus x_{285}^0x_{277}^2x_{273}^3x_{243}^8x_{211}^{13}x_{204}^{13}\end{aligned}$
    31 $\begin{aligned}x_{ 168}& = z_ {31} \oplus x_{111} \oplus x_{ 130} \oplus x_{ 258}^6 \oplus x_{ 253}^6 \oplus x_{218}^{13} \oplus x_{ 274}^3x_{248}^7 \oplus x_{ 278}^2x_{ 262}^6 \oplus x_{244}^8x_{266}^5\\ & \quad\oplus x_{ 286}^0x_{282}^1 \oplus x_{212}^{13}x_{270}^4 \oplus x_{205}^{13}\overline{x_{75}}\oplus x_{195}^{13}\overline{x_{60}} \oplus x_{286}^0x_{278}^2x_{274}^3x_{244}^8x_{212}^{13}x_{205}^{13}\end{aligned}$
    32 $\begin{aligned}x_{ 169}& = z_ {32} \oplus x_{112} \oplus x_{ 131} \oplus x_{ 259}^6 \oplus x_{ 254}^6 \oplus x_{219}^{13} \oplus x_{ 275}^3x_{249}^7 \oplus x_{ 279}^2x_{ 263}^6 \oplus x_{245}^8x_{267}^5\\ & \quad\oplus x_{ 287}^0x_{283}^1 \oplus x_{213}^{13}x_{271}^4 \oplus x_{206}^{13}\overline{x_{76}}\oplus x_{196}^{13}\overline{x_{61}} \oplus x_{287}^0x_{279}^2x_{275}^3x_{245}^8x_{213}^{13}x_{206}^{13}\end{aligned}$
    33 $\begin{aligned}x_{170}& = z_ {33} \oplus x_{113} \oplus x_{ 132} \oplus x_{ 260}^6 \oplus x_{ 255}^6 \oplus x_{220}^{13} \oplus x_{ 276}^3x_{250}^7 \oplus x_{ 280}^2x_{ 264}^6 \oplus x_{246}^8x_{268}^5\\ & \quad\oplus x_{ 288}^0x_{284}^1 \oplus x_{214}^{13}x_{272}^4 \oplus x_{207}^{13}x_{77}\oplus x_{197}^{13}\overline{x_{62}} \oplus x_{288}^0x_{280}^2x_{276}^3x_{246}^8x_{214}^{13}x_{207}^{13}\end{aligned}$
    34 $\begin{aligned} x_{171}& = z_ {34} \oplus x_{114} \oplus x_{ 133} \oplus x_{ 261}^6 \oplus x_{ 256}^6 \oplus x_{221}^{13} \oplus x_{ 277}^3x_{251}^7 \oplus x_{ 281}^2x_{ 265}^6 \oplus x_{247}^8x_{269}^5\\ & \quad\oplus x_{ 289}^0x_{285}^1 \oplus x_{215}^{13}x_{273}^4 \oplus x_{208}^{13}x_{78}\oplus x_{198}^{13}\overline{x_{63}} \oplus x_{289}^0x_{281}^2x_{277}^3x_{247}^8x_{215}^{13}x_{208}^{13}\end{aligned}$
     | Show Table
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    Table 5.  Possible tradeoffs for conditional BSW sampling resistance based TMDTO attack

    $ \delta $ $ D' $ $ T' $ $ M $ $ P $
    $ 30 $ $ 2^{104} $ $ 2^{99} $ $ 2^{122} $ $ 2^{152} $
    $ 32 $ $ 2^{106} $ $ 2^{103} $ $ 2^{118} $ $ 2^{150} $
    $ 34 $ $ 2^{108} $ $ 2^{107} $ $ 2^{114} $ $ 2^{148} $
     | Show Table
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