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❆❧❣❡❜r❛✐❝ ❈r②♣t♦❣r❛♣❤②✿ ■❞❡❛s✱ Pr♦♣♦s❛❧s✱ ❱✉❧♥❡r❛❜✐❧✐t✐❡s ❛♥❞ P♦ss✐❜✐❧✐t✐❡s

❱✐t❛❧② ❘♦♠❛♥✬❦♦✈ ❖▼❙❑✿ ❆▲▼❆❩✱ ❆♣r✐❧ ✷✵✶✺

❈♦♥t❡♥t

✶ ❆❧❣❡❜r❛✐❝ ✈❡rs✐♦♥s ♦❢ ❝❧❛ss✐❝❛❧ s❝❤❡♠❡s ✷ ❈r②♣t❛♥❛❧②s✐s ♦❢ ❛❧❣❡❜r❛✐❝ ✈❡rs✐♦♥s ♦❢ ❝❧❛ss✐❝❛❧ s❝❤❡♠❡s ✸ ❆ ♥♦♥❧✐♥❡❛r ✈❡rs✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ♠❡t❤♦❞ ✹ ❈r②♣t♦ ●❛❧❧❡r②

❉✐✣❡✲❍❡❧❧♠❛♥

▼❡r❦❧❡

❘❛❧♣❤ ▼❡r❦❧❡

❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ s❝❤❡♠❡ ❚❤❡ ❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ ✭✶✾✼✻✮ ❦❡② ❛❣r❡❡♠❡♥t s❝❤❡♠❡

P✉❜❧✐❝ ❞❛t❛✿ ④G ✕ ❣r♦✉♣✱ g ∈ G ⑥✳ ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ k ∈ Z✱ t❤❡♥ ♣✉❜❧✐❝s gk✳ ❇♦❜ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ l ∈ Z✱ t❤❡♥ ♣✉❜❧✐❝s gl✳ ❆❣r❡❡♠❡♥t✿ Alice : (gl)k = gkl = (gk)l : Bob

❊❧●❛♠❛❧

❊❧●❛♠❛❧

❊❧●❛♠❛❧ ✖ ▼❛ss❡②✲❖♠✉r❛✮ ❚❤❡ ❊❧●❛♠❛❧ ✭✶✾✽✵✮✕▼❛ss❡②✲❖♠✉r❛ ✭✶✾✽✷✮ ❝r②♣t♦s②st❡♠ ❢♦r ♠❡ss❛❣❡ ✭❦❡②✮ tr❛♥s♠✐ss✐♦♥

Pr✐✈❛t❡ ❞❛t❛ ✭❦❡②✮✿ ④g ∈ G⑥✳ P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r r s✉❝❤ t❤❛t gr = 1✳ ◆✉♠❜❡r r ♠❛② ❜❡ ❣✐✈❡♥ ❛s t❤❡ ♦r❞❡r |G| ♦❢ G✱ ♦r |g| ♦❢ g. ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ k ∈ Z, (k, r) = 1✱ t❤❡♥ ♣✉❜❧✐❝s gk✳ ❇♦❜ ❝❤♦♦s❡s l ∈ Z, (l, r) = 1✱ t❤❡♥ ❝♦♠♣✉t❡s ❛♥❞ ♣✉❜❧✐❝s (gk)l = gkl✳ ❆❧✐❝❡ ❝♦♠♣✉t❡s k−1(modr) ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s (gkl)k−1 = gl. ❚❤❡ tr❛♥s♠✐tt❡❞ ❦❡②✿ ❇♦❜ ❝♦♠♣✉t❡s l−1(modr), t❤❡♥ ❤❡ r❡❝♦✈❡rs t❤❡ tr❛♥s♠✐tt❡❞ ❦❡②✿ (gl)l−1 = g.

❊❧●❛♠❛❧ ❊❧●❛♠❛❧ ✭✶✾✽✺✮ ❝r②♣t♦s②st❡♠ ❢♦r ♠❡ss❛❣❡ ✭❦❡②✮ tr❛♥s♠✐ss✐♦♥

❆❧✐❝❡ s❡ts ♣✉❜❧✐❝ ❞❛t❛✿ ④G ✕ ❣r♦✉♣✱ g ∈ G⑥✳ ❆❧s♦ s❤❡ s❡ts ♣r✐✈❛t❡ ❞❛t❛ ✭❦❡②✮✿ 0 < a < |g|, ❛♥❞ ♦t❤❡r ♣✉❜❧✐❝ ❞❛t❛ ✭❡♥❝②♣❡r✐♥❣ ❦❡②✮✿ ga. ❇♦❜ ✇❛♥ts t♦ s❡♥❞ ❛ ♠❡ss❛❣❡ m ∈ G t♦ ❆❧✐❝❡✳ ❍❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ k ∈ Z, 0 < k < |g|✱ t❤❡♥ ♣✉❜❧✐❝s (gk, mgak)✳ ❚❤❡ tr❛♥s♠✐tt❡❞ ♠❡ss❛❣❡✿ ❆❧✐❝❡ ❝♦♠♣✉t❡s gak = (gk)a ❛♥❞ (gak)−1, t❤❡♥ r❡❝♦✈❡rs t❤❡ ♠❡ss❛❣❡ m = (mgak)(gak)−1.

P❧❛t❢♦r♠s ❛♥❞ ♦♣❡r❛t✐♦♥s✿ ♥✉♠❜❡r t❤❡♦r❡t✐❝ ❛♥❞ ❛❧❣❡❜r❛✐❝

P❧❛t❢♦r♠s ❛♥❞ ♦♣❡r❛t✐♦♥s✿ ♥✉♠❜❡r t❤❡♦r❡t✐❝ ❛♥❞ ❛❧❣❡❜r❛✐❝

❈❧❛ss✐❝ ♣❧❛t❢♦r♠s✿ G = F∗

pr ✕ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣r♦✉♣ ♦❢ ❛

✜♥✐t❡ ✜❡❧❞ Fpr ✱ ♦r G(E) ✕ t❤❡ ❣r♦✉♣ ♦❢ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ E ✭♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞✮✳ ❈❧❛ss✐❝ ♦♣❡r❛t✐♦♥s✿ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ✐♥✈♦❧✉t✐♦♥✱ ♦r ❛❞❞✐t✐♦♥ ❛♥❞ t❛❦✐♥❣ ♠✉❧t✐♣❧❡✳

• r♦✉♣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ♣❧❛t❢♦r♠s✿ G ✕ ❛❜str❛❝t ❣r♦✉♣

✭❆rt✐♥ ❜r❛✐❞ ❣r♦✉♣s✱ ♠❛tr✐① ❣r♦✉♣s ♦✈❡r ✜❡❧❞s ❛♥❞ r✐♥❣s✱ ♣♦❧②❝②❝❧✐❝ ❣r♦✉♣s✱ ✜♥✐t❡ p✲❣r♦✉♣s ❛r❡ ♠♦st ♣♦♣✉❧❛r✮✳

• r♦✉♣ ❜❛s❡❞ ♦♣❡r❛t✐♦♥s✿ ❘✐❣❤t ✭❧❡❢t✮ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥✈❡rs✐♦♥✱

✐♥✈♦❧✉t✐♦♥✱ ❝♦♥❥✉❣❛t✐♦♥✱ ❛❝t✐♦♥s ❜② ❡♥❞♦♠♦r♣❤✐s♠ ✭❛✉t♦♠♦r♣❤✐s♠✮✳ ❆❧❣❡❜r❛✐❝ ♦♣❡r❛t✐♦♥s✿ ♦♣❡r❛t✐♦♥s ❞❡r✐✈❡❞ ❢r♦♠ ❜❛s✐❝ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡ ❣✐✈❡♥ ❛❧❣❡❜r❛✐❝ ♣❧❛t❢♦r♠✱ ✐♥❝❧✉❞✐♥❣ ♠♦r♣❤✐s♠s✳

❚❤❡ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠ ✐♥ ❛ ♠❛tr✐① ❣r♦✉♣

❚❤❡ ❉✐s❝r❡t❡ ▲♦❣❛r✐t❤♠ Pr♦❜❧❡♠ ✐♥ GLn(Fq), q = pr.

g ∈ GLn(Fq), h = gk, k = loggh. ❋✐♥❞ t❤❡ ❏♦r❞❛♥ ❢♦r♠✿ J(g) = tgt−1. J(g) = Jr1(λ1) ⊕ ... ⊕ Jrt(λt), t

i=1 ri = n.

λ1, ..., λt ❛r❡ r♦♦ts ✭✐♥ ❡①t❡♥s✐♦♥s Fqni ♦❢ Fq✮ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ pg(x) = |g − λE| = (x − λ1)r1...(x − λt)rt = 0. ❍❡r❡ Js(λ) =     λ 1 ... λ 1 ... . . . . . ... λ     ✐s ❏♦r❞❛♥ ❜❧♦❝❦ ♦❢ s✐③❡ s.

❊✛❡❝t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❏♦r❞❛♥ ❢♦r♠

❊✛❡❝t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ J(g)✿ ■♥♣✉t✿ g ∈ GLn(Fq)✳ ❖✉t♣✉t✿ J(g)✳

✶ ❇② t❤❡ ❍❡ss❡♥❜❡r❣✬s ❛❧❣♦r✐t❤♠ ✭✇❤✐❝❤ ✐s ♠♦r❡ ❡✛❡❝t✐✈❡ ✐♥ t❤❡

❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ t❤❛♥ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ✐s O(n3)✮ ✇❡ ✜♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ pg(x).

✷ ❇② t❤❡ ♣r♦❜❛❜✐❧✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❇❡♥✲❖r✬s ❛❧❣♦r✐t❤♠ ✇❡ ❣❡t ❛

♣r❡s❡♥t❛t✐♦♥ pg(x) = f e1

1 ...f es s , ✇❤❡r❡ fi ✐s ✐rr❡❞✉❝✐❜❧❡

♣♦❧②♥♦♠✐❛❧ ♦✈❡r Fq ♦❢ ❞❡❣r❡❡ ni.

✸ Fqni = Fq[x]/ideal(fi(x))✳ ❲❡ ✜♥❞ r♦♦ts αij, 1 ≤ j ≤ ni, ♦❢ fi ✐♥

Fqni ✳ ◆❛♠❡❧②✱ αi1 = x, αij = xqi−1mod(fi(x)), 2 ≤ j ≤ ni.

✹ ❲❡ ✜♥❞ s✐③❡s ♦❢ ❏♦r❞❛♥ ❜❧♦❝❦s Jl, ❛♥❞ t❤❡♥ ✇❡ ❣❡t t❤❡ ❏♦r❞❛♥

❢♦r♠ J(g) = J1 ⊕ ... ⊕ Jt.

P♦❧②♥♦♠✐❛❧✐t② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❍❡ss❡♥❜❡r❣✬s ❛♥❞ ❇❡♥✲❖r✬s ❛❧❣♦r✐t❤♠s s♦❧✈❡ t❛s❦s ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❇❡❝❛✉s❡ ni ≤ n, ✇❡ ✉s❡ ✐♥ ❡❛❝❤ ♦❢ s ≤ n ✐t❡r❛t✐♦♥s ♦♥ t❤❡ st❡♣ ✸ log qni ≤ nlog q ♦♣❡r❛t✐♦♥s✳ ❍❡♥❝❡ t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤✐s ♣r♦❝❡❞✉r❡ ✐s ❡st✐♠❛t❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥ n ❛♥❞ log q✳

❘❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ❢♦r ❛ ♠❛tr✐① ❣r♦✉♣ ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ t♦ t❤❡ ♠✉❧t✐♣❧❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ❢♦r ✜♥✐t❡ ✜❡❧❞✭s✮

❘❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❉▲P ❢♦r ❛ ♠❛tr✐① ❣r♦✉♣ t♦ t❤❡ ♠✉❧t✐♣❧❡ ❉▲P ❢♦r ✜♥✐t❡ ✜❡❧❞s✿ ■♥♣✉t✿ h = gk, h, g ∈ GLn(Fq). ❖✉t♣✉t✿ k ∈ Z.

✶ ❋✐♥❞ t s✉❝❤ t❤❛t tgt−1 = J(g). ✷ ❈♦♠♣✉t❡ tht−1 = (tgt−1)k. ✸ αk

ij = βij, ✇❤❡r❡ βij ❛r❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ♦❢ tht−1.

■❞❡❛✿ ❆✳❏✳ ▼❡♥❡③❡s ❛♥❞ ❙✳❆✳ ❱❛♥st♦♥❡✱ ❆ ♥♦t❡ ♦♥ ❝②❝❧✐❝ ❣r♦✉♣s✱ ✜♥✐t❡ ✜❡❧❞s✱ ❛♥❞ t❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠✱ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✐♥ ❊♥❣✐♥❡❡r✐♥❣✱ ❈♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣✱ ✶✾✾✷✱ ✸✱ ✻✼✲✼✹✳ ❆✳❏✳ ▼❡♥❡③❡s ❛♥❞ ❨✳✲❍✳ ❲✉✱ ❚❤❡ ❞✐s❝r❡t❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✐♥

• ▲(n, q)✳ ❆rs ❈♦♠❜✐♥❛t♦r✐❛✱ ✶✾✾✼✱ ✹✼✱ ✷✸✲✸✷✳

❊①tr❛ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❏♦r❞❛♥ ❜❧♦❝❦s ♦❢ s✐③❡ ≥ 2.

❊①❛♠♣❧❡ α 1 α k = αk kαk−1 αk

• .

❊①tr❛ ❡q✉❛t✐♦♥✿ kαk−1 = β, ✇❤❡r❡ β ✐s 12✲❡♥tr② ♦❢ t−1ht. ❇② ♠✉❧t✐♣❧②✐♥❣ ♦❢ ✐ts ❜♦t❤ s✐❞❡s t♦ α✱ ❛♥❞ ❞✐✈✐❞✐♥❣ t♦ αk ✇❡ ♦❜t❛✐♥ k ∈ Fq, ♥❛♠❡❧②✱ kmod p, ✐✳❡✳✱ ✇❡ ❣❡t ❡①tr❛ ✉s❡❢✉❧ ✐♥❢♦r♠❛t✐♦♥✳

❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✐♥ ●▲n(Fq) ✐s ♥♦ ♠♦r❡ ❞✐✣❝✉❧t t❤❛♥ t❤❡ ❧♦❣❛r✐t❤♠ ♣r♦❜❧❡♠ ✐♥ ❛ s✉✐t❛❜❧❡ ❡①t❡♥s✐♦♥ Fqm ✇❤❡r❡ m ≤ n.

❖t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ s❝❤❡♠❡✿ ♠✐①❡❞ ✭✐♥✈♦❧✉t✐♦♥ ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✮ ✈❡rs✐♦♥

■♥✈♦❧✉t✐♦♥ ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥✿ P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G✱ ❡❧❡♠❡♥t g ∈ G✱ ❛♥❞ s✉❜❣r♦✉♣s H1, H2 ≤ G, s✉❝❤ t❤❛t [H1, H2] = 1✳ ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ♥✉♠❜❡r k ∈ Z ❛♥❞ ❡❧❡♠❡♥t a ∈ H1✱ ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s (gk)a✳ ❇♦❜ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ♥✉♠❜❡r l ∈ Z ❛♥❞ ❡❧❡♠❡♥t b ∈ H2✱ ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s ((gl)b)✳ ❙❤❛r❡❞ ❦❡② ✐s Alice : (((gl)b)k)a = (gkl)ab = (((gk)a)l)b : Bob

Pr❡❢❡r❡♥❝❡ ♦❢ t❤❡ ♠✐①❡❞ ✈❡rs✐♦♥

❇❡❝❛✉s❡ ❝♦♥❥✉❣❛t✐♦♥ ❝❛♥ ♣❡r♠✉t❡ ❏♦r❞❛♥ ❜❧♦❝❦s t❤❡ ▼❡♥❡③❡s✲❱❛♥st♦♥❡✲❲✉✬s ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ k ❛♥❞ l ❝❛♥ ♥♦t ❜❡ ❛♣♣❧✐❡❞✳

❖t❤❡r ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡✬s s❝❤❡♠❡✿ ✈❡rs✐♦♥s ✉s✐♥❣ ❛✉t♦♠r♣❤✐s♠s ♦r ❡♥❞♦♠♦r♣❤✐s♠s

❆❝t✐♦♥ ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦r ❡♥❞♦♠♦r♣❤✐s♠✿ P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G✱ ❡❧❡♠❡♥t g ∈ G✱ ❛♥❞ s✉❜❣r♦✉♣s H1, H2 ≤❆✉tG, s✉❝❤ t❤❛t [H1, H2] = 1✳ ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ❛✉t♦♠♦r♣❤✐s♠ ϕ ∈ H1✱ ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s ϕ(g)✳ ❇♦❜ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ❛✉t♦♠♦r♣❤✐s♠ ψ ∈❆✉tG✱ ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s ψ(g)✳ ❙❤❛r❡❞ ❦❡② ✐s Alice : ϕ(ψ(g)) = ψ(ϕ(g)) : Bob P❛rt✐❝✉❧❛r ❝❛s❡✿ ϕ, ψ ∈ ■♥♥G✳ ❚❤✐s ✐s ❉✐✣❡✲❍❡❧❧♠❛♥✲▼❡r❦❧❡ s❝❤❡♠❡✳ ❖♥❡ ❝❛♥ ✉s❡ ❊♥❞G ✐♥st❡❛❞ ♦❢ ❆✉tG✱ ❛♥❞ t✇♦ ❝♦♠♠✉t✐♥❣ ❡❧❡♠❡♥t✇✐s❡ s✉❜s❡♠✐❣r♦✉♣s H1, H2 ♦❢ ❊♥❞G✱ ❛s ✇❡❧❧✳

❈❤❡♦♥ ❛♥❞ ❏✉♥❣ ✈❡rs✉s ❑♦ ❛♥❞ ❛❧✳

■♥ ❬❆ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❜r❛✐❞ ❉✐✣❡✲❍❡❧❧♠❛♥ ❝♦♥❥✉❣❛❝② ♣r♦❜❧❡♠✱ ❆❞✈❛♥❝❡s ✐♥ ❈r②♣t♦❧♦❣②✱ ❈❘❨P❚❖✬✷✵✵✸✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ❈♦♠♣✳ ❙❝✐❡♥❝❡✱ ✼✲✶✹❪ ❈❤❡♦♥ ❛♥❞ ❏✉♥ ♣r♦♣♦s❡❞ ❛ ♣♦❧②♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ s♦❧✈✐♥❣ t❤❡ ❑♦ ❡t✳ ❛❧✳ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧ ✈✐❛ ❢❛✐t❤❢✉❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❆rt✐♥ ❜r❛✐❞ ❣r♦✉♣ Bn ♦♥ n str✐♥❣s ❜② ♠❛tr✐❝❡s✳

❈♦♥❥✉❣❛t✐♦♥ ✐♥st❡❛❞ ♦❢ ✐♥✈♦❧✉t✐♦♥

❈♦♥❥✉❣❛t✐♦♥ ✐♥st❡❛❞ ♦❢ ✐♥✈♦❧✉t✐♦♥✿

P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G ❛♥❞ ❡❧❡♠❡♥t g ∈ G✱ s✉❜❣r♦✉♣s H1 ❛♥❞ H2 s✉❝❤ t❤❛t [H1, H2] = 1. ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ❡❧❡♠❡♥t a ∈ H1✱ t❤❡♥ ♣✉❜❧✐❝s ga = aga−1. ❇♦❜ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ ❡❧❡♠❡♥t b ∈ H2✱ t❤❡♥ ♣✉❜❧✐❝s gb. ❙❤❛r❡❞ ❦❡② ✐s✿ Alice : (gb)a = gab = K = gba = (ga)b : Bob

❚❤❡ s❡❛r❝❤ ❝♦♥❥✉❣❛❝② ♣r♦❜❧❡♠

G s❤♦✉❧❞ ❜❡ ♥♦♥❛❜❡❧✐❛♥✳ ❙❡❝✉r✐t② ♦❢ t❤❡ s❝❤❡♠❡ ❜❛s❡s ♦♥ t❤❡ ❝♦♥❥✉❣❛❝② s❡❛r❝❤ ♣r♦❜❧❡♠✱ t❤❛t ✐s t♦ ✜♥❞ t❤❡ ❡❧❡♠❡♥t a ❜② g ❛♥❞ ga, ♦r✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ t♦ ✜♥❞ ❛♥ ❡❧❡♠❡♥t a′ ∈ H1 ❢♦r ✇❤✐❝❤ ♦♥❡ ❤❛s ga = ga′✳ ❚❤❡♥ ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ (gb)a′ = (ga′)b = (ga)b = gab = K.

❈r②♣t❛♥❛❧②s✐s ❜② ❈❤❡♦♥ ❛♥❞ ❏✉♥

❍♦✇ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ❝♦♥❥✉❣❛t✐♥❣ ❡❧❡♠❡♥t✿

✶ ❆♣♣❧② t❤❡ ▲❛✇r❡♥❝❡✲❑r❛♠♠❡r r❡♣r❡s❡♥t❛t✐♦♥

ϕ : Bn → GLn(n−1)/2(Z[t±1, q±1]) t♦ ❝♦♠♣✉t❡ ✐♠❛❣❡s ϕ(g), ϕ(a−1ga).

✷ ❋✐♥❞ ϕ(a) s✉❝❤ t❤❛t ϕ(a)ϕ(g)ϕ(a)−1 = ϕ(ga). ✸ ❋✐♥❞ t❤❡ ♣r❡✐♠❛❣❡ a ∈ Bn ♦❢ ϕ(a).

❉✐✣❝✉❧t✐❡s

❉✐✣❝✉❧t✐❡s✿

✶ ❉✐r❡❝t ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ●❛✉ss ❡❧✐♠✐♥❛t✐♦♥ s❤♦✉❧❞ ❞❡❛❧ ✇✐t❤

❝♦❡✣❝✐❡♥ts ❛s ❧❛r❣❡ ❛s 22n✮✳

✷ ❆ s♦❧✉t✐♦♥ ”ϕ(a)” ❝❛♥ ❜❡ ♦✉t ♦❢ ϕ(H1). ◆♦t✐❝❡✱ t❤❛t ϕ(a)

s❤♦✉❧❞ ❜❡ ✐♥✈❡rt✐❜❧❡✳ ❍❡♥❝❡✱ t❤✐s ❛♣♣r♦❛❝❤ ✐s ✐rr❡❛❧✳

❆❞✈❛♥❝❡❞ ❝r②♣t❛♥❛❧②s✐s ❜② ❈❤❡♦♥ ❛♥❞ ❏✉♥

❍♦✇ ♦♥❡ ❝❛♥ ✜♥❞ t❤❡ s❤❛r❡❞ ❦❡②✿

✶ ❆♣♣❧② t❤❡ ▲❛✇r❡♥❝❡✲❑r❛♠♠❡r r❡♣r❡s❡♥t❛t✐♦♥

ϕ : Bn → GLn(n−1)/2(Z[t±1, q±1]) t♦ ❝♦♠♣✉t❡ ✐♠❛❣❡s ϕ(g), ϕ(ga), varphi(gb).

✷ ❙♦❧✈❡ ❡q✉❛t✐♦♥s ϕ(ga)Y = Yϕ(g), ϕ(σi)Y = Yϕ(σi). ✸ ❋✐♥❞ ❛ ♣r❡✐♠❛❣❡ a ♦❢ ϕ(a) ✐♥ Bn✳

◆♦✇ ♦♥❡ ❤❛s s✐♠✐❧❛r ❞✐✣❝✉❧t✐❡s ✐♥ ❛♣♣❧②✐♥❣ ♦❢ t❤❡ ❝r②♣t❛♥❛❧②s✐s✳ ❖♥❡ ❤❛s t♦♦ ♠✉❝❤ ❡q✉❛t✐♦♥s ❛♥❞ ✉♥❦♥♦✇♥s✳ ■❢ ♦♥❡ ✜♥❞s ❛ s✐♥❣✉❧❛r s♦❧✉t✐♦♥✱ ❤❡ ♥❡❡❞s t♦ r❡♣❡❛t ♣r♦❝❡❞✉r❡✳ ❚❤✉s t❤✐s ❛❧❣♦r✐t❤♠ ✐s ♣r♦❜❛❜✐❧✐st✐❝✳ ■t ✐s ♣r❛❝t✐❝❛❧❧② ♥♦♥ r❡❛❧✐③❛❜❧❡✳

❚s❛❜❛♥

❚s❛❜❛♥

❚s❛❜❛♥ ✈❡rs✉s ❛❧❧

■♥ ❬Pr❛❝t✐❝❛❧ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♦❧✉t✐♦♥s ♦❢ s❡✈❡r❛❧ ♠❛❥♦r ♣r♦❜❧❡♠s ✐♥ ♥♦♥❝♦♠♠✉t❛t✐✈❡✲❛❧❣❡❜r❛✐❝ ❝r②♣t♦❣r❛♣❤②✱ ❈r②♣t♦❧♦❣② ❡Pr✐♥t ❆r❝❤✐✈❡✿ ❘❡♣♦rt ✷✵✶✹✴✵✹✶❪ ❇♦❛③ ❚s❛❜❛♥ ♣r♦✈✐❞❡❞ ♣r♦✈❛❜❧❡ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♦❧✉t✐♦♥s ♦❢ ❛ ♥✉♠❜❡r ♦❢ ♣r♦❜❧❡♠s ✐♥ ❛❧❣❡❜r❛✐❝ ❝r②♣t♦❣r❛♣❤②✳ ❍❡ ♥❛♠❡❞ t❤✐s ❛♣♣r♦❛❝❤ t❤❡ ❛❧❣❡❜r❛✐❝ s♣❛♥ ♠❡t❤♦❞✳ ◆♦✇ ✇❡ ❞❡♠♦♥str❛t❡ t❤❡ ♠❡t❤♦❞ ✇✐t❤ ❛♣♣❧②✐♥❣ ✐t t♦ t❤❡ ▼✳ ❆♥s❤❡❧ ❡t ❛❧✳ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧✳

▼✳ ❆♥s❤❡❧ ❡t ❛❧✳ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧

▼✳ ❆♥s❤❡❧ ❡t ❛❧✳ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧✿ P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G ❛♥❞ ❡❧❡♠❡♥ts a1, ..., ak, b1, ..., bk ∈ G. ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ❣r♦✉♣ ✇♦r❞ v(x1, ..., xk)✱ ❝♦♠♣✉t❡s a = v(a1, ..., ak), ❛♥❞ ♣✉❜❧✐❝s ba

1, ..., ba k.

❇♦❜ ❝❤♦♦s❡s ❛ ❣r♦✉♣ ✇♦r❞ w(x1, ..., xk)✱ ❝♦♠♣✉t❡s b = w(b1, ..., bk), ❛♥❞ ♣✉❜❧✐❝s ab

1, ..., ab k.

❚❤❡ s❤❛r❡❞ ❦❡② K ✐s t❤❡ ❝♦♠♠✉t❛t♦r [a, b] = aba−1b−1. ❆❧✐❝❡ ❝❛♥ ❝♦♠♣✉t❡ K ❛s av(ab

1, ..., ab k)−1. ❇♦❜ ❝♦♠♣✉t❡s K ❛s

w(ba

1, ..., ba k)b−1.

❚s❛❜❛♥✬s ❝r②♣t❛♥❛❧②s✐s ♦❢ ▼✳ ❆♥s❤❡❧ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧ ❚s❛❜❛♥✬s ❝r②♣t❛♥❛❧②s✐s ♦❢ ▼✳ ❆♥s❤❡❧ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧✿

G =❣♣(g1, ..., gk) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♠❛tr✐① ❣r♦✉♣ ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ Fq, q = pr. ❋♦r ❛ s❡t S ⊆ ▼n(Fq)✱ ❧❡t ❆❧❣(S) ❜❡ t❤❡ ❛❧❣❡❜r❛ ❣❡♥❡r❛t❡❞ ❜② S. ❚❤❡♥ ❆❧❣(G) ❂ s♣❛♥(G). ❆ ❜❛s✐s ❢♦r t❤❡ ✉♥❞❡r❧②✐♥❣ ✈❡❝t♦r s♣❛❝❡ ♦❢ ❆❧❣(G) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t✐♠❡ O(kn6).

❚s❛❜❛♥✬s ❝r②♣t❛♥❛❧②s✐s ♦❢ ▼✳ ❆♥s❤❡❧ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧

■♥♣✉t✿ a1, ..., ak, b1, ..., bk, ab

1, ..., ab k, ba 1, ..., ba k ∈ G, ✇❤❡r❡ a ∈

❣♣(a1, ..., ak), b ∈ ❣♣(b1, ..., bk) ❛r❡ ✉♥❦♥♦✇♥✳

✶ ❖✤✐♥❡✿ ●❡♥❡r❛t❡ ❜❛s❡s ❢♦r ❆❧❣(A) ❛♥❞ ❆❧❣(B). ▲❡t d ❜❡ t❤❡

♠❛①✐♠✉♠ ♦❢ t❤❡ s✐③❡s ♦❢ t❤❡s❡ ❜❛s❡s✳

✷ ❖♥❧✐♥❡✿

✭❛✮ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ♦♥ t❤❡ d ❝♦❡✣❝✐❡♥ts ❞❡t❡r♠✐♥❣ x ∈ ❆❧❣(A) : bi · x = x · ba

i ❢♦r i = 1, ..., k.

✭❜✮ ❋✐① ❛ ❜❛s✐s ❢♦r t❤❡ s♦❧✉t✐♦♥ s♣❛❝❡✱ ❛♥❞ ♣✐❝❦ r❛♥❞♦♠ s♦❧✉t✐♦♥s x ✉♥t✐❧ x ✐s ✐♥✈❡rt✐❜❧❡✳ ✭❝✮ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦♠♦❣❡♥❡♦✉s s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ♦♥ t❤❡ d ❝♦❡✣❝✐❡♥ts ❞❡t❡r♠✐♥❣ x ∈ ❆❧❣(B) : ai · y = y · ab

i ❢♦r i = 1, ..., k.

✭❞✮ ❋✐① ❛ ❜❛s✐s ❢♦r t❤❡ s♦❧✉t✐♦♥ s♣❛❝❡✱ ❛♥❞ ♣✐❝❦ r❛♥❞♦♠ s♦❧✉t✐♦♥s y ✉♥t✐❧ y ✐s ✐♥✈❡rt✐❜❧❡✳ ✭❡✮ ❖✉t♣✉t✿ K = xyx−1y−1.

❚❤❡ s♣❡❛❦❡r ❛♥❞ ▼②❛s♥✐❦♦✈ ✈❡rs✉s ❛❧❧

❰ñíîâíàÿ èäåÿ ìåòîäà ëèíåéíîãî ðàçëîæåíèÿ

❰áîçíà÷åíèÿ✿ V ✕ ïðîñòðàíñòâî êîíå÷íîé ðàçìåðíîñòè íàä ïîëåì F ñ áàçèñîì B = {v1, . . . , vr}✳ ❊♥❞(V) ✕ ïîëóãðóïïà ýíäîìîðôèçìîâ ïðîñòðàíñòâà V. Ýëåìåíòû v ∈ V✕ âåêòîðû îòíîñèòåëüíî áàçèñà B✳ Ýíäîìîðôèçìû a ∈ ❊♥❞(V) ✕ ìàòðèöû îòíîñèòåëüíî B✱ va ✕ îáðàç v îòíîñèòåëüíî a. ➘ëÿ ïîäìíîæåñòâ W ⊆ V è A ⊆ ❊♥❞(V) îáîçíà÷èì W A = {wa|w ∈ W, a ∈ A}✳ Ïîëàãàåì ❙♣(W) ïîäïðîñòðàíñòâî V✱ ïîðîæäåííîå W✱ A✕ ïîäìîíîèä✱ ïîðîæäåííûé A â ❊♥❞(V)✳ Ïðåäïîëàãàåì✱ ÷òî ýëåìåíòû ïîëÿ F çàäàíû â íåêîòîðîé êîíñòðóêòèâíîé ôîðìå✱ ïðè÷åì îïðåäåëåí ðàçìåð çàäàíèÿ✳ ❰ïåðàöèè â F ýôôåêòèâíû✱ ïðåäñòàâëÿþòñÿ çà ïîëèíîìèàëüíîå âðåìÿ îò ðàçìåðîâ íîðìàëüíûõ ôîðì✳

❰ñíîâíàÿ ëåììà

➘ëÿ α ∈ F ÷åðåç |α| îáîçíà÷àåòñÿ åãî ðàçìåð✳ ➘ëÿ v = (α1, . . . , αr) ∈ V ïîëàãàåì |v| = max |αi|. ➘ëÿ ìàòðèöû a = (αij) ∈ ❊♥❞(V) ïîëàãàåì |a| = max{|αij|}. ▲❡♠♠❛ ✭❰ñíîâíàÿ ëåììà✮ Ñóùåñòâóåò àëãîðèòì íàõîæäåíèÿ äëÿ äàííûõ êîíå÷íûõ ïîäìíîæåñòâ W ⊆ V è U ⊆ End(V) áàçèñà ïîäïðîñòðàíñòâà Sp(W U) â âèäå wa1

1 , . . . , wat t ✱ ãäå wi ∈ W è ai ✕ ïðîèçâåäåíèå

ýëåìåíòîâ èç U✳ ×èñëî èñïîëüçîâàííûõ îïåðàöèé íàä ýëåìåíòàìè ïîëÿ ïîëèíîìèàëüíî ïî r = dimF V è êîëè÷åñòâàì ýëåìåíòîâ W è U✳

➘îêàçàòåëüñòâî îñíîâíîé ëåììû

➚ëãîðèòì✿

✶ ❒åòîäîì èñêëþ÷åíèé ➹àóññà íàõîäèì ìàêñèìàëüíîå

ëèíåéíî íåçàâèñèìîå ✭ë✳í✳✮ ïîäìíîæåñòâî L0 ♦❢ W✳ ➬àìåòèì✱ ÷òî Sp(LU ) = Sp(W U)✳

✷ ➘îáàâëÿåì ê ìíîæåñòâó L0 ýëåìåíòû va, v ∈ L0, a ∈ U,

ïðîâåðÿÿ êàæäûé ðàç ë✳ í✳ ïîëó÷åííîãî ìíîæåñòâà✳ Òàêèì îáðàçîì áóäåò ïîñòðîåíî ìàêñèìàëüíîå ë✳ í✳ ïîäìíîæåñòâî L1 ìíîæåñòâà L0 ∪ LU

0 ðàñøèðÿþùåå L0✳ ➬àìåòèì✱ ÷òî

Sp(LU ) = Sp(LU

1

)✱ è ýëåìåíòû â L1 èìåþò ôîðìó w èëè wa✱ ãäå w ∈ W è a ∈ U✳ ❰òñþäà✱ åñëè L0 = L1✱ òî L0 ✕ áàçèñ â Sp(W U)✳

✸ ➴ñëè L0 = L1✱ òî ïîâòîðÿåì ïðîöåäóðó äëÿ L1 \ L0 è

íàõîäèì ìàêñèìàëüíîå ë✳ í✳ ïîäìíîæåñòâî L2 â L1 ∪ (L1 \ L0)U✱ ðàñøèðÿþùåå L1✳ Ñòðîèì L0 < L1 < . . . < Li â V✳ Òàê êàê ðàçìåðíîñòü r ïðîñòðàíñòâà V êîíå÷íà✱ ïîñëåäîâàòåëüíîñòü ñòàáèëèçèðóåòñÿ íà i ≤ r✳ Òîãäà Li ✕ áàçèñ â Sp(W U) èç ýëåìåíòîâ òðåáóåìîãî âèäà✳

❰öåíêà ñëîæíîñòè

×èñëî îïåðàöèé â ìåòîäå èñêëþ÷åíèé ➹àóññà îòíîñèòåëüíî ìàòðèöû ðàçìåðà n × r åñòü O(n2r)✳ Ñëåäîâàòåëüíî✱ òðåáóåòñÿ íå áîëåå O(n2r) øàãîâ ïîñòðîåíèÿ L0 èç W✱ ãäå n = |W| ✕ ÷èñëî ýëåìåíòîâ W✳ ➬àìåòèì✱ ÷òî |Lj| ≤ r äëÿ ëþáîãî j✳ Ïîýòîìó íàõîæäåíèå Lj+1 èñïîëüçóåò ìàòðèöó ñîîòâåòñòâóþùóþ Lj ∪ LU

j

ðàçìåðà íå áîëüøå r + r|U|✳ ➶åðõíÿÿ ãðàíèöà✿ O(r 3|U|2)✳ Òàê êàê èòåðàöèé ≤ r✱ îáùàÿ îöåíêà O(r 3|U|2 + r|W|2)✳ ❈♦r♦❧❧❛r② Ïðè ñäåëàííûõ ïðåäïîëîæåíèÿõ îòíîñèòåëüíî F àëãîðèòì îñíîâíîé ëåììû ðàáîòàåò çà ïîëèíîìèàëüíîå îò ðàçìåðîâ r = dimF V✱ |W|✱ |U|✱ è max{|w|, |u| | w ∈ W, u ∈ U} âõîäà âðåìÿ✳

➪àçîâàÿ ìîäåëü

Ïóñòü U1 è U2 ✕ êîíå÷íûå ïîäìíîæåñòâà ïîëóãðóïïû ýíäîìîðôèçìîâ ❊♥❞(V); ∀u1 ∈ U1, u2 ∈ U2 : u1u2 = u2u1; A =< U1 >, B =< U2 >, v ∈ V, a ∈ A, b ∈ B; ④a ∈ A, b ∈ B⑥ ✕ ñåêðåòíûå äàííûå✱ ④U1, U2, v, va, vb⑥ ✕ îòêðûòûå äàííûå✱ ❚❤❡♦r❡♠ Ïî äàííûì U1, U2, v, va, vb çà ïîëèíîìèàëüíîå âðåìÿ íàõîäèòñÿ âåêòîð vab = vba✳ ➪åç âû÷èñëåíèÿ a èëè b✦

➚ëãîðèòì

✶ Ïî U1 è v✱ èñïîëüçóÿ àëãîðèòì îñíîâíîé ëåììû ✭ñì✳ òàêæå

ñëåäñòâèå èç îñíîâíîé ëåììû✮✱ çà ïîëèíîìèàëüíîå âðåìÿ íàõîäèì áàçèñ va1, . . . , vat, ai ∈ A, ïðîñòðàíñòâà Sp(vA). ❒åòîäîì èñêëþ÷åíèÿ ➹àóññà ðàçëàãàåì va ïî ýòîìó áàçèñó✿ va = Σt

i=1αivai,

αi ∈ F.

✷ ❮àõîäèì vab :

vab = (va)b = (Σt

i=1αivai)b =

Σt

i=1αivaib = Σt i=1αivbai = Σt i=1αi(vb)ai.

❮åò íåîáõîäèìîñòè â íàõîæäåíèè íè a✱ íè b, ÷òîáû âû÷èñëèòü vab✳ ➬àìåòèì òàêæå✱ ÷òî íåò íåîáõîäèìîñòè çíàòü U2✱ äîñòàòî÷íî òîãî✱ ÷òî äëÿ íåêîòîðîãî b ∈ ❊♥❞(V) èìååì ∀(u ∈ U1)ub = bu✳

❚❤❡ s♣❡❛❦❡r ❛♥❞ ▼②❛s♥✐❦♦✈✬s ❝r②♣t❛♥❛❧②s✐s ♦❢ ❲❛♥❣ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧

❲❡ ♣r♦♣♦s❡ ♥❡✇ ♣r♦✈❛❜❧❡ ♣r❛❝t✐❝❛❧ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❜r❛✐❞ ❲❛♥❣✱ ❳✉✱ ▲✐✱ ▲✐♥ ❛♥❞ ❲❛♥❣ ❉♦✉❜❧❡ s❤✐❡❧❞❡❞ ♣✉❜❧✐❝ ❦❡② ❝r②♣t♦s②st❡♠s✳ ❲❡ s❤♦✇ t❤❛t ❛ ❧✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ❛tt❛❝❦ ❜❛s❡❞ ♦♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♠❡t❤♦❞ ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❛✉t❤♦r ✇♦r❦s ❢♦r t❤❡ ✐♠❛❣❡ ♦❢ ❜r❛✐❞s ✉♥❞❡r t❤❡ ▲❛✇r❡♥❝❡✲❑r❛♠♠❡r r❡♣r❡s❡♥t❛t✐♦♥ ❜② ✜♥❞✐♥❣ t❤❡ ❡①❝❤❛♥❣✐♥❣ ❦❡②s ✐♥ t❤❡ ❜♦t❤ t✇♦ ♠❛✐♥ ♣r♦t♦❝♦❧s ♣r♦♣♦s❡❞ ✐♥ ❬❳✳ ❲❛♥❣✱ ❈✳ ❳✉✱ ●✳ ▲✐✱ ❍✳ ▲✐♥ ❛♥❞ ❲✳ ❲❛♥❣✱ ❉♦✉❜❧❡ s❤✐❡❧❞❡❞ ♣✉❜❧✐❝ ❦❡② ❝r②♣t♦s②st❡♠s✱ ❈r②♣t♦❧♦❣② ❡Pr✐♥t ❆r❝❤✐✈❡✿ ❘❡♣♦rt ✷✵✶✹✴✺✺✽❪✳

❲❛♥❣ ❡t ❛❧✳ ♣r♦t♦❝♦❧

❲❛♥❣ ❡t ❛❧✳ ♣r♦t♦❝♦❧✿

P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G✱ ❡❧❡♠❡♥t h ∈ G, ❛♥❞ t✇♦ s✉❜❣r♦✉♣s A = ❣♣(a1, ..., an), B = ❣♣(b1, ..., bm) ♦❢ G✱ s✉❝❤ t❤❛t [A, B] = 1. ❆❧✐❝❡ ❝❤♦♦s❡s ❢♦✉r ❡❧❡♠❡♥ts c1, c2, d1, d2 ∈ A, ❝♦♠♣✉t❡s x = d1c1hc2d2, ❛♥❞ t❤❡♥ s❡♥❞s x t♦ ❇♦❜✳ ❇♦❜ ❝❤♦♦s❡s s✐① ❡❧❡♠❡♥ts f1, f2, g1, g2, g3, g4 ∈ B✱ ❝♦♠♣✉t❡s y = g1f1hf2g2 ❛♥❞ w = g3f1xf2g4✱ ❛♥❞ t❤❡♥ s❡♥❞s (y, w) t♦ ❆❧✐❝❡✳ ❆❧✐❝❡ ❝❤♦♦s❡s t✇♦ ❡❧❡♠❡♥ts d3, d4 ∈ A✱ ❝♦♠♣✉t❡s z = d3c1yc2d4 ❛♥❞ u = d−1

1 wd−1 2 ✱ ❛♥❞ t❤❡♥ s❡♥❞s (z, u) t♦

❇♦❜✳ ❇♦❜ s❡♥❞s v = g−1

1 zg−1 2

t♦ ❆❧✐❝❡✳ ❆❧✐❝❡ ❝♦♠♣✉t❡s KA = d−1

3 vd−1 4 ✳

❇♦❜ ❝♦♠♣✉t❡s KB = g−1

3 ug−1 4

= c1f1hf2c2 ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ KA ❛♥❞ t❤❡♥ K = KA = KB ✐s ❆❧✐❝❡ ❛♥❞ ❇♦❜✬s ❝♦♠♠♦♥ s❡❝r❡t ❦❡②✳

❆ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❲❛♥❣ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧

◆♦✇ ✇❡ s❤♦✇ ❤♦✇ t❤❡ ❝♦♠♠♦♥ s❡❝r❡t ❦❡② ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✳ ▲❡t BzB ❜❡ s✉❜s♣❛❝❡ ♦❢ V ❣❡♥❡r❛t❡❞ ❜② ❛❧❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ fzg ✇❤❡r❡ f, g ∈ B. ❲❡ ❝❛♥ ❝♦♥str✉❝t ❛ ❜❛s✐s {eizli : ei, li ∈ B} ♦❢ ✐t ✐♥ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❙✐♥❝❡ v ∈ BzB, ✇❡ ❝❛♥ ✇r✐t❡ ✐t ✐♥ t❤❡ ❢♦r♠ v =

r

• i=1

αieizli, αi ∈ F. ✭✶✮ ❆❧s♦ ✇❡ ❝♦♥str✉❝t ❜❛s❡s {e′

jhl′ j : e′ j, l′ j ∈ B} ❛♥❞ {e′′ kwl′′ k : e′′ k, l′′ k ∈ B}

♦❢ BwB. ❚❤❡♥ y =

s

• j=1

βje′

jhl′ j , βj ∈ F,

✭✷✮ x =

q

• k=1

γke′′

kwl′′ k , γk ∈ F.

✭✸✮

❆ ❝r②♣t❛♥❛❧②s✐s ♦❢ ❲❛♥❣ ❡t✳ ❛❧✳ ♣r♦t♦❝♦❧✿ r❡✈❡❛❧✐♥❣ ♦❢ ❛ s❡❝r❡t

◆♦✇ ✇❡ s✇❛♣ w ❜② u ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ✭✸✮✱ ❛♥❞ ♦❜t❛✐♥

q

• k=1

γke′′

kul′′ k = q

• k=1

γke′′

kd−1 1 wd−1 2 l′′ k =

d−1

1 ( q

• k=1

γke′′

kwl′′ k )d−1 2

= d−1

1 xd−1 2

= c1hc2. ❚❤❡♥ ✇❡ s✇❛♣ h ❜② c1hc2 ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ✭✷✮✳

s

• j=1

βjf ′′

j c1hc2g′′ j = c1( s

• j=1

βje′

jhl′ j )c2 = c1yc2 = c1g1f1hf2g2c2.

❆t ❧❛st ✇❡ s✇❛♣ z ❜② c1g1f1hf2g2c2 ✐♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ♦❢ ✭✶✮ ❛♥❞ ❣❡t

r

• i=1

αieic1g1f1hf2g2c2li = d−1

3 ( r

• i=1

αieizli)d−1

4

= c1f1hf2c2 = K.

❆❧❣❡❜r❛✐❝ ✈❡rs✐♦♥s✿ ❣❡♥❡r❛❧ s❝❤❡♠❡

Ïðåäëàãàåòñÿ îáùàÿ ñõåìà✳ Ïóñòü G ✕ íåêîòîðàÿ àëãåáðàè÷åñêàÿ ñèñòåìà ✭ãðóïïà✱ êîëüöî✱ ëóïà è ò✳ï✳✮✳ ➶ûäåëÿþòñÿ äâå êîíå÷íî ïîðîæä➻ííûå ïîëóãðóïïû îïåðàòîðîâ A è B✱ äåéñòâóþùèõ íà G. ×àñòî òðåáóåòñÿ✱ ÷òîáû ëþáîé îïåðàòîð α ∈ A áûë ïåðåñòàíîâî÷åí ñ ëþáûì îïåðàòîðîì β ∈ B. ➶ õîäå ðàáîòû ïðîòîêîëà ïóáëèêóþòñÿ äàííûå îòíîñèòåëüíî äåéñòâèÿ îïåðàòîðîâ íà ýëåìåíòû G. ✃îððåñïîíäåíòû ➚ëèñà è ➪îá íà îñíîâå ýòèõ äàííûõ è âûáðàííûõ èìè ñàìèìè ñåêðåòíûõ îïåðàòîðîâ ìîãóò âîññòàíîâèòü êàêîé✲òî ýëåìåíò èç

• G. ✃ðèïòîñòîéêîñòü àëãîðèòìà çàêëþ÷àåòñÿ â êîíå÷íîì èòîãå

îò íåâîçìîæíîñòè ðåàëüíî íàéòè ýòîò ðåçóëüòàò ïîñòîðîííåìó íàáëþäàòåëþ✱ íå âëàäåþùåìó ñåêðåòàìè✳

✃ðèïòîàíàëèç

❒åòîä ëèíåéíîãî ðàçëîæåíèÿ è îñíîâàííàÿ íà í➻ì àòàêà ïîçâîëÿþò ïðè óñëîâèè✱ ÷òî G âëîæåíà êàêèì✲òî ýôôåêòèâíûì îáðàçîì â êîíå÷íî ìåðíîå ëèíåéíîå ïðîñòðàíñòâî V, à îïåðàòîðû åñòåñòâåííî ïðîäîëæàþòñÿ äî ýíäîìîðôèçìîâ V, ðàñêðûâàòü ðåçóëüòàò✱ íå ðåøàÿ ñîîòâåòñòâóþùèõ àëãîðèòìè÷åñêèõ çàäà÷ ïîèñêà èñïîëüçîâàííûõ êëþ÷åé✳

❆ ♥❡✇ ❢✉♥❝t✐♦♥ ♦♥ ❣r♦✉♣s

❉❡✜♥✐t✐♦♥ ▲❡t K ❜❡ ❛ ❝❧❛ss ♦❢ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s✳ ❋✉♥❝t✐♦♥ ρ : K → N ∪ {∞} ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳ ▲❡t G ∈ K ❜❡ ❛ ❣r♦✉♣ ✐♥ K ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ✜①❡❞ ✜♥✐t❡ ❣❡♥❡r❛t✐♥❣ s❡t X. ❚❤❡♥✿ ❋♦r ❡❛❝❤ g ∈ G ✇❡ s❡t ρX(g) = l, ✇❤❡r❡ l ✐s t❤❡ ♠✐♥✐♠❛❧ ♥✉♠❜❡r s✉❝❤ t❤❛t t❤❡ ♥♦r♠❛❧ ❝❧♦s✉r❡ ncl(g) ♦❢ g ✐s ❣❡♥❡r❛t❡❞ ❜② ❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ gf, ✇❤❡r❡ t❤❡ ✇♦r❞ ❧❡♥❣t❤ ♦❢ f ✇✳r✳t✳ X ✐s ≤ l. ❲❡ s❡t ρX(G) = max{ρX(g) : g ∈ G}. ❚❤❡ ❢✉♥❝t✐♦♥ ρ(G) ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s ρX(G) ❢♦r ❛❧❧ ♣♦ss✐❜❧❡ X.

◆✐❧♣♦t❡♥t ❝❛s❡

■t ❝❧❡❛r t❤❛t ρ(A) = 0 ♦♥ ❡✈❡r② ❛❜❡❧✐❛♥ ❣r♦✉♣ A✳ ❚❤❡♦r❡♠ ▲❡t G ❜❡ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♥✐❧♣♦t❡♥t ❣r♦✉♣ ♦❢ ❝❧❛ss c✳ ❚❤❡♥ ρ(G) ≤ c − 1. Pr♦❜❧❡♠

• ✐✈❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦❢ ρ(G) ❢♦r ❛♥ ❛r❜✐tr❛r② ♣♦❧②❝②❝❧✐❝ ❣r♦✉♣ G.

❆♥ ❡①❛♠♣❧❡ ♦❢ ❦❡② tr❛♥s♠✐tt❡❞ s❝❤❡♠❡

▲❡t G ❜❡ ❛♥ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ❛♥❞ g ❜❡ ❛ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥t ♦❢ G ▲❡t A ❛♥❞ B ❜❡ t✇♦ ❡❧❡♠❡♥t✇✐s❡ ♣❡r♠✉t❛❜❧❡ s✉❜s❡♠✐❣r♦✉♣s ♦❢ ❊♥❞G. ❆❧✐❝❡ ❝❤♦♦s❡s α ∈ A ❛♥❞ ♣✉❜❧✐❝ α(g). ❇♦❜ ❝❤♦♦s❡s β ∈ B ❛♥❞ ♣✉❜❧✐❝s β(g). ❚❤❡ tr❛♥s♠✐tt❡❞ ❦❡②✿ K = α(β(g)) = β(α(g)).

❆ ♥♦♥ ❧✐♥❡❛r ❛tt❛❝❦

❲❡ ❝♦♠♣✉t❡ ❛ s❡t X ♦❢ ❣❡♥❡r❛t✐♥❣ ❡❧❡♠❡♥ts ♦❢ ❛ s✉❜s②st❡♠✱ ❣❡♥❡r❛t❡❞ ❜② ❛❧❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ λ(g), λ ∈ A. ■t ✐s ♦✛✲❧✐♥❡ ♣r♦❝❡❞✉r❡✳ ▲❡t X = {λi(g) : i = 1, ..., t}. ❚❤❡♥ ✇❡ ✇r✐t❡ α(g) ❛s ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❢♦r♠ α(g) = w(λ1(g), ..., λt(g)), ✇❤❡r❡ w ✐s ❛ t❡r♠✳ ❚❤❡♥ ✇❡ s✇❛♣ g ✇✐t❤ β(g)✱ t❤❡♥ ✇❡ ❣❡t w(λ1(β(g)), ..., λt(β(g)) = β(w(λ1(g), ..., λt(g)) = β(α(g)) = K.

❈r②♣t♦ ●❛❧❧❡r② ❈r②♣t♦ ●❛❧❧❡r②

❉✐✣❡

❉✐✣❡

❑❛❤r♦❜❛❡✐ ❛♥❞ ❇❛✉♠s❧❛❣

❉❡❧❛r❛♠ ❛♥❞ ●✐❧❜❡rt

▼②❛s♥✐❦♦✈

❆❧❡①❡✐

❙❤♣✐❧r❛✐♥

❱❧❛❞✐♠✐r

❯s❤❛❦♦✈

❙❛s❤❛

❘♦♠❛♥✬❦♦✈ ❛♥❞ ❏❛r❦♦✈❛

❱✐t❛❧② ❛♥❞ ◆❛st②❛

❚❤❡ ❡♥❞

ÑÏ➚Ñ➮➪❰✦ ❚❍❆◆❑ ❨❖❯✦ ❊❙❚❖❨ ❆●❘❆❉❊❈■❉❖✦