theory and practice of zk-snarks

Transcript

1. [L4/"3,Tͷཧ࿦ͱԠ༻ 0TVLF
   4VEP
   
   
   
   -BZFS9
   *OD
   /PW

2. "CPVU
   .F 0TVLF
   ![PPN@[PPN[P -BZFS9
   3FTFBDIFS
   
   &OHJOFFS
   ࠷ۙ͸4/"3,T45"3,T԰

3. "HFOEB w *OUSPEVDUJPO
   GPS
   [L4/"3,T
   w %JWF
   JOUP
   [L4/"3,T
   w 7FSJpBCMF
   GPSNBM
   WFSJpDBUJPO
   PG
   TNBSU
   DPOUSBDUT

4. *OUSPEVDUJPO
   GPS
   [L4/"3,T

5. ;FSP,OPXMFEHF
   1SPPG w ͋Δ໰୊ͷղΛ஌͍ͬͯΔ͜ͱΛʮೖྗ஋ʯΛ໌Β͔ʹͤͣʹ
 ʮ஌͍ͬͯΔࣄ࣮ʯ͚ͩΛূ໌͢Δɻ
   w ύεϫʔυࣗମ͸໌Β͔ʹ͠ͳ͍ͰɺύεϫʔυΛ஌͍ͬͯΔࣄ࣮ ͚ͩΛূ໌͢Δɻ
   w JOQVU
   YΛ໌Β͔ʹͤͣʹɹɹɹɹ
   ͱͳΔYΛ஌͍ͬͯΔ͜ͱΛূ໌ f(x)

   = y

6. ;FSP,OPXMFEHF
   1SPPG w $PNQMFUFOFTTɿ
   w 1SPWFSͷ͍࣋ͬͯΔ໰୊͕ਅͰ͋ΔͳΒ͹ɺ7FSJGZFS͸ਅͰ͋Δ͜ͱ͕ ඞͣΘ͔Δ͜ͱɻ
   w 4PVOEOFTTɿ
   w 1SPWFSͷ͍࣋ͬͯΔ໰୊ِ͕Ͱ͋ΔͳΒ͹ɺ7FSJpFS͸͔ͳΓߴ͍֬཰

   ͰͦΕِ͕Ͱ͋Δͱݟൈ͚Δ͜ͱɻ
   w ;FSPLOPXMFEHFɿ
   w ͋Δ໰୊ͷղΛ஌͍ͬͯΔ͜ͱΛʮೖྗ஋ʯΛ໌Β͔ʹͤͣʮ஌͍ͬͯ Δࣄ࣮ʯ͚ͩΛূ໌͢Δɻ

7. [L4/"3,T w [Lɿθϩ஌ࣝ
   w /PO*OUFSBDUJWFɿඇର࿩త
   w 4VTTJODUɿNTΦʔμʔͰݕূՄೳ
   w 1SPPG
   4J[F͕ඇৗʹখ͍͞ʢ਺ඦ#ZUFʣ

   ϒϩοΫνΣʔϯͱ૬ੑ͕ྑ͍

   1SPWFS 7FSJpFS 1SPPG

8. 4IJFMEFE
   5SBOTBDUJPO [$BTI 'SPN
   
   5P
   
   "NPVOU
   5Y ΦϯνΣʔϯͰ[L4/"3,TΛ༻͍ɺ5Y͕ਖ਼౰Ͱ͋Δࣄ࣮͚ͩΛ఻೻ 1SPWJOH

   7FSJGZJOH QSPPG

9. [L4/"3,T
   PO
   &UIFSFVN #Z[BOUJVN
   6QEBUFͰCO"EE
   CO4DBMBS.VM
   CO1BJSJOHͷ௥Ճ 4/"3,Tͷݕূʹඞཁͳପԁۂઢ CO ্ͷԋࢉΛ1SFDPNQJMF $POTUBOUJOPQMF
   6QEBUFͰ(BT
   $PTUͷ࡟ݮΛ༧ఆ &7.֎Ͱԋࢉ͢Δ͜ͱͰ(BTΛݻఆͯ͠͠·͏ SFG
   <>
   IUUQTFJQTFUIFSFVNPSH&*14FJQ

10. 0VUTPVSDJOH
    B
    DPNQVUBUJPO ܭࢉॲཧೳྗ͕ߴ͍P⒎DIBJOͷαʔόʔʹܭࢉΛҕৡ
    ൿಗԽٕज़Ͱ͸ͳ͘εέʔϦϯάٕज़ͱͯ͠4/"3,TΛ׆༻ ϒϩοΫνΣʔϯ 1SPPG 4UBUF
    4 4UBUF
    4A 1SPWFS

11. 4/"3,T
    
    45"3,T
    
    #VMMFUQSPPGT 5PYJDXBTUF
    GSFF 1SPPG
    UJNF 7FSJGZ
    UJNF 1SPPG
    TJ[F 4/"3,T /P T NT

    d# 45"3,T :FT dT dNT d # #VMMFUQSPPGT :FT dT dNT d# SFG
    <>
    [DBTI
    BU
    EFWDPO [DBTIͷTIJFMEFE
    USBOTBDUJPO

12. %JWF
    JOUP
    [L4/"3,T

13. 0WFSWJFX
    4/"3,T
    TDIFNF 1SPWFS 7FSJpFS 
    #BTFE
    PO
    1JOOPDIJP
    <1()3> GVODUJPO DJSDVJU
    
    HFOFSBUPS PODF LFZ
    HFOFSBUPS

    DJSDVJU QSPWJOH
    LFZ WFSJpDBUJPO
    LFZ BOZ
    OVNCFS
    PG
    UJNFT XJUOFTT
    
    NBQ
    TJNVMBUF QVCMJD
    
    JOQVU QSJWBUF
    JOQVU GVODUJPO BTTJHONFOU  TFUVQ QSPPG SFG
    <>4/"3,T
    GPS
    $
    
    7FSJGZJOH
    1SPHSBN
    &YFDVUJPOT
    4VDDJODUMZ
    BOE
    JO
    ;FSP
    ,OPXMFEHF QVCMJD
    
    JOQVU
    TJ[F εϚʔτίϯτϥΫτ

14. 4FUVQ
    
    3$4
    3BOL
    $POTUSBJOU
    4ZTUFN f(x) = x3 + x + 5 f(x)

    = 35 ͱͳΔ੍ݶΛຬͨ͢DPOTUSBJOUTΛ࡞͍ͬͯ͘       x SFG
    <>
    2VBESBUJD
    "SJUINFUJD
    1SPHSBNT
    GSPN
    ;FSP
    UP
    )FSP TB
    
    TC
    
    TD
    
     POF JOQVU PVUQVU D D D 
    ֻ͚ࢉPS଍͠ࢉͷ੍໿

15. 4FUVQ
    
    2"1T
    2BESBUJD
    "SJUINFUJD
    $JSDVJUT ଟ߲ࣜʹม׵͢Δ͜ͱͰશͯͷDPOTUSBJOUTΛҰؾʹܭࢉͯ͠͠·͏ SFG
    <>2VBESBUJD
    "SJUINFUJD
    1SPHSBNT
    GSPN
    ;FSP
    UP
    )FSP ͦΕͧΕͷDPOTUSBJOUTΛଟ্߲ࣜͷ఺ͱଊ͑ɺ ϥάϥϯδϡิؒͳͲͰ෮ݩ
    ͦΕͧΕ఺͔Β࣍ࣜΛ෮ݩ
    
    
    YΛ୅ೖ͢Ε͹લͷεϥΠυͱಉ͡ DPOTUSBJOUT਺ͷ࠷খଟ߲ࣜ; Y

     Y Y Y Y ͰׂΓ੾ΕΔ
    ݕূ͸ɺ͜ͷଟ߲ࣜΛςΩτʔͳYU͚ͩͰධՁͯ͠ߦ͏

16. 4FUVQ
    
    2"1T
    2BESBUJD
    "SJUINFUJD
    $JSDVJUT "
    QPMZOPNJBMT
    
    <
    
    
    >
    
    <
    
    

    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    > ࠷ऴతͳ2"1͸ଟ߲ࣜ܎਺ͷঢॱϕΫτϧ #
    QPMZOPNJBMT
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    > $
    QPMZOPNJBMT
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    >
    
    <
    
    
    > A1 (x) = 0.833x3 − 5x2 + 9.166x − 5 ࣮ࡍ͸༗ݶମԋࢉ

17. ,P&
    ,OPXMFEHF
    PG
    &YQPOFOU
    "TTVNQUJPO 1JOOPDIJPͰ͸҉߸ֶతͳ೿ੜԾఆΛར༻ͯ͠ূ໌Λੜ੒͢Δ ପԁۂઢ্ͷ఺ 1J
    2J
    3
    4 ʹ͍ͭͯɺൿີ஋Lʹରͯͦ͠ΕͧΕ2JL1Jͷͱ͖
    

    4L3ʹͳΔʢ3
    4 ͸ 1J
    2J ͷઢܗ࿨Ͱ͔͠ද͢͜ͱ͕Ͱ͖ͳ͍ 4
    
    L3Ͱ͋Δ͜ͱࣗମ͸LΛ஌ΒͣʹϖΞϦϯάͰݕূՄೳɻ LΛ஌͍ͬͯΔͱઢܗ࿨Ͱද͢͜ͱͳ͘ 3
    L3 ͕࡞Εͯ͠·͏ʜɻ
    ͜ͷൿີ஋L͕UPYJD
    XBTUFͰ͋ΓUSVTUFE
    TFUVQͷඞཁੑʹܨ͕Δ R = P1 i1 + P2 i2 + ⋅ ⋅ ⋅ , S = Q1 i1 + Q2 i2 + ⋅ ⋅ ⋅ e(R, Q) = e(P, S) J   ʜ

18. 7FSJpDBUJPO
    PG
    TOBSLT
    QSPPG ͭͷϖΞϦϯάʹΑΓݕূΛߦ͏ ྫ͑͹ɺ1SPWFS͕ଟ߲ࣜ" U ͷίϛοτϝϯτΛ஌͍ͬͯΔ͜ͱͷݕূ e(πa , VKa ) =

    e(G, π′ a ) 4FUVQ
 QSPWJOH
    LFZ
    
    
    
 WFSJGZJOH
    LFZ
    
    
 UPYJD
    XBTUF
    
    
    
    
    1SPPG PKa,i = Ai (t) ⋅ G t, ka VKa = ka G PK′ a,i = Ai (t) ⋅ G ⋅ ka πa = Σ ⃗ si PKa,i π′ a = Σ ⃗ si PK′ a,i " U ͷίϛοτϝϯτ͕ਖ਼͍͠ൣғͰ
    ઢܗ࿨͕࡞ΒΕ͍ͯͳ͍ͱ
    ,P&ʹΑΓݕূ͕௨Βͳ͍ P = G, Q = VKa , R = πa , S = π′ a T͸ݕূऀʹ͸෼͔Βͳ͍ FY
    TdT·Ͱ͔͠༩͑ͯͳ͍

19. 1()3
    
    XIBU`T
    MFGU w ಉ༷ʹɺଟ߲ࣜ#ɺ$ʹ͍ͭͯͷίϛοτϝϯτݕূ
    w ͦΕͧΕͷXJUOFTTൣғΛૢ࡞͢Δ͜ͱͰݕূΛෆਖ਼ʹύε͠Α͏ͱ͍ͯ͠ͳ͍͔
    w ଟ߲ࣜ"ɺ#ɺ$ͷઢܗ࿨܎਺͕ͦΕͧΕಉ͔͡
    w ͦΕͧΕҟͳΔXJUOFTTΛ༩͑Δ͜ͱͰݕূΛෆਖ਼ʹύε͠Α͏ͱ͍ͯ͠ͳ͍͔
    w

    "
    
    #
    
    $
    
    )
    
    ;Λຬ͔ͨ͢
    w ͜ΕΛຬͨ͢͜ͱ͕ʮݩͷ໰୊ʯ͕ਅͰ͋Δ͜ͱΛࣔ͢
    w QVCMJD
    JOQVUʹجͮ͘ݕূ伴ͷੜ੒
    w ύϥϝʔλʔͷΑ͏ʹಇ͘
    ʢݕূ伴αΠζ͕૿͑ͯ͠·͏ʣ
    w 1SPWFSͷBTTJHONFOUͷൿಗԽ
    w ଞͷ"`
    #`
    $`
    )`͕"
    #
    $
    )Ͱ͸ͳ͍৘ใͷൿಗԽ
    w ଟ߲ࣜ"ɺ#ɺ$ʹͦΕͧΕʹϥϯμϜ஋ΛՃ͑Δ
    w ඇରশϖΞϦϯάʹΑΔޮ཰ԽͳͲ
    <#$57>

20. 7FSJpBCMF
    'PSNBM
    7FSJpDBUJPO
    PG
    4NBSU
    $POUSBDUT

21. 4/"3,T
    PWFS
    'PSNBM
    7FSJpDBUJPO w ໰୊ఆٛ
    w εϚʔτίϯτϥΫτͷηΩϡϦςΟ͕େ͖ͳ՝୊
    w ܗࣜతݕূʹΑΔεϚʔτίϯτϥΫτͷ࢓༷ఆٛʹΑΔղܾࡦ
    w PODIBJOͰ͸ͦͷεϚʔτίϯτϥΫτ͕ܗࣜతݕূ͞Ε͔ͨ൑அ͢Δ͢΂͕ ͳ͍
    

    w ղܾࡦ
    w ܗࣜతݕূΛ4/"3,TΛ༻͍ͯP⒎DIBJOʹΞ΢τιʔε͠ɺͦͷίϯτϥΫτ ͕͔֬ʹܗࣜతݕূ͞Εͨࣄ࣮ΛPODIBJOʹه࿥͢Δ
    w ͭͷϓϩτίϧྫ
    w ίϯτϥΫτ͕ݺͼग़͢ίϯτϥΫτͷܗࣜతݕূνΣοΫ
    w Ϣʔβʔ͕ݺͼग़͢ίϯτϥΫτͷܗࣜతݕূνΣοΫ

22. 1SPUPDPM
    FYNBQMFT w ίϯτϥΫτࢀরͷηΩϡϦςΟ޲্
    w PODIBJOͷίϯτϥΫτ͕$"--͢ΔίϯτϥΫτ͕ܗࣜతݕূ͞ΕͯΔࣄ࣮Λূ໌ ͢Δ
    w 4UBCMF
    $PJO΍EFSJWBUJWFT
    TZTUFNʹ୲อ͞Ε͍ͯΔ&3$τʔΫϯʹPWFSqPX΍ SFFOUSBODZ͕ͳ͍ͱܗࣜతݕূ͞ΕͨίϯτϥΫτͷΈ୲อՄೳʹ͢Δϓϩτίϧ
    w

    ΢ΥϨοτϒϥ΢βࢀরͷηΩϡϦςΟ޲্
    w ΢ΥϨοτ΍%BQQϒϥ΢β͕ίϯτϥΫτΛݺͼग़͢ͱ͖ʹɺܗࣜతݕূ͞Ε͍ͯ Δ͔νΣοΫ
    w ੬ऑੑ͚ͩͰͳ͘ɺ1MBTNB΍4UBUF
    $IBOOFMίϯτϥΫτ͕ʮඪ४ن֨ʯʹԊͬͯ ͍Δ͔ܗࣜతݕূΛߦ͏
    w ϒϩοΫνΣʔϯʹ͓͚Δ
    IUUQT
    ͷΑ͏ʹಈ࡞͢Δ
    w ʮ8BSOJOH
    ΞΫηε͍ͯ͠Δ1MBTNBίϯτϥΫτ͸ඪ४ن֨ʹ४ڌ͍ͯ͠·ͤ Μɻʯ

23. 0WFSWJFX
    PG
    TOBSLT
    PWFS
    '7 'PSNBM
    
    7FSJpDBUJPO
    1SPHSBN 3". $JSDVJU 7FSJpFS
    
    $POUSBDU 4FUVQ 1SPWFS

    0UIFS
    $POUSBDUT EBQQ
    CSPXTFST 'SPOUFOE #BDLFOE 0ODIBJO 0⒎DIBJO

24. 5JOZ3". 3".
    3BOEPN
    "DDFTT
    .BDIJOF
    ϨδελͱϝϞϦ͕͋ΔܭࢉϞσϧ SFG
    IUUQTXXXTDJQSMBCPSHEPD5JOZ3".TQFDQEG ϓϩάϥϜʹରͯ͠ɺਖ਼͘͠$16ͷঢ়ଶભҠͤ͞Δ͜ͱͷz੍ݶz͕DJSDVJUͱͳΔ
    VOCPVOEFEͳλΠϜεςοϓ΍࠶ؼΛؚΉϓϩάϥϜ΋DJSDVJUʹͰ͖Δ൓໘ɺ
    DJSDVJUͷαΠζ͕ͱͯ΋େ͖͘ͳͬͯ͠·͏ɻ
    શͯͷPQDPEFʹରͯ͠DJSDVJU͕ධՁ͞ΕΔͨΊɺ
    5JOZ3".Ͱ͸PQDPEFΛݸʹ5JOZʹʢͦΕͰ΋΍͹͍ʣɻ