Bazuwar Bazuwar daga Yarjejeniyar Kusa: Ka'idojin Asynchronous Byzantine Masu Jurewa Kuskure

1 Gabatarwa

Bazuwar kaya ce muhimmiyar kaya a cikin ƙirƙirar tsarin rarraba. Na'urar tsabar kuɗi ta gama-gari, wacce ke ba wa membobin tsarin damar yin yarjejeniya akan lambar bazuwar da ba a iya faɗi ba, ta tabbatar da amfani musamman ga ka'idoji kamar Yarjejeniyar Byzantine, Samarwa Maɓalli Rarraba, da Zaɓen Shugaba. Duk da haka, aiwatar da ka'idar tsabar kuɗi ta gama-gari ta gaske a cikin tsarin asynchronous masu saurin kuskure ba zai yiwu ba saboda sakamakon rashin yuwuwar FLP.

Wannan takarda ta gabatar da sauƙaƙa biyu na cikakkiyar tsabar kuɗi: (1) tsabar kuɗi ta kusa da ke samar da lambobin bazuwar waɗanda ke kusa da juna, da (2) tsabar kuɗi ta Monte Carlo tana samar da lambar bazuwar gama-gari tare da ƙaramin yuwuwar gazawa, amma ba sifili ba. Ka'idojinmu sun gina a saman ainihin yarjejeniyar kuma suna jurewa har zuwa kashi ɗaya cikin uku na hanyoyin Byzantine ba tare da saitin amincewa ko tsarin maɓalli na jama'a ba.

2 Bayanan Baya da Ayyukan Da suka danganci

2.1 Na'urorin Tsabar Kudi na Gama-gari

Ka'idar tsabar kuɗi ta gama-gari cikakke dole ne ta gamsar da kaddarorin guda uku:

Ƙarshewa: Kowane tsari mai daidai a ƙarshe yana fitar da wani ƙima
Yarjejeniya: Babu hanyoyin daidai guda biyu da suka fitar da ƙima daban-daban
Bazuwar: Dole ne ƙimar da aka fitar ta zama rarraba cikin daidaito akan yankin D, |D| ≥ 2

Aiwatar da baya ko dai suna ɗauka tsarin aiki tare da saitin amincewa ko kuma tsarin aiki na ɓangare. Aikinmu yana mai da hankali kan tsarin asynchronous ba tare da irin waɗannan zato ba.

2.2 Yarjejeniyar Kusa

Yarjejeniyar kusa tana ba wa hanyoyin damar yanke shawara kan ƙimomin da ke kusa da juna a cikin ƙayyadaddun haƙuri ε. Aikin haɗuwa ana iya bayyana shi kamar haka:

$v_i^{r+1} = \frac{\sum_{j \in S} v_j^r}{|S|}$ inda $S$ shine saitin ƙimomin da aka karɓa a cikin kewayon da aka yarda da shi

Wannan ainihin yana samar da tushen aiwatar da tsabar kuɗi ta kusa.

3 Zane na Ka'idoji

3.1 Tsabar Kudi na Kusa

Ka'idar mu ta tsabar kuɗi ta kusa tana tabbatar da cewa duk hanyoyin daidai suna fitar da ƙimomin da ke cikin nisa ε na juna bayan k zagaye. Ka'idar tana haɗuwa cikin sauri mai ma'ana tare da adadin zagayen:

$\epsilon_k \leq \epsilon_0 \cdot \alpha^k$ inda $\alpha < 1$ shine ƙimar haɗuwa

Algorithm yana ci gaba a cikin zagayen asynchronous, tare da kowane tsari yana watsa kimantawarsa na yanzu kuma yana amfani da aikin yarjejeniya kusa da ƙimomin da aka karɓa.

3.2 Tsabar Kudi ta Monte Carlo

Tsabar kuɗi ta Monte Carlo tana ba da garantin yarjejeniya tare da yuwuwar 1-δ don kowane ƙaramin δ > 0. Yuwuwar gazawa tana raguwa cikin ma'ana tare da adadin zagayen:

$P[\text{gazawa}] \leq e^{-\beta k}$ don wasu m $\beta > 0$

Wannan ka'idar ta haɗu da yarjejeniya kusa tare da dabarun ɓoyayyen bayanai don cimma kaddarorin da ake so ba tare da saitin amincewa ba.

4 Bincike na Fasaha

4.1 Tushen Lissafi

Tsaron ka'idojinmu ya dogara ne da kaddarorin haɗuwa na yarjejeniyar kusa a gaban kurakuran Byzantine. Don tsarin tare da hanyoyin n da f < n/3 gazawar Byzantine, mun tabbatar:

$\lim_{k \to \infty} \max_{i,j \in \text{daidai}} |v_i^k - v_j^k| = 0$

Ƙimar haɗuwa ta dogara da tsarin hanyar sadarwa da takamaiman aikin yarjejeniya da aka yi amfani da shi.

4.2 Binciken Tsaro

Ka'idojinmu suna da ƙarfi a kan maƙiyan Byzantine masu daidaitawa waɗanda ke sarrafa har zuwa f < n/3 hanyoyin. Hujjojin tsaro suna bin tsarin kwaikwayo, suna nuna cewa babu yanayi da zai iya bambanta tsakanin ka'idar ta gaske da aikin manufa.

5 Sakamakon Gwaji

Mun kimanta ka'idojinmu a cikin kwaikwayayyun hanyoyin sadarwa na asynchronous tare da bambancin adadin hanyoyin (n = 10 zuwa 100) da ƙimar gazawar Byzantine (f < n/3). Sakamakon ya nuna:

Haɗuwa mai ma'ana zuwa yarjejeniya a cikin zagaye 5-10 don sigogi na al'ada
Hadaddiyar sadarwa na O(n³log n) don yarjejeniyar Byzantine binary
Gagarumin ci gaba akan maganganun O(n⁴) na baya

Pseudocode mai zuwa yana kwatanta ainihin zagayen yarjejeniyar kusa:

function ApproximateAgreementRound(value, round):
    broadcast("PROPOSE", value, round)
    received = wait_for_messages(n - f, round)
    valid_values = filter_within_range(received, value - ε, value + ε)
    new_value = median(valid_values)  // ko matsakaita don yankuna masu ci gaba
    return new_value

6 Cikakkun Bayanai na Aiwa

Aiwatar mu tana amfani da na'urorin ɓoyayyen bayanai na yau da kullun ciki har da ayyukan hash da sa hannun dijital. Tsarin algorithm na core:

class MonteCarloCommonCoin:
    def __init__(self, n, f, delta):
        self.n = n  # jimillar hanyoyin
        self.f = f  # matsakaicin gazawar Byzantine
        self.delta = delta  # yuwuwar gazawa
        self.round = 0
        
    def generate_coin(self):
        while True:
            self.round += 1
            estimate = self.approximate_agreement_round()
            if self.consensus_achieved(estimate):
                return self.finalize_output(estimate)
            if self.round > self.required_rounds():
                return self.fallback_output()

7 Aikace-aikace da Hanyoyin Gaba

Aikace-aikace na Yanzu:

Yarjejeniyar Byzantine tare da hadaddiyar sadarwa O(n³log n)
Matsalar Rukunin Bazuwar da ke tsaka-tsaki ta amfani da tsabar kuɗi ta kusa tare da lambobin Gray
Zaɓen shugaba a cikin tsarin blockchain marasa izini

Hanyoyin Bincike na Gaba:

Daidaituwa ka'idoji don ɓoyayyen bayanai masu jurewa quantum
Ƙaddamawa zuwa saitunan membobi masu ƙarfi
Inganta don yanayin hanyar sadarwa ta duniya tare da ɓangarorin aiki tare
Haɗin kai tare da gine-ginen blockchain masu tsattsage

8 Nassoshi

Fischer, M. J., Lynch, N. A., & Paterson, M. S. (1985). Rashin yiwuwar yarjejeniya rarraba tare da tsari ɗaya mai kuskure. Jaridar ACM.
Rabin, M. O. (1983). Janar-janar na Byzantine da aka zaɓa bazuwar. Taron kan Tushen Kimiyyar Kwamfuta.
Cachin, C., Kursawe, K., & Shoup, V. (2000). Oracle na bazuwar a Constantinople: Yarjejeniyar Byzantine asynchronous mai amfani ta amfani da ɓoyayyen bayanai. PODC.
Miller, A., Xia, Y., Croman, K., Shi, E., & Song, D. (2016). Badger na zuma na ka'idojin BFT. CCS.
Abraham, I., Malkhi, D., & Spiegelman, A. (2019) Yarjejeniyar Byzantine asynchronous ingantacce mai inganci. PODC.

Bincike na Asali

Wannan bincike yana ba da gagarumin gudummawa ga fagen samar da bazuwar rarraba ta hanyar magance ƙayyadaddun iyakokin tsarin asynchronous. Gabatar da tsabar kuɗi na kusa da na Monte Carlo yana wakiltar hanya mai ma'ana don kewaye rashin yuwuwar FLP, kama da yadda tsarin blockchain mai amfani kamar Bitcoin ya samo asali daga ra'ayoyin yarjejeniya na ka'idar zuwa aiwatar da aiki.

Nasarar da marubutan suka samu na hadaddiyar sadarwa O(n³log n) don yarjejeniyar Byzantine binary tana wakiltar gagarumin ci gaba akan maganganun O(n⁴) na baya. Wannan ci gaban ya yi daidai da yanayin binciken tsarin rarraba mai fa'ida, inda rage yawan kuɗin sadarwa yake da muhimmanci don turawa mai amfani. Irin wannan damuwa game da inganci ya haifar da ci gaba a wasu yankuna, kamar ingantacciyar hanyoyin cibiyoyin sadarwa masu adawa (GANs) a cikin injinan koyo, inda CycleGAN ya nuna yadda za a iya sanya ra'ayoyin ka'idar su zama masu amfani ta hanyar yin la'akari da ƙirar algorithm.

Haɗin yarjejeniyar kusa tare da lambobin Gray don magance matsalar Rukunin Bazuwar da ke tsaka-tsaki yana da sabon salo musamman. Wannan hanya tana nuna yadda za a iya sake amfani da ra'ayoyin kimiyyar kwamfuta na gargajiya don ƙalubalen tsarin rarraba na zamani. Dabarar tana da kamanceceniya da aikace-aikacen ka'idar lamba a cikin tsarin ajiya rarraba, inda rarraba bayanai mai inganci da dawo da su suka dogara da sifofin lissafi tare da takamaiman kaddarorin tsaka-tsaki.

Daga mahangar tsaro, juriya a kan maƙiyan Byzantine masu daidaitawa ba tare da saitin amincewa ko PKI ba abin lura ne. Wannan ya yi daidai da yanayin tsarin rarraba na yanzu waɗanda ke ba da fifikon rage amincewa. Hanyar tana raba kamanceceniya ta falsafa tare da tsarin hujja maras sani, inda dabarun ɓoyayyen bayanai ke ba da damar tantancewa ba tare da bayyana bayanan da ke ƙasa ko dogaro ga hukumomin da aka amince da su ba.

Ƙimar haɗuwa mai ma'ana da aka nuna a cikin ka'idoji yana nuna yuwuwar aikace-aikace fiye da abubuwan amfani na gaggawa da aka tattauna. Irin wannan kaddarorin haɗuwa sun tabbatar da ƙima a cikin algorithms ingantawa da koyo, inda saurin haɗuwa zuwa yarjejeniya ke ba da damar lissafin rarraba mai inganci. Kamar yadda aka lura a cikin bincike daga cibiyoyi kamar Laboratory na Kimiyyar Kwamfuta da Injiniyan Wwayo na MIT, irin waɗannan kaddarorin suna da ƙima musamman a cikin yanayin lissafi na gefe tare da ƙuntatawa albarkatun.

Aikin gaba zai iya bincika haɗin kai zuwa ɓoyayyen bayanai na homomorphic da lissafi mai tsaro na ɗimbin ɓangarori, inda samar da bazuwar rarraba ke taka muhimmiyar rawa. Hakanan dabarun da aka haɓaka a cikin wannan takarda na iya samun aikace-aikace a cikin tsarin koyo na tarayya, inda daidaita bazuwar a cikin nodes masu rarraba ba tare da ikon tsakiya ba yana gabatar da ƙalubale iri ɗaya da waɗanda aka magance a cikin wannan bincike.