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

❊❧●❛♠❛❧ ❊❧●❛♠❛❧

❊❧●❛♠❛❧ ✖ ▼❛ss❡②✲❖♠✉r❛✮ ❚❤❡ ❊❧●❛♠❛❧ ✭✶✾✽✵✮✕▼❛ss❡②✲❖♠✉r❛ ✭✶✾✽✷✮ ❝r②♣t♦s②st❡♠ ❢♦r ♠❡ss❛❣❡ ✭❦❡②✮ tr❛♥s♠✐ss✐♦♥ Pr✐✈❛t❡ ❞❛t❛ ✭❦❡②✮✿ ④ g ∈ G ⑥✳ P✉❜❧✐❝ ❞❛t❛✿ ❣r♦✉♣ G ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r r s✉❝❤ t❤❛t g r = 1 ✳ ◆✉♠❜❡r r ♠❛② ❜❡ ❣✐✈❡♥ ❛s t❤❡ ♦r❞❡r | G | ♦❢ G ✱ ♦r | g | ♦❢ g . ❆❧✐❝❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ k ∈ Z , ( k , r ) = 1 ✱ t❤❡♥ ♣✉❜❧✐❝s g k ✳ ❇♦❜ ❝❤♦♦s❡s l ∈ Z , ( l , r ) = 1 ✱ t❤❡♥ ❝♦♠♣✉t❡s ❛♥❞ ♣✉❜❧✐❝s ( g k ) l = g kl ✳ ❆❧✐❝❡ ❝♦♠♣✉t❡s k − 1 ( mod r ) ❛♥❞ t❤❡♥ ♣✉❜❧✐❝s ( g kl ) k − 1 = g l . ❚❤❡ tr❛♥s♠✐tt❡❞ ❦❡②✿ ❇♦❜ ❝♦♠♣✉t❡s l − 1 ( mod r ) , t❤❡♥ ❤❡ r❡❝♦✈❡rs t❤❡ tr❛♥s♠✐tt❡❞ ❦❡②✿ ( g l ) l − 1 = g .

❊❧●❛♠❛❧ ❊❧●❛♠❛❧ ✭✶✾✽✺✮ ❝r②♣t♦s②st❡♠ ❢♦r ♠❡ss❛❣❡ ✭❦❡②✮ tr❛♥s♠✐ss✐♦♥ ❆❧✐❝❡ s❡ts ♣✉❜❧✐❝ ❞❛t❛✿ ④ G ✕ ❣r♦✉♣✱ g ∈ G ⑥✳ ❆❧s♦ s❤❡ s❡ts ♣r✐✈❛t❡ ❞❛t❛ ✭❦❡②✮✿ 0 < a < | g | , ❛♥❞ ♦t❤❡r ♣✉❜❧✐❝ ❞❛t❛ ✭❡♥❝②♣❡r✐♥❣ ❦❡②✮✿ g a . ❇♦❜ ✇❛♥ts t♦ s❡♥❞ ❛ ♠❡ss❛❣❡ m ∈ G t♦ ❆❧✐❝❡✳ ❍❡ ❝❤♦♦s❡s ❛ ♣r✐✈❛t❡ k ∈ Z , 0 < k < | g | ✱ t❤❡♥ ♣✉❜❧✐❝s ( g k , mg ak ) ✳ ❚❤❡ tr❛♥s♠✐tt❡❞ ♠❡ss❛❣❡✿ ❆❧✐❝❡ ❝♦♠♣✉t❡s g ak = ( g k ) a ❛♥❞ ( g ak ) − 1 , t❤❡♥ r❡❝♦✈❡rs t❤❡ ♠❡ss❛❣❡ m = ( mg ak )( g ak ) − 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 ∗ p r ✕ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❣r♦✉♣ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ F p r ✱ ♦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♦❜❧❡♠ ✐♥ GL n ( F q ) , q = p r . g ∈ GL n ( F q ) , h = g k , k = log g h . ❋✐♥❞ t❤❡ ❏♦r❞❛♥ ❢♦r♠✿ J ( g ) = tgt − 1 . J ( g ) = J r 1 ( λ 1 ) ⊕ ... ⊕ J r t ( λ t ) , � t i = 1 r i = n . λ 1 , ..., λ t ❛r❡ r♦♦ts ✭✐♥ ❡①t❡♥s✐♦♥s F q ni ♦❢ F q ✮ ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ p g ( x ) = | g − λ E | = ( x − λ 1 ) r 1 ... ( x − λ t ) r t = 0 . ❍❡r❡   λ 1 0 ... 0 0 λ 1 ... 0   J s ( λ ) =   . . . . .   ... λ 0 0 0 ✐s ❏♦r❞❛♥ ❜❧♦❝❦ ♦❢ s✐③❡ s .

❊✛❡❝t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❏♦r❞❛♥ ❢♦r♠ ❊✛❡❝t✐✈❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ J ( g ) ✿ ■♥♣✉t✿ g ∈ GL n ( F q ) ✳ ❖✉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 ( n 3 ) ✮ ✇❡ ✜♥❞ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ♣♦❧②♥♦♠✐❛❧ p g ( x ) . ✷ ❇② t❤❡ ♣r♦❜❛❜✐❧✐st✐❝ ♣♦❧②♥♦♠✐❛❧ ❇❡♥✲❖r✬s ❛❧❣♦r✐t❤♠ ✇❡ ❣❡t ❛ ♣r❡s❡♥t❛t✐♦♥ p g ( x ) = f e 1 1 ... f e s s , ✇❤❡r❡ f i ✐s ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ♦✈❡r F q ♦❢ ❞❡❣r❡❡ n i . ✸ F q ni = F q [ x ] / ideal ( f i ( x )) ✳ ❲❡ ✜♥❞ r♦♦ts α ij , 1 ≤ j ≤ n i , ♦❢ f i ✐♥ F q ni ✳ ◆❛♠❡❧②✱ α i 1 = x , α ij = x q i − 1 mod ( f i ( x )) , 2 ≤ j ≤ n i . ✹ ❲❡ ✜♥❞ s✐③❡s ♦❢ ❏♦r❞❛♥ ❜❧♦❝❦s J l , ❛♥❞ t❤❡♥ ✇❡ ❣❡t t❤❡ ❏♦r❞❛♥ ❢♦r♠ J ( g ) = J 1 ⊕ ... ⊕ J t .

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❡ n i ≤ n , ✇❡ ✉s❡ ✐♥ ❡❛❝❤ ♦❢ s ≤ n ✐t❡r❛t✐♦♥s ♦♥ t❤❡ st❡♣ ✸ log q n i ≤ 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 = g k , h , g ∈ GL n ( F q ) . ❖✉t♣✉t✿ k ∈ Z . ✶ ❋✐♥❞ t s✉❝❤ t❤❛t tgt − 1 = J ( g ) . ✷ ❈♦♠♣✉t❡ tht − 1 = ( tgt − 1 ) k . ✸ α k ij = β ij , ✇❤❡r❡ β ij ❛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✐❛✱ ✶✾✾✼✱ ✹✼✱ ✷✸✲✸✷✳