HAL Id: inria-00419094
https://hal.inria.fr/inria-00419094
Submitted on 22 Sep 2009
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
A Software Implementation of ECM for NFS
Alexander Kruppa
To cite this version:
Alexander Kruppa. A Software Implementation of ECM for NFS. [Research Report] RR-7041, INRIA.
2009. <inria-00419094>
a p p o r t
d e r e c h e r c h e
ISRN
INRIA/RR--7041--FR+ENG
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
A Software Implementation of ECM for NFS
Alexander Kruppa
N° 7041
Centre de recherche INRIA Nancy – Grand Est
❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙
❆❧❡①❛♥❞❡r ❑r✉♣♣❛
❚❤è♠❡ ✿ ❆❧❣♦r✐t❤♠✐q✉❡✱ ❝❛❧❝✉❧ ❝❡rt✐✜é ❡t ❝r②♣t♦❣r❛♣❤✐❡ ➱q✉✐♣❡✲Pr♦❥❡t ❈❆❈❆❖ ❘❛♣♣♦rt ❞❡ r❡❝❤❡r❝❤❡ ♥➦ ✼✵✹✶ ✖ ❙❡♣t❡♠❜r❡ ✷✵✵✾ ✖ ✸✾ ♣❛❣❡s ❆❜str❛❝t✿ ❚❤❡ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ✭❊❈▼✮ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ r❡❧❛t✐♦♥ ❝♦❧❧❡❝t✐♦♥ ♣❤❛s❡ ♦❢ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ✭◆❋❙✮ t♦ ❤❡❧♣ ✐❞❡♥t✐❢② s♠♦♦t❤ ✐♥t❡❣❡rs✳ ❚❤✐s r❡q✉✐r❡s r❛♣✐❞❧② ✜♥❞✐♥❣ s♠❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ❢♦r ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣♦s✐t❡s✱ ❡❛❝❤ ♦❢ ❛ ❢❡✇ ♠❛❝❤✐♥❡ ✇♦r❞s ✐♥ s✐③❡✳ ❲❡ ♣r❡s❡♥t ❛ s♦❢t✲ ✇❛r❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ t❤❛t ✐s ♦♣t✐♠✐③❡❞ ❢♦r ❤✐❣❤ t❤r♦✉❣❤♣✉t ♦♣❡r❛t✐♦♥ ❛♥❞ ❝♦♠♣❛r❡ ✐t ✇✐t❤ r❡❝❡♥t❧② ♣r♦♣♦s❡❞ ❤❛r❞✇❛r❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ ❊❈▼✳ ❑❡②✲✇♦r❞s✿ ■♥t❡❣❡r ❢❛❝t♦r✐♥❣✱ ❊❧❧✐♣t✐❝ ❈✉r✈❡s✱ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡❯♥❡ ✐♠♣❧é♠❡♥t❛t✐♦♥ ❞✬❊❈▼ ♣♦✉r ❧❡ ❝r✐❜❧❡
❛❧❣é❜r✐q✉❡
❘és✉♠é ✿ ▲✬❛❧❣♦r✐t❤♠❡ ❊❈▼ ❞❡ ❢❛❝t♦r✐s❛t✐♦♥ ❞✬❡♥t✐❡r ❡st ✉t✐❧✐sé ❞❛♥s ❧❡ ❝r✐❜❧❡ ❛❧❣é❜r✐q✉❡ ✭◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡✱ ◆❋❙✮ ♣♦✉r ✐❞❡♥t✐✜❡r ❧❡s r❡❧❛t✐♦♥s ❢r✐❛❜❧❡s✳ ❈❡❧❛ ♥é❝❡ss✐t❡ ❞❡ tr♦✉✈❡r r❛♣✐❞❡♠❡♥t ❞❡s ♣❡t✐ts ❢❛❝t❡✉rs ♣r❡♠✐❡rs ❞❛♥s ✉♥ ❣r❛♥❞ ♥♦♠❜r❡ ❞✬❡♥t✐❡rs✱ ❝❤❛❝✉♥ ❢❛✐s❛♥t q✉❡❧q✉❡s ♠♦ts✲♠❛❝❤✐♥❡✳ ◆♦✉s ♣rés❡♥t♦♥s ✉♥❡ ✐♠♣❧é♠❡♥t❛t✐♦♥ ❧♦❣✐❝✐❡❧❧❡ ❞✬❊❈▼ q✉✐ ❡st ♦♣t✐♠✐sé❡ ♣♦✉r ❝❡ ❝❛s ♣ré❝✐s ❡t ❧❛ ❝♦♠✲ ♣❛r♦♥s á ❞❡s ✐♠♣❧é♠❡♥t❛t✐♦♥s ré❝❡♥t❡s ❡♥ ♠❛tér✐❡❧✳ ▼♦ts✲❝❧és ✿ ❢❛❝t♦r✐s❛t✐♦♥ ❞❡s ❡♥t✐❡rs✱ ❝♦✉r❜❡s ❡❧❧✐♣t✐q✉❡s✱ ❝r✐❜❧❡ ❛❧❣é❜r✐q✉❡❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✸
✶ ■♥tr♦❞✉❝t✐♦♥
❚❤❡ s✐❡✈✐♥❣ st❡♣ ♦❢ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡ ❬✶✺❪ ✐❞❡♥t✐✜❡s ✐♥t❡❣❡r ♣❛✐rs (a, b) ✇✐t❤ a ⊥ b s✉❝❤ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ t✇♦ ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s Fi(a, b), i ∈ {1, 2}, ❛r❡ ❜♦t❤ s♠♦♦t❤✱ ✇❤❡r❡ t❤❡ s✐❡✈✐♥❣ ♣❛r❛♠❡t❡rs ❛r❡ ❝❤♦s❡♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s♠♦♦t❤♥❡ss ❝r✐t❡r✐♦♥✳ ❚②♣✐❝❛❧❧② t❤❡ t✇♦ ♣♦❧②♥♦♠✐❛❧s ❡❛❝❤ ❤❛✈❡ ❛ ✏❢❛❝t♦r ❜❛s❡ ❜♦✉♥❞✑ Bi, ❛ ✏❧❛r❣❡ ♣r✐♠❡ ❜♦✉♥❞✑ Li, ❛♥❞ ❛ ♣❡r♠✐ss✐❜❧❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❧❛r❣❡ ♣r✐♠❡s ki❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡♠✱ s♦ t❤❛t Fi(a, b)✐s ❝♦♥s✐❞❡r❡❞ s♠♦♦t❤ ✐❢ ✐t ❝♦♥t❛✐♥s ♦♥❧② ♣r✐♠❡ ❢❛❝t♦rs ✉♣ t♦ Bi ❡①❝❡♣t ❢♦r ✉♣ t♦ ki ♣r✐♠❡ ❢❛❝t♦rs ❣r❡❛t❡r t❤❛♥ Bi✱ ❜✉t ♥♦♥❡ ❡①❝❡❡❞✐♥❣ Li✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ ♦❢ t❤❡ ❘❙❆✲ ✶✺✺ ❝❤❛❧❧❡♥❣❡ ♥✉♠❜❡r ❬✺❪ ✭❛ ❤❛r❞ ✐♥t❡❣❡r ♦❢ 512✲❜✐t✮ t❤❡ ✈❛❧✉❡s B = 224✱ L = 109 ❛♥❞ k = 2 ✇❡r❡ ✉s❡❞ ❢♦r ❜♦t❤ ♣♦❧②♥♦♠✐❛❧s✳ ❑❧❡✐♥❥✉♥❣ ❬✶✹❪ ❣✐✈❡s ❛♥ ❡st✐♠❛t❡ ❢♦r t❤❡ ❝♦st ♦❢ ❢❛❝t♦r✐♥❣ ❛ 1024✲❜✐t ❘❙❆ ❦❡② ❜❛s❡❞ ♦♥ t❤❡ ♣❛r❛♠❡t❡rs B1= 1.1·109, B2= 3 · 108,❛♥❞ L1= L2= 242✇✐t❤ k1= 5❛♥❞ k2= 4. ❚❤❡ ❝♦♥tr✐❜✉t✐♦♥ ♦❢ t❤❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s t♦ ❡❛❝❤ ♣♦❧②♥♦♠✐❛❧ ✈❛❧✉❡ Fi(a, b) ❢♦r ❛ s❡t ♦❢ (a, b) ♣❛✐rs ✐s ❛♣♣r♦①✐♠❛t❡❞ ✇✐t❤ ❛ s✐❡✈✐♥❣ ♣r♦❝❡❞✉r❡✱ ✇❤✐❝❤ ❡st✐♠❛t❡s r♦✉❣❤❧② ✇❤❛t t❤❡ s✐③❡ ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✈❛❧✉❡s ✇✐❧❧ ❜❡ ❛❢t❡r ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s ❤❛✈❡ ❜❡❡♥ ❞✐✈✐❞❡❞ ♦✉t✳ ■❢ t❤❡s❡ ❡st✐♠❛t❡s ❢♦r ❛ ♣❛rt✐❝✉❧❛r (a, b) ♣❛✐r ❛r❡ s♠❛❧❧ ❡♥♦✉❣❤ t❤❛t ❜♦t❤ Fi(a, b) ✈❛❧✉❡s ♠✐❣❤t ❜❡ s♠♦♦t❤✱ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✈❛❧✉❡s ❛r❡ ❝♦♠♣✉t❡❞✱ t❤❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s ❛r❡ ❞✐✈✐❞❡❞ ♦✉t✱ ❛♥❞ t❤❡ t✇♦ ❝♦❢❛❝t♦rs ci ❛r❡ t❡st❡❞ t♦ s❡❡ ✐❢ t❤❡② s❛t✐s❢② t❤❡ s♠♦♦t❤♥❡ss ❝r✐t❡r✐♦♥✳ ■❢ ♦♥❧② ♦♥❡ ❧❛r❣❡ ♣r✐♠❡ ✐s ♣❡r♠✐tt❡❞✱ ♥♦ ❢❛❝t♦r✐♥❣ ♥❡❡❞s t♦ ❜❡ ❝❛rr✐❡❞ ♦✉t ❛t ❛❧❧ ❢♦r t❤❡ ❧❛r❣❡ ♣r✐♠❡s✿ ✐❢ ci > Li ❢♦r ❡✐t❤❡r i, t❤✐s (a, b) ♣❛✐r ✐s ❞✐s❝❛r❞❡❞✳ ❙✐♥❝❡ ❣❡♥❡r❛❧❧② Li < B2i ❛♥❞ ❛❧❧ ♣r✐♠❡ ❢❛❝t♦rs ❜❡❧♦✇ Bi ❤❛✈❡ ❜❡❡♥ r❡♠♦✈❡❞✱ ❛ ❝♦❢❛❝t♦r ci≤ Li ✐s ♥❡❝❡ss❛r✐❧② ♣r✐♠❡ ❛♥❞ ♥❡❡❞ ♥♦t ❜❡ ❢❛❝t♦r❡❞✳ ■❢ ✉♣ t♦ t✇♦ ❧❛r❣❡ ♣r✐♠❡s ❛r❡ ♣❡r♠✐tt❡❞✱ ❛♥❞ t❤❡ ❝♦❢❛❝t♦r ci✐s ❝♦♠♣♦s✐t❡ ❛♥❞ t❤❡r❡❢♦r❡ ❣r❡❛t❡r t❤❛♥ t❤❡ ❧❛r❣❡ ♣r✐♠❡ ❜♦✉♥❞ ❜✉t ❜❡❧♦✇ L2 i ✭♦r ❛ s✉✐t❛❜❧② ❝❤♦s❡♥ t❤r❡s❤♦❧❞ s♦♠❡✇❤❛t ❧❡ss t❤❡♥ L2 i✮✱ ✐t ✐s ❢❛❝t♦r❡❞✳ ❙✐♥❝❡ t❤❡ ♣r✐♠❡ ❢❛❝t♦rs ✐♥ ci ❛r❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② Bi, ❛♥❞ Li ✐s t②♣✐❝❛❧❧② ❧❡ss t❤❛♥ B1.5i ✱ t❤❡ ❢❛❝t♦rs ❝❛♥ ❜❡ ❡①♣❡❝t❡❞ ♥♦t t♦ ❜❡ ✈❡r② ♠✉❝❤ s♠❛❧❧❡r t❤❛♥ t❤❡ sq✉❛r❡ r♦♦t ♦❢ t❤❡ ❝♦♠♣♦s✐t❡ ♥✉♠❜❡r✳ ❚❤✐s ✇❛② t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ s♣❡❝✐❛❧ ♣✉r♣♦s❡ ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ✇❤❡♥ s♠❛❧❧ ❞✐✈✐s♦rs ✭❝♦♠♣❛r❡❞ t♦ t❤❡ ❝♦♠♣♦s✐t❡ s✐③❡✮ ❛r❡ ♣r❡s❡♥t ❞♦❡s ♥♦t ❝♦♠❡ ✐♥t♦ ❣r❡❛t ❡✛❡❝t✱ ❛♥❞ ❣❡♥❡r❛❧ ♣✉r♣♦s❡ ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ❧✐❦❡ ❙◗❯❋❖❋ ♦r ▼P◗❙ ♣❡r❢♦r♠ ✇❡❧❧✳ ■♥ ♣r❡✈✐♦✉s ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ ◗❙ ❛♥❞ ◆❋❙✱ ✈❛r✐♦✉s ❛❧❣♦r✐t❤♠s ❢♦r ❢❛❝t♦r✐♥❣ ❝♦♠♣♦s✐t❡s ♦❢ t✇♦ ♣r✐♠❡ ❢❛❝t♦rs ❤❛✈❡ ❜❡❡♥ ✉s❡❞✱ ✐♥❝❧✉❞✐♥❣ ❙◗❯❋❖❋ ❛♥❞ P♦❧❧❛r❞✲❘❤♦ ✐♥ ❬✾✱ ❝❤❛♣t❡r ✸✳✻❪✱ ❛♥❞ P✕✶✱ ❙◗❯❋❖❋✱ ❛♥❞ P♦❧❧❛r❞✲❘❤♦ ✐♥ ❬✹✱ ➓✸❪✳ ■❢ ♠♦r❡ t❤❛♥ t✇♦ ❧❛r❣❡ ♣r✐♠❡s ❛r❡ ❛❧❧♦✇❡❞✱ t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ s♣❡❝✐❛❧ ♣✉r♣♦s❡ ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ♣❛②s ♦✛✳ ●✐✈❡♥ ❛ ❝♦♠♣♦s✐t❡ ❝♦❢❛❝t♦r ci > L2i✱ ✇❡ ❦♥♦✇ t❤❛t ✐t ❝❛♥ ❜❡ s♠♦♦t❤ ♦♥❧② ✐❢ ✐t ❤❛s ❛t ❧❡❛st t❤r❡❡ ♣r✐♠❡ ❢❛❝t♦rs✱ ♦❢ ✇❤✐❝❤ ❛t ❧❡❛st ♦♥❡ ♠✉st ❜❡ ❧❡ss t❤❛♥ c1/3 i . ■❢ ✐t ❤❛s ♥♦ s✉❝❤ s♠❛❧❧ ❢❛❝t♦r✱ t❤❡ ❝♦❢❛❝t♦r ✐s ♥♦t s♠♦♦t❤✱ ❛♥❞ ✐ts ❢❛❝t♦r✐③❛t✐♦♥ ✐s ♥♦t ❛❝t✉❛❧❧② r❡q✉✐r❡❞✱ ❛s t❤✐s (a, b) ♣❛✐r ✇✐❧❧ ❜❡ ❞✐s❝❛r❞❡❞✳ ❍❡♥❝❡ ❛♥ ❡❛r❧②✲❛❜♦rt str❛t❡❣② ❝❛♥ ❜❡ ❡♠♣❧♦②❡❞ t❤❛t ✉s❡s s♣❡❝✐❛❧✲♣✉r♣♦s❡ ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ✉♥t✐❧ ❡✐t❤❡r ❛ ❢❛❝t♦r ✐s ❢♦✉♥❞ ❛♥❞ t❤❡ ♥❡✇ ❝♦❢❛❝t♦r ❝❛♥ ❜❡ t❡st❡❞ ❢♦r s♠♦♦t❤♥❡ss✱ ♦r ❛❢t❡r ❛ ♥✉♠❜❡r ♦❢ ❢❛❝t♦r✐♥❣ ❛tt❡♠♣ts ❤❛✈❡ ❢❛✐❧❡❞✱ t❤❡ ❝♦❢❛❝t♦r ♠❛② ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ♥♦t s♠♦♦t❤ ✇✐t❤ ❤✐❣❤ ♣r♦❜❛❜✐❧✐t② s♦ t❤❛t t❤✐s (a, b) ♣❛✐r ❝❛♥ ❜❡ ❞✐s❝❛r❞❡❞✳ ❙✉✐t❛❜❧❡ ❝❛♥❞✐❞❛t❡s ❢♦r ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ❢♦r t❤✐s ♣✉r♣♦s❡ ❛r❡ t❤❡ P✕✶ ♠❡t❤♦❞✱ t❤❡ P✰✶ ♠❡t❤♦❞✱ ❛♥❞ t❤❡ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ✭❊❈▼✮✳ ❆❧❧ ❤❛✈❡ ✐♥ ❝♦♠♠♦♥ t❤❛t ❛ ♣r✐♠❡ ❢❛❝t♦r p ✐s ❢♦✉♥❞ ✐❢ t❤❡ ♦r❞❡r ♦❢ s♦♠❡ ❣r♦✉♣ ❞❡✜♥❡❞ ♦✈❡r✹ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛ Fp✐s ✐ts❡❧❢ s♠♦♦t❤✳ ❆ ❜❡♥❡✜❝✐❛❧ ♣r♦♣❡rt② ✐s t❤❛t ❢♦r ❊❈▼✱ ❛♥❞ t♦ ❛ ❧❡ss❡r ❡①t❡♥t ❢♦r P✰✶✱ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❝❤♦s❡♥ s♦ t❤❛t t❤❡ ❣r♦✉♣ ♦r❞❡r ❤❛s ❦♥♦✇♥ s♠❛❧❧ ❢❛❝t♦rs✱ ♠❛❦✐♥❣ ✐t ♠♦r❡ ❧✐❦❡❧② s♠♦♦t❤✳ ❚❤✐s ✐s ♣❛rt✐❝✉❧❛r❧② ❡✛❡❝t✐✈❡ ✐❢ t❤❡ ♣r✐♠❡ ❢❛❝t♦r t♦ ❜❡ ❢♦✉♥❞✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ❣r♦✉♣ ♦r❞❡r✱ ✐s s♠❛❧❧✳ ❆❧t❤♦✉❣❤ t❤❡ P✕✶ ❛♥❞ P✰✶ ♠❡t❤♦❞s ❜② t❤❡♠s❡❧✈❡s ❤❛✈❡ ❛ r❡❧❛t✐✈❡❧② ♣♦♦r ❛s②♠♣t♦t✐❝ ❛❧❣❡❜r❛✐❝ ❝♦♠♣❧❡①✐t② ✐♥ O(√p) ✭❛ss✉♠✐♥❣ ❛♥ ❛s②♠♣t♦t✐❝❛❧❧② ❢❛st st❛❣❡ ✷ ❛s ❞❡s❝r✐❜❡❞ ✐♥ ❬✷✸❪ ❢♦r ❡①❛♠♣❧❡✮✱ t❤❡② ✜♥❞ s✉r♣r✐s✐♥❣❧② ♠❛♥② ♣r✐♠❡s ✐♥ ❢❛r ❧❡ss t✐♠❡✱ ♠❛❦✐♥❣ t❤❡♠ ✉s❡❢✉❧ ❛s ❛ ✜rst q✉✐❝❦ tr② t♦ ❡❧✐♠✐♥❛t❡ ❡❛s② ❝❛s❡s ❜❡❢♦r❡ ❊❈▼ ❜❡❣✐♥s✳ ■♥ ❢❛❝t✱ P✕✶ ❛♥❞ P✰✶ ♠❛② ❜❡ ✈✐❡✇❡❞ ❛s ❜❡✐♥❣ ❡q✉✐✈❛❧❡♥t t♦ ❧❡ss ❡①♣❡♥s✐✈❡ ❊❈▼ ❛tt❡♠♣ts ✭❜✉t ❛❧s♦ ❧❡ss ❡✛❡❝t✐✈❡✱ ❞✉❡ t♦ ❢❡✇❡r ❦♥♦✇♥ ❢❛❝t♦rs ✐♥ t❤❡ ❣r♦✉♣ ♦r❞❡r✮✳ ❆♥♦t❤❡r ✇❡❧❧✲❦♥♦✇♥ s♣❡❝✐❛❧✲♣✉r♣♦s❡ ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠ ✐s P♦❧❧❛r❞✬s ✏❘❤♦✑ ♠❡t❤♦❞ ❬✷✺❪ ✇❤✐❝❤ ❧♦♦❦s ❢♦r ❛ ❝♦❧❧✐s✐♦♥ ♠♦❞✉❧♦ p ✐♥ ❛♥ ✐t❡r❛t❡❞ ♣s❡✉❞♦✲r❛♥❞♦♠ ❢✉♥❝t✐♦♥ ♠♦❞✉❧♦ N✱ ✇❤❡r❡ p ✐s ❛ ♣r✐♠❡ ❢❛❝t♦r ♦❢ N ✇❡ ❤♦♣❡ t♦ ✜♥❞✳ ❲❤❡♥ ❝❤♦♦s✐♥❣ ♥♦ ❧❡ss t❤❛♥ p2 log(2)n + 0.28 ✐♥t❡❣❡rs ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ❢r♦♠ [1, n]✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❝❤♦♦s✐♥❣ ❛t ❧❡❛st ♦♥❡ ✐♥t❡❣❡r ♠♦r❡ t❤❛♥ ♦♥❝❡ ✐s ❛t ❧❡❛st 0.5, ✇❡❧❧ ❦♥♦✇♥ ❛s t❤❡ ❇✐rt❤❞❛② P❛r❛❞♦① ✇❤✐❝❤ st❛t❡s t❤❛t ✐♥ ❛ ❣r♦✉♣ ♦❢ ♦♥❧② ✷✸ ♣❡♦♣❧❡✱ t✇♦ s❤❛r❡ ❛ ❜✐rt❤❞❛② ✇✐t❤ ♠♦r❡ t❤❛♥ 50% ♣r♦❜❛❜❧✐❧✐t②✳ ❋♦r t❤❡ ❘❤♦ ♠❡t❤♦❞✱ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s t♦ ✜♥❞ ❛ ♣r✐♠❡ ❢❛❝t♦r p ✐s ✐♥ O √p , ❛♥❞ ✐♥ ❝❛s❡ ♦❢ P♦❧❧❛r❞✬s ♦r✐❣✐♥❛❧ ❛❧❣♦r✐t❤♠✱ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ❢♦r ♣r✐♠❡s p ❛r♦✉♥❞ 230 ✐s ❝❧♦s❡ t♦ 215 ≈ √p, ✇❤❡r❡ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❛❦❡s t❤r❡❡ ♠♦❞✉❧❛r sq✉❛r✐♥❣s ❛♥❞ ❛ ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❢♦r ❛♥ ❛✈❡r❛❣❡ ♦❢ ≈ 130000 ♠♦❞✉❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥s ✇❤❡♥ ❝♦✉♥t✐♥❣ sq✉❛r✐♥❣s ❛s ♠✉❧t✐♣❧✐❝❛t✐♦♥s✳ ❇r❡♥t ❬✷❪ ❣✐✈❡s ❛♥ ✐♠♣r♦✈❡❞ ✐t❡r❛t✐♦♥ ✇❤✐❝❤ r❡❞✉❝❡s t❤❡ ♥✉♠❜❡r ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❜② ❛❜♦✉t 25% ♦♥ ❛✈❡r❛❣❡✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ P✕✶✱ P✰✶✱ ❛♥❞ ❊❈▼ ❞♦❡s ❜❡tt❡r ♦♥ ❛✈❡r❛❣❡✳ ❋✉rt❤❡r♠♦r❡✱ tr②✐♥❣ t❤❡ P♦❧❧❛r❞✲❘❤♦ ♠❡t❤♦❞ ✇✐t❤ ♦♥❧② ❛ ❧♦✇ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ❜❡❢♦r❡ ♠♦✈✐♥❣ ♦♥ t♦ ♦t❤❡r ❢❛❝t♦r✐♥❣ ❛❧❣♦r✐t❤♠s ❤❛s ❛ ♥❡❣❧✐❣✐❜❧❡ ♣r♦❜✲ ❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ✖ ❛♠♦♥❣ t❤❡ 4798396 ♣r✐♠❡s ✐♥ [230, 230+ 108]✱ ♦♥❧② 3483 ❛r❡ ❢♦✉♥❞ ✇✐t❤ ❛t ♠♦st 1000 ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧ P♦❧❧❛r❞✲❘❤♦ ❛❧❣♦r✐t❤♠ ✇✐t❤ ♣s❡✉❞♦✲r❛♥❞♦♠ ♠❛♣ x 7→ x2+ 1❛♥❞ st❛rt✐♥❣ ✈❛❧✉❡ x 0= 2✳ ❋♦r P✕✶✱ t❤❡r❡ ❛r❡ 1087179♣r✐♠❡s p ✐♥ t❤❡ s❛♠❡ r❛♥❣❡ ✇❤❡r❡ t❤❡ ❧❛r❣❡st ♣r✐♠❡ ❢❛❝t♦r ♦❢ p − 1 ❞♦❡s ♥♦t ❡①❝❡❡❞ 1000, ❛♥❞ ❡①♣♦♥❡♥t✐❛t✐♥❣ ❜② t❤❡ ♣r♦❞✉❝t ♦❢ ❛❧❧ ♣r✐♠❡s ❛♥❞ ♣r✐♠❡ ♣♦✇❡rs ✉♣ t♦ B r❡q✉✐r❡s ♦♥❧② B/ log(2)+O√B≈ 1.44B sq✉❛r✐♥❣s✱ ❝♦♠♣❛r❡❞ t♦ 4 ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♣❡r ✐t❡r❛t✐♦♥ ❢♦r t❤❡ ♦r✐❣✐♥❛❧ P♦❧❧❛r❞✲❘❤♦ ❛❧❣♦r✐t❤♠✳ ❇② ✉s✐♥❣ ❛ st❛❣❡ ✷ ❢♦r P✕✶✱ ✐ts ❛❞✈❛♥t❛❣❡ ✐♥❝r❡❛s❡s ❢✉rt❤❡r✳ ❋✐❣✉r❡ ✶ s❤♦✇s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ♣r✐♠❡ ❢❛❝t♦r ♦❢ p − 1 ❛♥❞ t❤❡ r❡q✉✐r❡❞ ♥✉♠❜❡r ♦❢ P♦❧❧❛r❞✲❘❤♦ ✐t❡r❛t✐♦♥s ❢♦r ✜♥❞✐♥❣ p✱ r❡s♣❡❝t✐✈❡❧②✱ ❢♦r ♣r✐♠❡s p ✐♥ [230, 230+ 108]. ❚❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❧❛r❣❡st ♣r✐♠❡ ❢❛❝t♦r ♦❢ p + 1 ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❛t ♦❢ p − 1✱ ✉♣ t♦ st❛t✐st✐❝❛❧ ♥♦✐s❡✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t ✉♥❧✐❦❡ P✕✶ ❛♥❞ P✰✶✱ t❤❡ P♦❧❧❛r❞✲❘❤♦ ♠❡t❤♦❞ ✐s ♥♦t s✉✐t❛❜❧❡ ❢♦r r❡♠♦✈✐♥❣ ✏❡❛s② ♣✐❝❦✐♥❣s✳✑ ❚❤✐s r❡s❡❛r❝❤ r❡♣♦rt ❞❡s❝r✐❜❡s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ tr✐❛❧ ❞✐✈✐s✐♦♥ ❢♦r ❝♦♠✲ ♣♦s✐t❡s ♦❢ ❛ ❢❡✇ ♠❛❝❤✐♥❡ ✇♦r❞s✱ ❛s ✇❡❧❧ ❛s t❤❡ P✕✶✱ P✰✶✱ ❛♥❞ ❊❧❧✐♣t✐❝ ❈✉r✈❡ ▼❡t❤♦❞ ♦❢ ❢❛❝t♦r✐③❛t✐♦♥ ❢♦r s♠❛❧❧ ❝♦♠♣♦s✐t❡s ♦❢ ♦♥❡ ♦r t✇♦ ♠❛❝❤✐♥❡ ✇♦r❞s✱ ❛✐♠❡❞ ❛t ❢❛❝t♦r✐♥❣ ❝♦❢❛❝t♦rs ❛s ♦❝❝✉r ❞✉r✐♥❣ t❤❡ s✐❡✈✐♥❣ ♣❤❛s❡ ♦❢ t❤❡ ◆✉♠❜❡r ❋✐❡❧❞ ❙✐❡✈❡✳
❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✺ 0 5000 10000 15000 20000 25000 30000 35000 40000 0 100 200 300 400 500 600 700 800 900 1000 Number of primes n Pollard rho p-1 0 5000 10000 15000 20000 25000 30000 35000 40000 10 20 30 40 50 60 70 80 90 100 Number of primes n Pollard rho p-1 ❋✐❣✉r❡ ✶✿ ◆✉♠❜❡r ♦❢ ♣r✐♠❡s p ✐♥ 230, 230+ 108 ✇❤❡r❡ t❤❡ ❧❛r❣❡st ♣r✐♠❡ ❢❛❝✲ t♦r ♦❢ p − 1✱ r❡s♣❡❝t✐✈❡❧② t❤❡ ♥✉♠❜❡r ♦❢ P♦❧❧❛r❞✲❘❤♦ ✐t❡r❛t✐♦♥s t♦ ✜♥❞ p✱ ✐s ✐♥ [100n, 100n + 99], n ∈ N. ❚❤❡ ❧❡❢t ❣r❛♣❤ s❤♦✇s 0 ≤ n ≤ 1000✱ t❤❡ r✐❣❤t ❣r❛♣❤ s❤♦✇s ❛ ③♦♦♠ ♦♥ 0 ≤ n ≤ 100✳
✻ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛
✷ ❚r✐❛❧ ❉✐✈✐s✐♦♥
❇❡❢♦r❡ ❢❛❝t♦r✐♥❣ ♦❢ t❤❡ ♥♦♥✲s✐❡✈❡❞ ❝♦❢❛❝t♦r ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✈❛❧✉❡s ✐♥t♦ ❧❛r❣❡ ♣r✐♠❡s ❝❛♥ ❝♦♠♠❡♥❝❡✱ t❤❡ ❝♦❢❛❝t♦r ♥❡❡❞s t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ❞✐✈✐❞✐♥❣ ♦✉t ❛❧❧ t❤❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✳ ❋♦r ♠❡❞✐✉♠ s✐③❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✱ s❛② ❧❛r❣❡r t❤❛♥ ❛ ❢❡✇ ❤✉♥❞r❡❞ ♦r ❛ ❢❡✇ t❤♦✉s❛♥❞✱ ❛ s✐❡✈✐♥❣ t❡❝❤♥✐q✉❡ ✭✑r❡✲s✐❡✈✐♥❣✑✮ ❝❛♥ ❜❡ ✉s❡❞ ❛❣❛✐♥ t❤❛t st♦r❡s t❤❡ ♣r✐♠❡s ✇❤❡♥ r❡✲s✐❡✈✐♥❣ ❤✐ts ❛ ❧♦❝❛t✐♦♥ ♣r❡✈✐♦✉s❧② ♠❛r❦❡❞ ❛s ✑❧✐❦❡❧② s♠♦♦t❤✳✑ ❋♦r ❧❛r❣❡ ❢❛❝t♦r ❜❛s❡ ♣r✐♠❡s✱ s❛② ❧❛r❣❡r t❤❛♥ ❛ ❢❡✇ t❡♥ t❤♦✉s❛♥❞✱ t❤❡ ♥✉♠❜❡r ♦❢ ❤✐ts ✐♥ t❤❡ s✐❡✈❡ ❛r❡❛ ✐s s♠❛❧❧ ❡♥♦✉❣❤ t❤❛t t❤❡ ♣r✐♠❡s ❝❛♥ ❜❡ st♦r❡❞ ❞✉r✐♥❣ t❤❡ ✐♥✐t✐❛❧ s✐❡✈✐♥❣ ♣r♦❝❡ss ✐ts❡❧❢✳ ❋♦r t❤❡ s♠❛❧❧❡st ♣r✐♠❡s✱ ❤♦✇❡✈❡r✱ r❡✲s✐❡✈✐♥❣ ✐s ✐♥❡✣❝✐❡♥t✱ ❛♥❞ ❛ tr✐❛❧ ❞✐✈✐s✐♦♥ t❡❝❤♥✐q✉❡ s❤♦✉❧❞ ❜❡ ✉s❡❞✳ ❚❤✐s ❙❡❝t✐♦♥ ❡①❛♠✐♥❡s ❛ ❢❛st tr✐❛❧ ❞✐✈✐s✐♦♥ r♦✉t✐♥❡✱ ❜❛s❡❞ ♦♥ ✐❞❡❛s ❜② ▼♦♥t❣♦♠❡r② ❛♥❞ ●r❛♥❧✉♥❞ ❬✶✸❪ ❬✷✶❪✱ t❤❛t ♣r❡❝♦♠♣✉t❡s s❡✈❡r❛❧ ✈❛❧✉❡s ♣❡r ❝❛♥❞✐❞❛t❡ ♣r✐♠❡ ❞✐✈✐s♦r t♦ s♣❡❡❞ ✉♣ t❤❡ ♣r♦❝❡ss✳✷✳✶ ❚r✐❛❧ ❞✐✈✐s✐♦♥ ❛❧❣♦r✐t❤♠
●✐✈❡♥ ♠❛♥② ❝♦♠♣♦s✐t❡ ✐♥t❡❣❡rs Ni✱ 0 ≤ i < n✱ ✇❡ ✇❛♥t t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♣r✐♠❡s ❢r♦♠ s♦♠❡ s❡t P = {pj, 0 ≤ j < k} ♦❢ s♠❛❧❧ ♦❞❞ ♣r✐♠❡s ❞✐✈✐❞❡ ❡❛❝❤ Ni✳ ❲❡ ❛ss✉♠❡ n ≫ k✳ ❊❛❝❤ Ni ✐s ❛ ♠✉❧t✐✲✇♦r❞ ✐♥t❡❣❡r ♦❢ ✉♣ t♦ ℓ + 1 ✇♦r❞s✱ Ni=Pℓj=0ni,jβj✱ ✇❤❡r❡ β ✐s t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❜❛s❡ ✭❡✳❣✳✱ β = 232 ♦r β = 264✮ ❛♥❞ ℓ ✐s ♦♥ t❤❡ ♦r❞❡r ♦❢ ✑❛ ❢❡✇✱✑ s❛② ℓ ≤ 4✳ ❋♦r ❡❛❝❤ ♣r✐♠❡ p ∈ P ✱ ✇❡ ♣r❡❝♦♠♣✉t❡ wj = βjmod p ❢♦r 1 ≤ j ≤ ℓ✱ p✐♥✈= p−1 (mod β) ❛♥❞ p❧✐♠= j β−1 p k✳ ❈♦♥s✐❞❡r ❛ ♣❛rt✐❝✉❧❛r ✐♥t❡❣❡r N = Pℓ j=0njβj✱ ❛♥❞ ❛ ♣❛rt✐❝✉❧❛r ♣r✐♠❡ p ∈ P✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❞♦❡s ❛ s❡♠✐✲r❡❞✉❝t✐♦♥ ♠♦❞✉❧♦ p t♦ ♦❜t❛✐♥ ❛ s✐♥❣❧❡✲ ✇♦r❞ ✐♥t❡❣❡r ❝♦♥❣r✉❡♥t t♦ N (mod p)✱ t❤❡♥ t❡sts t❤✐s s✐♥❣❧❡✲✇♦r❞ ✐♥t❡❣❡r ❢♦r ❞✐✈✐s✐❜✐❧✐t② ❜② p✳ ❚♦ ❞♦ s♦✱ ✇❡ ❝♦♠♣✉t❡ r = n0+Pℓj=1njwj ≤ (β − 1)(ℓ(p − 1) + 1)✳ ❚♦ s✐♠♣❧✐❢② t❤❡ ♥❡①t st❡♣s✱ ✇❡ r❡q✉✐r❡ p <qβ ℓ✳ ❊✈❡♥ ❢♦r β = 232✱ ℓ = 4✱ t❤✐s ❣✐✈❡s p < 32768✇❤✐❝❤ ✐s ❡❛s✐❧② s✉✣❝✐❡♥t ❢♦r tr✐❛❧ ❞✐✈✐s✐♦♥ ✐♥ ◆❋❙✳ ❲✐t❤ t❤✐s ❜♦✉♥❞ ♦♥ p✱ ✇❡ ❤❛✈❡ r < (β −1)(√βℓ −ℓ+1)✳ ❲❡ t❤❡♥ ❞❡❝♦♠♣♦s❡ r ✐♥t♦ r = r1β + r0✱ ✇❤❡r❡ 0 ≤ r0 < β✳ ❚❤✐s ✐♠♣❧✐❡s r1 < √βℓ✱ ❛♥❞ r1w1 ≤ r1(p − 1) <√βℓ q β ℓ − 1 = β −√βℓ✳ ❚❤❡ ❛❧❣♦r✐t❤♠ t❤❡♥ ❞♦❡s ❛♥♦t❤❡r r❡❞✉❝t✐♦♥ st❡♣ ❜② s = r1w1+ r0✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ s = s1β + s0 < 2β − p✱ s♦ t❤❛t ❛ ✜♥❛❧ r❡❞✉❝t✐♦♥ st❡♣ t = s0+ s1w1 < β ♣r♦❞✉❝❡s ❛ ♦♥❡✲✇♦r❞ r❡s✉❧t✳ ❙✐♥❝❡ r1(p−1) < β−√βℓ✱ s < 2β−√βℓ−1 < 2β−p✳ ❙✐♥❝❡ s1 ✐s ❡✐t❤❡r 0 ♦r 1✱ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❛❞❞✐t✐♦♥ ✐♥ s0+ s1w1 ✐s r❡❛❧❧② ❥✉st ❛ ❝♦♥❞✐t✐♦♥❛❧ ❛❞❞✐t✐♦♥✳ ◆♦✇ ✇❡ ❤❛✈❡ ❛ ♦♥❡✲✇♦r❞ ✐♥t❡❣❡r t ✇❤✐❝❤ ✐s ❞✐✈✐s✐❜❧❡ ❜② p ✐❢ ❛♥❞ ♦♥❧② ✐❢ N ✐s✳ ❚♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r p | t✱ ✇❡ ✉s❡ t❤❡ ✐❞❡❛ ❢r♦♠ ❬✶✸✱ ➓✾❪ t♦ ❝♦♠♣✉t❡ u = tp−1mod β✱ ✉s✐♥❣ t❤❡ ♣r❡❝♦♠♣✉t❡❞ p✐♥✈ = p−1 (mod β)✳ ■❢ p | t✱ t/p ✐s ❛♥ ✐♥t❡❣❡r < β ❛♥❞ s♦ t❤❡ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ mod β ♠✉st ♣r♦❞✉❝❡ t❤❡ ❝♦rr❡❝t u = t/p✳ ❚❤❡r❡ ❛r❡ jβ−1 p + 1k ♠✉❧t✐♣❧❡s ♦❢ p ✭✐♥❝❧✉❞✐♥❣ ✵✮ ❧❡ss t❤❛♥ β✱ ✉♥❞❡r ❞✐✈✐s✐♦♥ ❜② p t❤❡s❡ ♠❛♣ t♦ t❤❡ ✐♥t❡❣❡rsh0, . . . ,jβ−1p ki✳ ❙✐♥❝❡ p ✐s ❝♦♣r✐♠❡ t♦ β✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② p−1 (mod β)✐s ❛ ❜✐❥❡❝t✐✈❡ ♠❛♣✱ s♦ ❛❧❧ ♥♦♥✲♠✉❧t✐♣❧❡s ♦❢ p ♠✉st ♠❛♣ t♦ t❤❡ r❡♠❛✐♥✐♥❣ ✐♥t❡❣❡rshjβ−1 p k + 1, β − 1i✳ ❍❡♥❝❡ t❤❡ t❡st ❢♦r ❞✐✈✐s✐❜✐❧✐t②❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✼ ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❛ ♦♥❡✲✇♦r❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t p✐♥✈✱ ❛♥❞ ♦♥❡ ❝♦♠♣❛r✐s♦♥ t♦ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t p❧✐♠= jβ−1 p k✳
✷✳✷ ■♠♣❧❡♠❡♥t❛t✐♦♥
❚❤❡ ❛❧❣♦r✐t❤♠ ✐s q✉✐t❡ s✐♠♣❧❡ t♦ ✐♠♣❧❡♠❡♥t ♦♥ ❛♥ ①✽✻ ❈P❯✱ ✇❤✐❝❤ ♦✛❡rs t❤❡ t✇♦✲✇♦r❞ ♣r♦❞✉❝t ♦❢ t✇♦ ♦♥❡✲✇♦r❞ ❛r❣✉♠❡♥ts ❜② ❛ s✐♥❣❧❡ ▼❯▲ ✐♥str✉❝t✐♦♥✳ ■t ♠✐❣❤t r✉♥ ❛s s❤♦✇♥ ✐♥ ❆❧❣♦r✐t❤♠ ✶✱ ✇❤❡r❡ x1✱ x0❛r❡ r❡❣✐st❡rs t❤❛t t❡♠♣♦r❛r✐❧② ❤♦❧❞ t✇♦✲✇♦r❞ ♣r♦❞✉❝ts✳ ❆ ♣❛✐r ♦❢ r❡❣✐st❡rs ❤♦❧❞✐♥❣ ❛ t✇♦✲✇♦r❞ ✈❛❧✉❡ r1β + r0 ✐s ✇r✐tt❡♥ ❛s r1: r0.❚❤❡ ✈❛❧✉❡s r0,1✱ s0,1✱ ❛♥❞ t0❝❛♥ ❛❧❧ ✉s❡ t❤❡ s❛♠❡ r❡❣✐st❡rs✱ ✇r✐tt❡♥ r0,1 ❤❡r❡✳ ❚❤❡ ❧♦♦♣ ♦✈❡r j s❤♦✉❧❞ ❜❡ ✉♥r♦❧❧❡❞✳ ■♥♣✉t✿ ▲❡♥❣t❤ ℓ N =Pℓ i=0niβi, 0 ≤ ni< β ❖❞❞ ♣r✐♠❡ p <qβ ℓ wj = βjmod p❢♦r 1 ≤ j ≤ ℓ p✐♥✈= p−1mod β p❧✐♠=jβ−1p k ❖✉t♣✉t✿ 1 ✐❢ p | N✱ 0 ♦t❤❡r✇✐s❡ r0:= n0❀ r1:= 0❀ ❢♦r 1 ≤ j ≤ ℓ ❞♦ x1: x0= nj· wj❀ r1: r0= r1: r0+ x1: x0❀ x0= r1· w1❀ r0= (r0+ x0) mod β❀ ✐❢ ❧❛st ❛❞❞✐t✐♦♥ s❡t ❝❛rr② ✢❛❣ t❤❡♥ r0= (r0+ w1) mod β❀ r0= r0· p✐♥✈❀ ✐❢ r0≤ p❧✐♠ t❤❡♥ r❡t✉r♥ 1❀ ❡❧s❡ r❡t✉r♥ 0❀ ❆❧❣♦r✐t❤♠ ✶✿ Ps❡✉❞♦✲❝♦❞❡ ❢♦r tr✐❛❧ ❞✐✈✐s✐♦♥ ♦❢ ♥✉♠❜❡rs ♦❢ ✉♣ t♦ ℓ + 1 ✇♦r❞s✳ ❚❤✐s ❝♦❞❡ ✉s❡s ℓ ♠✉❧t✐♣❧✐❝❛t✐♦♥s ♦❢ t✇♦ ✇♦r❞s t♦ ❛ t✇♦✲✇♦r❞ ♣r♦❞✉❝t✳ ❚❤❡s❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♦♥❡ ❛♥♦t❤❡r✱ s♦ t❤❡② ❝❛♥ ♦✈❡r❧❛♣ ♦♥ ❛ ❈P❯ ✇✐t❤ ♣✐♣❡❧✐♥❡❞ ♠✉❧t✐♣❧✐❡r✳ ❖♥ ❛♥ ❆t❤❧♦♥✻✹✱ ❖♣t❡r♦♥✱ ❛♥❞ P❤❡♥♦♠ ❈P❯s✱ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❝❛♥ st❛rt ❡✈❡r② ✷ ❝❧♦❝❦ ❝②❝❧❡s✱ t❤❡ ❧♦✇ ✇♦r❞ ♦❢ t❤❡ ♣r♦❞✉❝t ✐s ❛✈❛✐❧❛❜❧❡ ❛❢t❡r ✹ ❝❧♦❝❦ ❝②❝❧❡s✱ t❤❡ ❤✐❣❤ ✇♦r❞ ❛❢t❡r ✺ ❝❧♦❝❦ ❝②❝❧❡s✳ ❚❤✉s ✐♥ ❝❛s❡ ♦❢ ℓ = 4✱ t❤❡ ❧❛t❡♥❝② ❢♦r t❤❡ ✜rst 4 ♣r♦❞✉❝ts ❛♥❞ ❜✉✐❧❞✐♥❣ t❤❡✐r s✉♠ s❤♦✉❧❞ ❜❡ ✶✷ ❝②❝❧❡s✳ ❚❤❡ t✇♦ r❡♠❛✐♥✐♥❣ ♠✉❧t✐♣❧✐❡s✱ t❤❡ ❛❞❞✐t✐♦♥s ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ♠♦✈❡s s❤♦✉❧❞ ❜❡ ♣♦ss✐❜❧❡ ✐♥ ❛❜♦✉t ✶✶ ❝②❝❧❡s✱ ❣✐✈✐♥❣ ❛ t❤❡♦r❡t✐❝❛❧ t♦t❛❧ ❝♦✉♥t ♦❢ ❛❜♦✉t ✷✸ ❝❧♦❝❦ ❝②❝❧❡s ❢♦r tr✐❛❧ ❞✐✈✐❞✐♥❣ ❛ ✺ ✇♦r❞ ✐♥t❡❣❡r ❜② ❛ s♠❛❧❧ ♣r✐♠❡✳ ❉❛t❛ ♠♦✈❡♠❡♥t ❢r♦♠ ❝❛❝❤❡ ♠❛② ✐♥tr♦❞✉❝❡ ❛❞❞✐t✐♦♥❛❧ ❧❛t❡♥❝②✳✷✳✸ ❯s❡ ✐♥ ◆❋❙
●✐✈❡♥ ❛ s✐❡✈❡ r❡❣✐♦♥ ♦❢ s✐③❡ s ✇✐t❤ ❡✈❡r② d✲t❤ ❡♥tr② ❛ s✐❡✈❡ r❡♣♦rt✱ tr✐❛❧ ❞✐✈✐❞✐♥❣ ❜② t❤❡ ♣r✐♠❡ p ❢♦r ❛❧❧ s✐❡✈❡ r❡♣♦rts ❤❛s ❝♦st O(s/d)✱ ✇❤✐❧❡ r❡s✐❡✈✐♥❣ ❤❛s ❝♦st✽ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛ O(rs/p)✱ ✇❤❡r❡ r ✐s t❤❡ ♥✉♠❜❡r ♦❢ r♦♦ts ♠♦❞✉❧♦ p t❤❡ s✐❡✈❡❞ ♣♦❧②♥♦♠✐❛❧ ❤❛s✳ ❍❡♥❝❡ ✇❤❡t❤❡r tr✐❛❧ ❞✐✈✐s✐♦♥ ♦r r❡s✐❡✈✐♥❣ ✐s ♣r❡❢❡r❛❜❧❡ ✇✐❧❧ ❞❡♣❡♥❞ ♦♥ p dr✱ ✇❤❡r❡ t❤♦s❡ p ✇✐t❤ p dr < c❢♦r s♦♠❡ t❤r❡s❤♦❧❞ c s❤♦✉❧❞ ✉s❡ tr✐❛❧ ❞✐✈✐s✐♦♥✳ ❆s ♣r✐♠❡s ❛r❡ ❞✐✈✐❞❡❞ ♦✉t ♦❢ N✱ t❤❡ ♥✉♠❜❡r ♦❢ ✇♦r❞s ✐♥ N ♠❛② ❞❡❝r❡❛s❡✱ ♠❛❦✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❛❧ ❞✐✈✐s✐♦♥ ❢❛st❡r✳ ■t ♠✐❣❤t ❜❡ ✇♦rt❤✇❤✐❧❡ t♦ tr② t♦ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ N ❛s q✉✐❝❦❧② ❛s ♣♦ss✐❜❧❡✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ♣r✐♠❡ p ❞✐✈✐❞❡s N ♠❛② ❜❡ ❡st✐♠❛t❡❞ ❛s r/p✱ t❤❡ s✐③❡ ❞❡❝r❡❛s❡ ❛s log(p)✱ s♦ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t tr✐❛❧ ❞✐✈✐s✐♦♥ ❜② p ✇✐❧❧ ❞❡❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇♦r❞s ✐♥ N ♠❛② ❜❡ ❡st✐♠❛t❡❞ ❛s ❜❡✐♥❣ ♣r♦♣♦rt✐♦♥❛❧ t♦ r log(p)/p✳ ❋♦r tr✐❛❧ ❞✐✈✐s✐♦♥✱ t❤❡ ❝❛♥❞✐❞❛t❡ ❞✐✈✐s♦rs p ❝❛♥ ❜❡ s♦rt❡❞ s♦ t❤❛t t❤✐s ❡st✐♠❛t❡ ✐s ❞❡❝r❡❛s✐♥❣✳ ❚❤✐s ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t❡ ❞♦❡s ♥♦t t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ❢❛❝t t❤❛t N, ❜❡✐♥❣ ❛ s✐❡✈❡ r❡♣♦rt✱ ✐s ❧✐❦❡❧② s♠♦♦t❤✱ ❛♥❞ ✉♥❞❡r t❤✐s ❝♦♥❞✐t✐♦♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t p ❞✐✈✐❞❡s N ✐♥❝r❡❛s❡s ❜② ❇❛②❡s✬ t❤❡♦r❡♠✱ ♠♦r❡ s♦ ❢♦r ❧❛r❣❡r p t❤❛♥ ❢♦r s♠❛❧❧ ♦♥❡s✳
✷✳✹ ❚❡st✐♥❣ s❡✈❡r❛❧ ♣r✐♠❡s ❛t ♦♥❝❡
❆❧❣♦r✐t❤♠ ✶ r❡❞✉❝❡s t❤❡ ✐♥♣✉t ♥✉♠❜❡r t♦ ❛ ♦♥❡✲✇♦r❞ ✐♥t❡❣❡r ✇❤✐❝❤ ✐s ❝♦♥❣r✉❡♥t t♦ N (mod p)✱ t❤❡♥ t❡sts ❞✐✈✐s✐❜✐❧✐t② ❜② p ♦❢ t❤❛t ♦♥❡✲✇♦r❞ ✐♥t❡❣❡r✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ❞♦ t❤❡ r❡❞✉❝t✐♦♥ st❡♣ ❢♦r ❝♦♠♣♦s✐t❡ ❝❛♥❞✐❞❛t❡ ❞✐✈✐s♦rs q✱ t❤❡♥ t❡st ❞✐✈✐s✐❜✐❧✐t② ♦❢ t❤❡ r❡s✉❧t✐♥❣ ♦♥❡✲✇♦r❞ ✐♥t❡❣❡r ❢♦r ❛❧❧ p | q✳ ❚❤✐s ✇❛②✱ ❢♦r ✐♥t❡❣❡rs ❝♦♥s✐st✐♥❣ ♦❢ s❡✈❡r❛❧ ✇♦r❞s✱ t❤❡ ❡①♣❡♥s✐✈❡ r❡❞✉❝t✐♦♥ ♥❡❡❞s t♦ ❜❡ ❞♦♥❡ ♦♥❧② ♦♥❝❡ ❢♦r ❡❛❝❤ q✱ t❤❡ r❡❧❛t✐✈❡❧② ❝❤❡❛♣ ❞✐✈✐s✐❜✐❧✐t② t❡st ❢♦r ❡❛❝❤ p✳ ❚❤✐s ✐s ❛ttr❛❝t✐✈❡ ✐❢ t❤❡ ❜♦✉♥❞ q <pβ/ℓ✐s ♥♦t t♦♦ s♠❛❧❧✳ ❲✐t❤ w = 264✱ ℓ = 4✱ ✇❡ ❝❛♥ ✉s❡ q < 2147483648✱ ✇❤✐❝❤ ❛❧❧♦✇s ❢♦r s❡✈❡r❛❧ s♠❛❧❧ ♣r✐♠❡s ✐♥ q✳ ❋♦r ✐♥t❡❣❡rs N ✇✐t❤ ❛ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ✇♦r❞s✱ ✐t ♠❛② ❜❡ ✇♦rt❤✇❤✐❧❡ t♦ ✐♥tr♦❞✉❝❡ ❛♥ ❛❞❞✐t✐♦♥❛❧ r❡❞✉❝t✐♦♥ st❡♣ ✭❢♦r ❡①❛♠♣❧❡✱ ✉s✐♥❣ ▼♦♥t❣♦♠❡r②✬s ❘❊❉❈ ❢♦r ❛ r✐❣❤t✲t♦✲❧❡❢t r❡❞✉❝t✐♦♥✮ t♦ r❡❧❛① t❤❡ ❜♦✉♥❞ ♦♥ q t♦✱ ❡✳❣✳✱ q < w/ℓ✱ s♦ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ♣r✐♠❡s ✐♥ q ❝❛♥ ❜❡ ❞♦✉❜❧❡❞ ❛t t❤❡ ❝♦st ♦❢ ♦♥❧② t✇♦ ❛❞❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡s✳ ■♥ ◆❋❙✱ ✐❢ t❤❡ ♣r✐♠❡s ❢♦✉♥❞ ❜② r❡✲s✐❡✈✐♥❣ ❤❛✈❡ ❜❡❡♥ ❞✐✈✐❞❡❞ ♦✉t ❛❧r❡❛❞② ❜❡❢♦r❡ tr✐❛❧ ❞✐✈✐s✐♦♥ ❜❡❣✐♥s✱ t❤❡ Ni♠❛② ♥♦t ❜❡ ❧❛r❣❡ ❡♥♦✉❣❤ t♦ ♠❛❦❡ t❤✐s ❛♣♣r♦❛❝❤ ✇♦rt❤✇❤✐❧❡✳✷✳✺ P❡r❢♦r♠❛♥❝❡ ♦❢ tr✐❛❧ ❞✐✈✐s✐♦♥
❚♦ ♠❡❛s✉r❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ tr✐❛❧ ❞✐✈✐s✐♦♥ ❝♦❞❡✱ ✇❡ ❞✐✈✐❞❡ 107❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs ♦❢ 1, . . . , 5 ✇♦r❞s ❜② t❤❡ ✜rst n = 256, 512, 1024, ❛♥❞ 2048 ♦❞❞ ♣r✐♠❡s ♦♥ ❛ 2 ●❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯✱ s❡❡ ❋✐❣✉r❡ ✷✳ ❚❤❡ ❤✐❣❤❡r t✐♠✐♥❣s ♣❡r tr✐❛❧ ❞✐✈✐s✐♦♥ ❢♦r n = 256 ❛r❡ ❞✉❡ t♦ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❝♦st ♦❢ ❞✐✈✐❞✐♥❣ ♦✉t ❢♦✉♥❞ ❞✐✈✐s♦rs✱ ✇❤✐❝❤ ❤❛s ❛ ❣r❡❛t❡r r❡❧❛t✐✈❡ ❝♦♥tr✐❜✉t✐♦♥ ❢♦r s♠❛❧❧❡r ♣r✐♠❡s ✇❤✐❝❤ ❞✐✈✐❞❡ ♠♦r❡ ❢r❡q✉❡♥t❧②✳ ❚❤❡ t✐♠✐♥❣ ❢♦r ℓ = 4, n = 2048 ✐s ❝❧♦s❡ t♦ t❤❡ ♣r❡❞✐❝t❡❞ 23 ❝❧♦❝❦ ❝②❝❧❡s✳ ❚❤❡ s✉❞❞❡♥ ✐♥❝r❡❛s❡ ❢♦r n = 2048 ✐♥ t❤❡ ❝❛s❡ ♦❢ N ✇✐t❤ ♦♥❡ ✇♦r❞ ✐s ❞✉❡ t♦ ❝❛❝❤✐♥❣✿ ✇✐t❤ 7 st♦r❡❞ ✈❛❧✉❡s ✭p✱ p✐♥✈✱ p❧✐♠✱ w1,...,4✮ ♦❢ 8 ❜②t❡s ❡❛❝❤✱ n = 2048 ❤❛s ❛ t❛❜❧❡ ♦❢ ♣r❡❝♦♠♣✉t❡❞ ✈❛❧✉❡s ♦❢ s✐③❡ 112❦❇✱ ✇❤✐❝❤ ❡①❝❡❡❞s t❤❡ ❧❡✈❡❧✲✶ ❞❛t❛ ❝❛❝❤❡ s✐③❡ ♦❢ 64❦❇ ♦❢ t❤❡ P❤❡♥♦♠✳ ❋♦r ❧❛r❣❡ s❡ts ♦❢ ❝❛♥❞✐❞❛t❡ ♣r✐♠❡s✱ t❤❡ s❡q✉❡♥t✐❛❧ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ♣r❡❝♦♠♣✉t❡❞ ❞❛t❛ ❝❛✉s❡ ❢r❡q✉❡♥t ♠✐ss❡s ✐♥ t❤❡ ❧❡✈❡❧✲✶ ❝❛❝❤❡✱ ❛♥❞ t❤❡ tr✐❛❧ ❞✐✈✐s✐♦♥s ❢♦r N ♦❢ ♦♥❧② ♦♥❡ ✇♦r❞ ❛r❡ ❢❛st ❡♥♦✉❣❤ t❤❛t tr❛♥s❢❡r r❛t❡ ❢r♦♠ t❤❡ ❧❡✈❡❧✲✷ ❝❛❝❤❡ ❧✐♠✐ts t❤❡ ❡①❡❝✉t✐♦♥✳ ❚❤✐s ❝♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❜② ❝♦♠♣✉t✐♥❣ ❢❡✇❡r wi❝♦♥st❛♥ts ✭✐✳❡✳✱ ❝❤♦♦s✐♥❣ ❛ s♠❛❧❧❡r ℓ✮ ✐❢ t❤❡ N ❛r❡ ❦♥♦✇♥ t♦ ❜❡ s♠❛❧❧✱ ♦r st♦r✐♥❣ t❤❡ wi✐♥ s❡♣❛r❛t❡ ❛rr❛②s r❛t❤❡r t❤❛♥ ✐♥t❡r❧❡❛✈❡❞✱ s♦ t❤❛t t❤❡ wi ❢♦r ❧❛r❣❡r i ❞♦ ♥♦t ♦❝❝✉♣② ❝❛❝❤❡ ✇❤✐❧❡ t❤❡ N ♣r♦❝❡ss❡❞ ❛r❡ s♠❛❧❧✳ ❙✐♥❝❡ t❤❡ ✈❛❧✉❡ ♦❢ p ✐s ♥♦t ❛❝t✉❛❧❧② ♥❡❡❞❡❞ ❞✉r✐♥❣ t❤❡ tr✐❛❧ ❞✐✈✐s✐♦♥✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✾ ◆✉♠❜❡r ♦❢ ✇♦r❞s ✐♥ N n ✶ ✷ ✸ ✹ ✺ ✷✺✻ ✻✳✽ ✭✷✳✻✮ ✶✺✳✸ ✭✻✳✵✮ ✷✵✳✽ ✭✽✳✶✮ ✷✼✳✺ ✭✶✵✳✼✮ ✸✷✳✹ ✭✶✷✳✻✮ ✺✶✷ ✶✶✳✸ ✭✷✳✷✮ ✷✽✳✷ ✭✺✳✺✮ ✸✽✳✽ ✭✼✳✻✮ ✺✷✳✵ ✭✶✵✳✷✮ ✻✶✳✸✷ ✭✶✷✳✵✮ ✶✵✷✹ ✷✶✳✸ ✭✷✳✶✮ ✺✹✳✾ ✭✺✳✹✮ ✼✺✳✾ ✭✼✳✹✮ ✶✵✷✳✵ ✭✶✵✳✵✮ ✶✷✵✳✼ ✭✶✶✳✽✮ ✷✵✹✽ ✽✺✳✹ ✭✹✳✶✮ ✶✵✽✳✹ ✭✺✳✸✮ ✶✹✾✳✽ ✭✼✳✸✮ ✷✵✵✳✽ ✭✾✳✽✮ ✷✸✼✳✽ ✭✶✶✳✻✮ ❋✐❣✉r❡ ✷✿ ❚✐♠❡ ✐♥ s❡❝♦♥❞s ❢♦r tr✐❛❧ ❞✐✈✐s✐♦♥ ♦❢ 107 ❝♦♥s❡❝✉t✐✈❡ ✐♥t❡❣❡rs ❜② t❤❡ ✜rst n ♦❞❞ ♣r✐♠❡s ♦♥ ❛ 2 ●❍③ ❆▼❉ P❤❡♥♦♠ ❈P❯✳ ❚✐♠❡ ♣❡r tr✐❛❧ ❞✐✈✐s♦♥ ✐♥ ♥❛♥♦s❡❝♦♥❞s ✐♥ ♣❛r❡♥t❤❡s❡s✳ ❛✈♦✐❞ st♦r✐♥❣ ✐t ❛♥❞ r❡❝♦♠♣✉t✐♥❣ ✐t✱ ❡✳❣✳✱ ❢r♦♠ p✐♥✈✇❤❡♥ ✐t ♥❡❡❞s t♦ ❜❡ r❡♣♦rt❡❞ ❛s ❛ ❞✐✈✐s♦r✳
✸ ▼♦❞✉❧❛r ❛r✐t❤♠❡t✐❝
❚❤❡ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❛r❡ r❡❧❛t✐✈❡❧② ✐♥❡①♣❡♥s✐✈❡ ✇❤❡♥ ♠♦❞✉❧✐ ❛♥❞ r❡s✐❞✉❡s ♦❢ ♦♥❧② ❛ ❢❡✇ ♠❛❝❤✐♥❡ ✇♦r❞s ❛r❡ ❝♦♥s✐❞❡r❡❞✱ ❛♥❞ s❤♦✉❧❞ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❛ ✇❛② t❤❛t ❧❡ts t❤❡ ❝♦♠♣✐❧❡r ♣❡r❢♦r♠ ✐♥✲❧✐♥✐♥❣ ♦❢ s✐♠♣❧❡ ❛r✐t❤♠❡t✐❝ ❢✉♥❝t✐♦♥s t♦ ❛✈♦✐❞ ✉♥♥❡❝❡ss❛r② ❢✉♥❝t✐♦♥ ❝❛❧❧ ♦✈❡r❤❡❛❞ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t ❜❡t✇❡❡♥ r❡❣✲ ✐st❡rs✱ ♠❡♠♦r② ❛♥❞ st❛❝❦ ❞✉❡ t♦ t❤❡ ❝❛❧❧✐♥❣ ❝♦♥✈❡♥t✐♦♥s ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ ❛♥❞ ❛r❝❤✐t❡❝t✉r❡✳ ▼❛♥② s✐♠♣❧❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❝❛♥ ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❡❛s✐❧② ❛♥❞ ❡✣❝✐❡♥t❧② ✉s✐♥❣ ❛ss❡♠❜❧② ❧❛♥❣✉❛❣❡✱ ❜✉t ❛r❡ ❝✉♠❜❡rs♦♠❡ t♦ ✇r✐t❡ ✐♥ ♣✉r❡ ❈ ❝♦❞❡✱ ❡s♣❡❝✐❛❧❧② ✐❢ ♠✉❧t✐✲✇♦r❞ ♣r♦❞✉❝ts ♦r ❝❛rr② ♣r♦♣❛❣❛t✐♦♥ ❛r❡ ✐♥✈♦❧✈❡❞✳ ❚❤❡ ●◆❯ ❈ ❝♦♠♣✐❧❡r ♦✛❡rs ❛ ✈❡r② ✢❡①✐❜❧❡ ♠❡t❤♦❞ ♦❢ ✐♥❥❡❝t✐♥❣ ❛ss❡♠❜❧② ❝♦❞❡ ✐♥t♦ ❈ ♣r♦❣r❛♠s✱ ✇✐t❤ ❛♥ ✐♥t❡r❢❛❝❡ t❤❛t t❡❧❧s t❤❡ ❝♦♠♣✐❧❡r ❛❧❧ ❝♦♥str❛✐♥ts ♦♥ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❞❛t❛ ♦❢ t❤❡ ❛ss❡♠❜❧② ❜❧♦❝❦ s♦ t❤❛t ✐t ❝❛♥ ♣❡r❢♦r♠ ♦♣t✐♠✐③❛t✐♦♥ ♦♥ t❤❡ ❝♦❞❡ s✉rr♦✉♥❞✐♥❣ t❤❡ ❛ss❡♠❜❧② st❛t❡♠❡♥ts✳ ❇② ❞❡✜♥✐♥❣ s♦♠❡ ❝♦♠♠♦♥❧② ✉s❡❞ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ✐♥ ❛ss❡♠❜❧②✱ ♠✉❝❤ ♦❢ t❤❡ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❈✱ ❧❡tt✐♥❣ t❤❡ ❝♦♠♣✐❧❡r ❤❛♥❞❧❡ r❡❣✐st❡r ❛❧❧♦❝❛t✐♦♥ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t✳ ❚❤❡ r❡s✉❧t✐♥❣ ❝♦❞❡ ✐s ✉s✉❛❧❧② ♥♦t ♦♣t✐♠❛❧✱ ❜✉t q✉✐t❡ ✉s❡❛❜❧❡✳ ❋♦r t❤❡ ♠♦st t✐♠❡✲ ❝r✐t✐❝❛❧ ♦♣❡r❛t✐♦♥s✱ ✇r✐t✐♥❣ ❤❛♥❞✲♦♣t✐♠✐③❡❞ ❛ss❡♠❜❧② ❝♦❞❡ ♦✛❡rs ❛♥ ❛❞❞✐t✐♦♥❛❧ s♣❡❡❞ ✐♠♣r♦✈❡♠❡♥t✳ ❋♦r t❤❡ ♣r❡s❡♥t ✇♦r❦✱ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ❢♦r ♠♦❞✉❧✐ ♦❢ 1 ♠❛❝❤✐♥❡ ✇♦r❞ ❛♥❞ ♦❢ 2 ♠❛❝❤✐♥❡ ✇♦r❞s ✇✐t❤ t❤❡ t✇♦ ♠♦st s✐❣♥✐✜❝❛♥t ❜✐ts ③❡r♦ ✐s ✐♠♣❧❡♠❡♥t❡❞✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛r✐t❤♠❡t✐❝ ❢♦r ♠♦❞✉❧✐ ♦❢ 3 ♠❛❝❤✐♥❡ ✇♦r❞s ✐s ✐♥ ♣r♦❣r❡ss✳✸✳✶ ❆ss❡♠❜❧② s✉♣♣♦rt
❚♦ ❣✐✈❡ ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛♥ ❡❧❡♠❡♥t❛r② ❢✉♥❝t✐♦♥ t❤❛t ✐s ✐♠♣❧❡♠❡♥t❡❞ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ s♦♠❡ ❛ss❡♠❜❧② ❝♦❞❡✱ ✇❡ ❡①❛♠✐♥❡ ♠♦❞✉❧❛r ❛❞❞✐t✐♦♥ ✇✐t❤ ❛ ♠♦❞✉❧✉s ♦❢ 1 ♠❛❝❤✐♥❡ ✇♦r❞✳ ❚❤✐s ✐s ❛♠♦♥❣ t❤❡ ♠♦st s✐♠♣❧❡ ♦♣❡r❛t✐♦♥s ♣♦ss✐❜❧❡✱ ❜✉t ✉s❡❢✉❧ ❛s ❛♥ ❡①❛♠♣❧❡✳ ▲❡t ❛ ✏r❡❞✉❝❡❞ r❡s✐❞✉❡✑ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ♣♦s✐t✐✈❡ ♠♦❞✉❧✉s m ♠❡❛♥ ❛♥ ✐♥t❡❣❡r r❡♣r❡s❡♥t❛t✐✈❡ 0 ≤ r < m ♦❢ t❤❡ r❡s✐❞✉❡ ❝❧❛ss r (mod m). ▼♦❞✉❧❛r ❛❞❞✐t✐♦♥ ♦❢✶✵ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛ t✇♦ r❡❞✉❝❡❞ r❡s✐❞✉❡s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s (a + b) mod m = ( a + b − m ✐❢ a + b − m ≥ 0 a + b ♦t❤❡r✇✐s❡✳ ■❢ ❛♥② ♠♦❞✉❧✉s m < β ✐s ♣❡r♠✐tt❡❞✱ ✇❤❡r❡ β ✐s t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❜❛s❡✱ t❤❡♥ t❤❡ ♣r♦❜❧❡♠ t❤❛t a + b ♠✐❣❤t ♦✈❡r✢♦✇ t❤❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❛r✐s❡s✳ ❖♥❡ ❝♦✉❧❞ t❡st ❢♦r t❤✐s ❝❛s❡✱ t❤❡♥ t❡st ✐❢ a + b ≥ m✱ ❛♥❞ s✉❜tr❛❝t m ✐❢ ❡✐t❤❡r ✐s tr✉❡✱ ❜✉t t❤✐s ♥❡❝❡ss✐t❛t❡s t✇♦ t❡sts✳ ❲✐t❤ ❛ s❧✐❣❤t r❡❛rr❛♥❣❡♠❡♥t✱ ✇❡ ❝❛♥ ❞♦ ✇✐t❤ ♦♥❡✿ r := a + b❀ ✶ s := a − m❀ ✷ t := s + b❀ ✸ ✐❢ ❧❛st ❛❞❞✐t✐♦♥ s❡t ❝❛rr② ✢❛❣ t❤❡♥ ✹ r ✿❂ t❀ ✺ ❆❧❧ ❛r✐t❤♠❡t✐❝ ✐♥ t❤✐s ❝♦❞❡ ✐s ❛ss✉♠❡❞ ♠♦❞✉❧♦ t❤❡ ✇♦r❞ ❜❛s❡ β, ✐✳❡✳✱ t❤❡ ✐♥t❡❣❡rs ✐♥ r, s, ❛♥❞ t ❛r❡ r❡❞✉❝❡❞ r❡s✐❞✉❡s ♠♦❞✉❧♦ β✳ ■♥ ❧✐♥❡ ✷✱ s✐♥❝❡ a ✐s r❡❞✉❝❡❞ ♠♦❞✉❧♦ m✱ t❤❡ s✉❜tr❛❝t✐♦♥ a − m ♥❡❝❡ss❛r✐❧② ♣r♦❞✉❝❡s ❛ ❜♦rr♦✇✱ s♦ t❤❛t s = a − m + β✳ ■♥ ❧✐♥❡ ✸✱ ✐❢ s + b < β✱ t❤❡♥ t❤✐s ❛❞❞✐t✐♦♥ ❞♦❡s ♥♦t ♣r♦❞✉❝❡ ❛ ❝❛rr②✱ ❛♥❞ t = a + b − m + β < β✱ ✐✳❡✳✱ a + b − m < 0✳ ■❢ s + b ≥ β✱ t❤❡ ❛❞❞✐t✐♦♥ ❞♦❡s ♣r♦❞✉❝❡ ❛ ❝❛rr②✱ ❛♥❞ 0 ≤ t = s + b − β = a + b − m. ❍❡♥❝❡ t ✐s t❤❡ ♣r♦♣❡r r❡s✉❧t ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛ ❝❛rr② ♦❝❝✉rs ✐♥ ❧✐♥❡ ✸✱ t♦ ♠❛❦❡ ✉♣ ❢♦r t❤❡ ❜♦rr♦✇ ♦❢ ❧✐♥❡ ✷✳ ▲✐♥❡s ✶ ❛♥❞ ✷ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❝❛♥ ❜❡ ❡①❡❝✉t❡❞ ✐♥ ♣❛r❛❧❧❡❧✱ ❧❡❛❞✐♥❣ t♦ ❛ ❞❡♣❡♥❞❡♥t ❝❤❛✐♥ ♦❢ ❧❡♥❣t❤ ✸✳ ❲❡ r❡q✉✐r❡ a < m ❢♦r ❝♦rr❡❝t♥❡ss✱ ✐❢ b ≥ m, t❤❡ r❡s✉❧t st✐❧❧ s❛t✐s✜❡s r ≡ a + b (mod m) ❛♥❞ r < b, ❜✉t ♥♦t ♥❡❝❡ss❛r✐❧② r < m. ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ❈ ✇✐t❤ ❛ ●❈❈ ①✽✻ ❛ss❡♠❜❧② ❜❧♦❝❦ s❤♦✇♥ ❜❡❧♦✇✳ ❚❤❡ ✈❛❧✉❡ ♦❢ s✱ s❤♦✇♥ s❡♣❛r❛t❡❧② ❢♦r ❝❧❛r✐t② ❛❜♦✈❡✱ ✐s st♦r❡❞ ✐♥ t ❤❡r❡✳ r ❂ ❛ ✰ ❜❀ t ❂ ❛ ✲ ♠❀ ❴❴❛s♠❴❴ ✭ ✧❛❞❞ ✪✷✱ ✪✶❭♥❭t✧ ✴✯ t ✿❂ t ✰ ❜ ✯✴ ✧❝♠♦✈❝ ✪✶✱ ✪✵❭♥❭t✧ ✴✯ ✐❢ ✭❝❛rr②✮ r ✿❂ t ✯✴ ✿ ✧✰r✧ ✭r✮✱ ✧✰✫r✧ ✭t✮ ✿ ✧❣✧ ✭❜✮ ✿ ✧❝❝✧ ✮❀ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ t ❛♥❞ r ❛r❡ ❞♦♥❡ ✐♥ ❈✱ t♦ ❣✐✈❡ t❤❡ ❝♦♠♣✐❧❡r s♦♠❡ s❝❤❡❞✉❧✐♥❣ ❢r❡❡❞♦♠✳ ❙✐♥❝❡ ❈ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ❞✐r❡❝t ❛❝❝❡ss t♦ t❤❡ ❝❛rr② ✢❛❣✱ t❤❡ ❛❞❞✐t✐♦♥ t := t+b ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ ❛ss✐❣♥♠❡♥t ❛r❡ ❞♦♥❡ ✐♥ ❛ss❡♠❜❧②✳ ❚❤❡ ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❞❛t❛ ♣❛ss❡❞ t♦ t❤❡ ❛ss❡♠❜❧② ❜❧♦❝❦ st❛t❡ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ r ❛♥❞ t ♠✉st r❡s✐❞❡ ✐♥ r❡❣✐st❡rs ✭✧r✧✮ s✐♥❝❡ t❤❡ t❛r❣❡t ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠♦✈❡ ✐♥str✉❝t✐♦♥ ❝♠♦✈❝ ♠✉st ❜❡ ❛ r❡❣✐st❡r✱ ❛♥❞ ❛t ❧❡❛st ♦♥❡ ♦❢ s♦✉r❝❡ ♦r t❛r❣❡t ♦❢ t❤❡ ❛❞❞✐t✐♦♥ ✐♥str✉❝t✐♦♥ ❛❞❞ ♠✉st ❜❡ ❛ r❡❣✐st❡r✳ ❲❡ ❛❧❧♦✇ t❤❡ ✈❛r✐❛❜❧❡ ❜ t♦ ❜❡ ♣❛ss❡❞ ✐♥ ❛ r❡❣✐st❡r✱ ✐♥ ♠❡♠♦r② ♦r ❛s ❛♥ ✐♠♠❡❞✐❛t❡ ♦♣❡r❛♥❞ ✭✧❣✧✱ ✏❣❡♥❡r❛❧✑ ❝♦♥str❛✐♥t✱ ❢♦r ①✽✻❴✻✹ t❤❡ ❝♦rr❡❝t ❝♦♥str❛✐♥t ✐s ✧r♠❡✧ s✐♥❝❡ ✐♠♠❡❞✐❛t❡ ❝♦♥st❛♥ts ❛r❡ ♦♥❧② ✸✷ ❜✐t ✇✐❞❡✮✱ ✇❤✐❝❤ ✐s t❤❡ s♦✉r❝❡ ♦♣❡r❛♥❞ t♦ t❤❡ ❛❞❞ ✐♥str✉❝t✐♦♥✳ ❚❤❡ ✧✰✧ ♠♦❞✐✜❡r t❡❧❧s t❤❛t t❤❡ ✈❛❧✉❡s ✐♥ r ❛♥❞ t ✇✐❧❧ ❜❡ ♠♦❞✐✜❡❞✱ ❛♥❞ t❤❡ ✧✫✧ ♠♦❞✐✜❡r t❡❧❧s t❤❛t t ♠❛② ❜❡ ♠♦❞✐✜❡❞ ❜❡❢♦r❡ t❤❡ ❡♥❞ ♦❢ t❤❡ ❛ss❡♠❜❧② ❜❧♦❝❦ ❛♥❞ t❤✉s ♥♦
❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✶ ♦t❤❡r ✐♥♣✉t ✈❛r✐❛❜❧❡ s❤♦✉❧❞ ❜❡ ♣❛ss❡❞ ✐♥ t❤❡ r❡❣✐st❡r ❛ss✐❣♥❡❞ t♦ t✱ ❡✈❡♥ ✐❢ t❤❡✐r ✈❛❧✉❡s ❛r❡ ❦♥♦✇♥ t♦ ❜❡ ✐❞❡♥t✐❝❛❧✳ ❋✐♥❛❧❧②✱ ✧❝❝✧ t❡❧❧s t❤❡ ❝♦♠♣✐❧❡r t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✢❛❣s r❡❣✐st❡r ♠❛② ❝❤❛♥❣❡✳ ❚❤❡s❡ ❝♦♥str❛✐♥ts ♣r♦✈✐❞❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥ t❤❡ ❝♦♠♣✐❧❡r ♥❡❡❞s t♦ ❜❡ ❛❜❧❡ t♦ ✉s❡ t❤❡ ❛ss❡♠❜❧② ❜❧♦❝❦ ❝♦rr❡❝t❧②✱ ✇❤✐❧❡ ❧❡❛✈✐♥❣ ❡♥♦✉❣❤ ✢❡①✐❜✐❧✐t② t❤❛t ✐t ❝❛♥ ♦♣t✐♠✐③❡ r❡❣✐st❡r ❛❧❧♦❝❛t✐♦♥ ❛♥❞ ❞❛t❛ ♠♦✈❡♠❡♥t✱ ❝♦♠♣❛r❡❞ t♦✱ ❡✳❣✳✱ ❝♦♠♣✐❧❡rs t❤❛t r❡q✉✐r❡ ❛❧❧ ♣❛r❛♠❡t❡rs t♦ ❛ss❡♠❜❧② ❜❧♦❝❦s ✐♥ ❛ ✜①❡❞ s❡t ♦❢ r❡❣✐st❡rs✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ s♦❧✉t✐♦♥ ✐s t♦ ❝♦♠♣✉t❡ r := b − (m − a) ❛♥❞ ❛❞❞✐♥❣ m ✐❢ t❤❡ ♦✉t❡r s✉❜tr❛❝t✐♦♥ ♣r♦❞✉❝❡❞ ❛ ❜♦rr♦✇✳ ❍♦✇❡✈❡r✱ t❤✐s r❡q✉✐r❡s ❛ ❝♦♥❞✐t✐♦♥❛❧ ❛❞❞✐t✐♦♥ r❛t❤❡r t❤❛♥ ❛ ❝♦♥❞✐t✐♦♥❛❧ ♠♦✈❡✳ ❙✐♠✐❧❛r t♦ t❤❡ ♠♦❞✉❧❛r ❛❞❞✐t✐♦♥✱ ✈❛r✐♦✉s ❢✉♥❝t✐♦♥s s✉❝❤ ❛s ♠♦❞✉❧❛r s✉❜✲ tr❛❝t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❢♦r ♦♥❡ ❛♥❞ t✇♦✲✇♦r❞ ♠♦❞✉❧✐✱ t✇♦✲✇♦r❞ ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛♥❞ ❜✐♥❛r② s❤✐❢t✱ ❛♥❞ ❞✐✈✐s✐♦♥ ✇✐t❤ ❛ t✇♦✲✇♦r❞ ❞✐✈✲ ✐❞❡♥❞ ✭✉s❡❞✱ ❢♦r ❡①❛♠♣❧❡✱ ❢♦r ♣r❡♣❛r✐♥❣ ❛ r❡s✐❞✉❡ ❢♦r ✉s❡ ✇✐t❤ ❘❊❉❈ ♠♦❞✉❧❛r r❡❞✉❝t✐♦♥ ✇✐t❤ ❛ t✇♦✲✇♦r❞ ♠♦❞✉❧✉s✱ s❡❡ ✸✳✷✮ ❛r❡ ✇r✐tt❡♥ ❛s ❢✉♥❝t✐♦♥s ✇✐t❤ ❛s✲ s❡♠❜❧② s✉♣♣♦rt✳ ❆s ♦♣t✐♠✐③❛t✐♦♥ ❡✛♦rt ♣r♦❣r❡ss❡s✱ ♠♦r❡ t✐♠❡✲❝r✐t✐❝❛❧ ❢✉♥❝t✐♦♥s ❝✉rr❡♥t❧② ✇r✐tt❡♥ ✐♥ ❈ ✇✐t❤ ❛ss❡♠❜❧② ♠❛❝r♦s ✇✐❧❧ ❜❡ r❡♣❧❛❝❡❞ ❜② ❞❡❞✐❝❛t❡❞ ❛s✲ s❡♠❜❧② ❝♦❞❡✳
✸✳✷ ▼♦❞✉❧❛r r❡❞✉❝t✐♦♥ ✇✐t❤ ❘❊❉❈
▼♦♥t❣♦♠❡r② ♣r❡s❡♥t❡❞ ✐♥ ❬✶✼❪ ❛ ♠❡t❤♦❞ ❢♦r ❢❛st ♠♦❞✉❧❛r r❡❞✉❝t✐♦♥✳ ●✐✈❡♥ ❛♥ ✐♥t❡❣❡r 0 ≤ a < βm✱ ❢♦r ♦❞❞ ♠♦❞✉❧✉s m ♦❢ ♦♥❡ ♠❛❝❤✐♥❡ ✇♦r❞ ❛♥❞ ♠❛✲ ❝❤✐♥❡ ✇♦r❞ ❜❛s❡ β ✭❤❡r❡ ❛ss✉♠❡❞ ❛ ♣♦✇❡r ♦❢ 2✮✱ ❛♥❞ ❛ ♣r❡❝♦♠♣✉t❡❞ ❝♦♥st❛♥t m✐♥✈= −m−1mod β✱ ✐t ❝♦♠♣✉t❡s ❛♥ ✐♥t❡❣❡r 0 ≤ r < m ✇❤✐❝❤ s❛t✐s✜❡s rβ ≡ a (mod m). ■t ❞♦❡s s♦ ❜② ❝♦♠♣✉t✐♥❣ t❤❡ ♠✐♥✐♠❛❧ ♥♦♥✲♥❡❣❛t✐✈❡ tm s✉❝❤ t❤❛t a + tm ≡ 0 (mod β), t♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❞✐✈✐s✐♦♥ ❜② β ✐s ✈❡r② ✐♥✲ ❡①♣❡♥s✐✈❡✳ ❙✐♥❝❡ t < β✱ (a + tm)/β < 2m, ❛♥❞ ❛t ♠♦st ♦♥❡ ✜♥❛❧ s✉❜tr❛❝t✐♦♥ ♦❢ m❡♥s✉r❡s r < m. ❍❡ ❝❛❧❧s t❤❡ ❛❧❣♦r✐t❤♠ t❤❛t ❝❛rr✐❡s ♦✉t t❤✐s r❡❞✉❝t✐♦♥ ✏❘❊❉❈✱✑ s❤♦✇♥ ✐♥ ❆❧❣♦r✐t❤♠ ✷✳ ■♥♣✉t✿ m✱ t❤❡ ♠♦❞✉❧✉s β✱ t❤❡ ✇♦r❞ ❜❛s❡ a < βm✱ ✐♥t❡❣❡r t♦ r❡❞✉❝❡ m✐♥✈< β s✉❝❤ t❤❛t mm✐♥✈≡ −1 (mod β) ❖✉t♣✉t✿ r < m ✇✐t❤ rβ ≡ a (mod m) t := a · m✐♥✈mod β❀ r := (a + t · m)/β❀ ✐❢ r ≥ m t❤❡♥ r := r − m❀ ❆❧❣♦r✐t❤♠ ✷✿ ❆❧❣♦r✐t❤♠ ❘❊❉❈ ❢♦r ♠♦❞✉❧❛r r❡❞✉❝t✐♦♥ ✇✐t❤ ♦♥❡✲✇♦r❞ ♠♦❞✲ ✉❧✉s✳ ❆❧❧ ✈❛r✐❛❜❧❡s t❛❦❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡r ✈❛❧✉❡s✳ ❚❤❡ r❡❞✉❝❡❞ r❡s✐❞✉❡ ♦✉t♣✉t ❜② t❤✐s ❛❧❣♦r✐t❤♠ ✐s ♥♦t ✐♥ t❤❡ s❛♠❡ r❡s✐❞✉❡ ❝❧❛ss mod m ❛s t❤❡ ✐♥♣✉t✱ ❜✉t t❤❡ r❡s✐❞✉❡ ❝❧❛ss ❣❡ts ♠✉❧t✐♣❧✐❡❞ ❜② β−1 (mod m) ✐♥ t❤❡ ♣r♦❝❡ss✳ ❚♦ ♣r❡✈❡♥t ❛❝❝✉♠✉❧❛t✐♥❣ ♣♦✇❡rs ♦❢ β−1 (mod m)❛♥❞ ❤❛✈✐♥❣ ✉♥✲ ❡q✉❛❧ ♣♦✇❡rs ♦❢ β ✇❤❡♥✱ ❡✳❣✳✱ ❛❞❞✐♥❣ ♦r ❝♦♠♣❛r✐♥❣ r❡s✐❞✉❡s✱ ❛♥② r❡s✐❞✉❡ ♠♦❞✉❧♦ m✐s ❝♦♥✈❡rt❡❞ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ✜rst✱ ❜② ♠✉❧t✐♣❧②✐♥❣ ✐t ❜② β ❛♥❞ r❡❞✉❝✐♥❣ ✭✇✐t❤♦✉t ❘❊❉❈✮ ♠♦❞✉❧♦ m✱ ✐✳❡✳✱ t❤❡ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛✶✷ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛
r❡s✐❞✉❡ a (mod m) ✐s aβ (mod m)✳ ❚❤✐s ✇❛②✱ ✐❢ t✇♦ r❡s✐❞✉❡s ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ aβ (mod m) ❛♥❞ bβ (mod m) ❛r❡ ♠✉❧t✐♣❧✐❡❞ ❛♥❞ r❡❞✉❝❡❞ ✈✐❛ ❘❊❉❈✱ t❤❡♥ ❘❊❉❈(aβbβ) ≡ abβ (mod m) ✐s t❤❡ ♣r♦❞✉❝t ✐♥ ▼♦♥t❣♦♠❡r② r❡♣✲ r❡s❡♥t❛t✐♦♥✳ ❚❤✐s ❡♥s✉r❡s t❤❡ ❡①♣♦♥❡♥t ♦❢ β ✐♥ t❤❡ r❡s✐❞✉❡s ❛❧✇❛②s st❛②s 1✱ ❛♥❞ s♦ ❛❧❧♦✇s ❛❞❞✐t✐♦♥✱ s✉❜tr❛❝t✐♦♥✱ ❛♥❞ ❡q✉❛❧✐t② t❡sts ♦❢ r❡s✐❞✉❡s ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥✳ ❙✐♥❝❡ β ⊥ m✱ ✇❡ ❛❧s♦ ❤❛✈❡ aβ ≡ 0 (mod m) ✐❢ ❛♥❞ ♦♥❧② ✐❢ a ≡ 0 (mod m)✱ ❛♥❞ gcd(aβ, m) = gcd(a, m). ❙✐♥❝❡ β = 232♦r 264✐s ❛♥ ✐♥t❡❣❡r sq✉❛r❡✱
t❤❡ ❏❛❝♦❜✐ s②♠❜♦❧ s❛t✐s✜❡s aβ m = a m. ❋♦r ♠♦❞✉❧✐ m ♦❢ ♠♦r❡ t❤❛♥ ♦♥❡ ♠❛❝❤✐♥❡ ✇♦r❞✱ s❛② m < βk✱ ❛ ♣r♦❞✉❝t ♦❢ t✇♦ r❡❞✉❝❡❞ r❡s✐❞✉❡s ♠❛② ❡①❝❡❡❞ β✱ ❜✉t ✐s ❜❡❧♦✇ mβk✳ ❚❤❡ r❡❞✉❝t✐♦♥ ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t ✐♥ t✇♦ ✇❛②s✿ ♦♥❡ ❡ss❡♥t✐❛❧❧② ♣❡r❢♦r♠s t❤❡ ♦♥❡✲✇♦r❞ ❘❊❉❈ r❡❞✉❝t✐♦♥ k t✐♠❡s✱ ♣❡r❢♦r♠✐♥❣ O k2 ♦♥❡✲✇♦r❞ ♠✉❧t✐♣❧✐❡s✱ t❤❡ ♦t❤❡r r❡♣❧❛❝❡s ❛r✐t❤♠❡t✐❝ ♠♦❞✉❧♦ β ✐♥ ❘❊❉❈ ❜② ❛r✐t❤♠❡t✐❝ ♠♦❞✉❧♦ βk✱ ♣❡r❢♦r♠✐♥❣ O(1) k✲✇♦r❞ ♠✉❧t✐✲ ♣❧✐❝❛t✐♦♥s✳ ■♥ ❡✐t❤❡r ❝❛s❡✱ ❛ ❢✉❧❧ r❡❞✉❝t✐♦♥ ✇✐t❤ ✭r❡♣❡❛t❡❞ ♦♥❡✲✇♦r❞ ♦r ❛ s✐♥❣❧❡ ♠✉❧t✐✲✇♦r❞✮ ❘❊❉❈ ❞✐✈✐❞❡s t❤❡ r❡s✐❞✉❡ ❝❧❛ss ♦❢ t❤❡ ♦✉t♣✉t ❜② βk, ❛♥❞ t❤❡ ❝♦♥✲ ✈❡rs✐♦♥ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ♠✉st ♠✉❧t✐♣❧② ❜② βk ❛❝❝♦r❞✐♥❣❧②✳ ❚❤❡ ❢♦r♠❡r ♠❡t❤♦❞ ❤❛s ❧♦✇❡r ♦✈❡r❤❡❛❞ ❛♥❞ ✐s ♣r❡❢❡r❛❜❧❡ ❢♦r s♠❛❧❧ ♠♦❞✉❧✐✱ t❤❡ ❧❛tt❡r ❝❛♥ ✉s❡ ❛s②♠♣t♦t✐❝❛❧❧② ❢❛st ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✐❢ t❤❡ ♠♦❞✉❧✉s ✐s ❧❛r❣❡✳ ❆s ✐♥ ♦✉r ❛♣♣❧✐❝❛t✐♦♥ t❤❡ ♠♦❞✉❧✐ ❛r❡ q✉✐t❡ s♠❛❧❧✱ ♥♦ ♠♦r❡ t❤❛♥ t✇♦ ♠❛❝❤✐♥❡ ✇♦r❞s✱ ✇❡ ✉s❡ t❤❡ ❢♦r♠❡r ♠❡t❤♦❞✳ ❇❡❢♦r❡ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ✇✐t❤ ❘❊❉❈ ❢♦r ❛ ♣❛rt✐❝✉❧❛r m ❝❛♥ ❜❡❣✐♥✱ t❤❡ ❝♦♥st❛♥t m✐♥✈ ♥❡❡❞s t♦ ❜❡ ❝♦♠♣✉t❡❞✳ ■❢ β ✐s ❛ ♣♦✇❡r ♦❢ 2✱ ❍❡♥s❡❧ ❧✐❢t✐♥❣ ♠❛❦❡s t❤✐s ❝♦♠♣✉t❛t✐♦♥ ✈❡r② ❢❛st✳ ❚♦ s♣❡❡❞ ✐t ✉♣ ❢✉rt❤❡r✱ ✇❡ tr② t♦ ❣✉❡ss ❛♥ ❛♣♣r♦①✐✲ ♠❛t✐♦♥ t♦ m✐♥✈s♦ t❤❛t ❛ ❢❡✇ ❧❡❛st s✐❣♥✐✜❝❛♥t ❜✐ts ❛r❡ ❝♦rr❡❝t✱ t❤✉s s❛✈✐♥❣ ❛ ❢❡✇ ◆❡✇t♦♥ ✐t❡r❛t✐♦♥s✳ ❚❤❡ sq✉❛r❡ ♦❢ ❛♥② ♦❞❞ ✐♥t❡❣❡r ✐s ❝♦♥❣r✉❡♥t t♦ 1 (mod 8)✱ s♦ m✐♥✈ ≡ m (mod 8). ❚❤❡ ❢♦✉rt❤ ❜✐t ♦❢ m✐♥✈ ✐s ❡q✉❛❧ t♦ t❤❡ ❜✐♥❛r② ❡①❝❧✉s✐✈❡✲♦r ♦❢ t❤❡ s❡❝♦♥❞✱ t❤✐r❞✱ ❛♥❞ ❢♦✉rt❤ ❜✐t ♦❢ m✱ ❜✉t ♦♥ ♠❛♥② ♠✐❝r♦♣r♦❝❡ss♦rs ❛♥ ❛❧✲ t❡r♥❛t✐✈❡ s✉❣❣❡st✐♦♥ ❢r♦♠ ▼♦♥t❣♦♠❡r② ❬✷✷❪ ✐s s❧✐❣❤t❧② ❢❛st❡r✿ (3m) ❳❖❘ 2 ❣✐✈❡s t❤❡ ❧♦✇ 5 ❜✐ts ♦❢ m✐♥✈❝♦rr❡❝t❧②✳ ❊❛❝❤ ◆❡✇t♦♥ ✐t❡r❛t✐♦♥ x 7→ 2x − x2m❞♦✉❜❧❡s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦rr❡❝t ❜✐ts✱ s♦ t❤❛t ✇✐t❤ ❡✐t❤❡r ❛♣♣r♦①✐♠❛t✐♦♥✱ 3 ✐t❡r❛t✐♦♥s ❢♦r β = 232 ♦r 4 ❢♦r β = 264 s✉✣❝❡✳ ❈♦♥✈❡rt✐♥❣ r❡s✐❞✉❡s ♦✉t ♦❢ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ q✉✐❝❦❧② ✇✐t❤ ❘❊❉❈✱ ❜✉t ❝♦♥✈❡rt✐♥❣ t❤❡♠ t♦ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ r❡✲ q✉✐r❡s ❛♥♦t❤❡r ♠♦❞✉❧❛r r❡❞✉❝t✐♦♥ ❛❧❣♦r✐t❤♠✳ ■❢ s✉❝❤ ❝♦♥✈❡rs✐♦♥s ❛r❡ t♦ ❜❡ ❞♦♥❡ ❢r❡q✉❡♥t❧②✱ ✐t ♣❛②s t♦ ♣r❡❝♦♠♣✉t❡ ℓ = β2mod m✱ s♦ t❤❛t ❘❊❉❈(aℓ) = aβ mod m❛❧❧♦✇s ✉s✐♥❣ ❘❊❉❈ ❢♦r t❤❡ ♣✉r♣♦s❡✳ ■♥ s♦♠❡ ❝❛s❡s✱ t❤❡ ✜♥❛❧ ❝♦♥❞✐t✐♦♥❛❧ s✉❜tr❛❝t✐♦♥ ♦❢ m ✐♥ ❘❊❉❈ ❝❛♥ ❜❡ ♦♠✐t✲ t❡❞✳ ■❢ a < m✱ t❤❡♥ a + tm < mβ s✐♥❝❡ t < β, s♦ r = (a + tm)/β < m ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ ✇❤❡♥ ❝♦♥✈❡rt✐♥❣ r❡s✐❞✉❡s ♦✉t ♦❢ ▼♦♥t❣♦♠❡r② ❢♦r♠✱ ♦r ✇❤❡♥ ❞✐✈✐s✐♦♥ ❜② ❛ ♣♦✇❡r ♦❢ 2 ♠♦❞✉❧♦ m ✐s ❞❡s✐r❡❞✳
✸✳✸ ▼♦❞✉❧❛r ✐♥✈❡rs❡
❚♦ ❝♦♠♣✉t❡ ❛ ♠♦❞✉❧❛r ✐♥✈❡rs❡ r ≡ a−1 (mod m) ❢♦r ❛ ❣✐✈❡♥ r❡❞✉❝❡❞ r❡s✐❞✉❡ a ❛♥❞ ♦❞❞ ♠♦❞✉❧✉s m ✇✐t❤ a ⊥ m✱ ✇❡ ✉s❡ ❛ ❜✐♥❛r② ❡①t❡♥❞❡❞ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦r✐t❤♠✳ ▼♦❞✉❧❛r ✐♥✈❡rs❡s ❛r❡ ✉s❡❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ st❛❣❡ ✷ ❢♦r t❤❡ P✕✶ ❛❧❣♦r✐t❤♠✱ ❛♥❞ ❢♦r ✐♥✐t✐❛❧✐s❛t✐♦♥ ♦❢ st❛❣❡ ✶ ♦❢ ❊❈▼ ✭❡①❝❡♣t ❢♦r ❛ s❡❧❡❝t ❢❡✇ ❝✉r✈❡s ✇❤✐❝❤ ❤❛✈❡ s✐♠♣❧❡ ❡♥♦✉❣❤ ♣❛r❛♠❡t❡rs t❤❛t t❤❡② ❝❛♥ ❜❡ ✐♥✐t✐❛❧✐s❡❞ ✉s✐♥❣ ♦♥❧② ❞✐✈✐s✐♦♥ ❜② s♠❛❧❧ ❝♦♥st❛♥ts✮✳ ❖✉r ❝♦❞❡ ❢♦r ❛ ♠♦❞✉❧❛r ✐♥✈❡rs❡ t❛❦❡s ❛❜♦✉t 0.5µs ❢♦r ♦♥❡✲ ✇♦r❞ ♠♦❞✉❧✐✱ ✇❤✐❝❤ ✐♥ ❝❛s❡ ♦❢ P✕✶ ✇✐t❤ s♠❛❧❧ B1 ❛♥❞ B2 ♣❛r❛♠❡t❡rs ❛❝❝♦✉♥ts❆ ❙♦❢t✇❛r❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❊❈▼ ❢♦r ◆❋❙ ✶✸ ❢♦r s❡✈❡r❛❧ ♣❡r❝❡♥t ♦❢ t❤❡ t♦t❛❧ r✉♥✲t✐♠❡✱ s❤♦✇✐♥❣ t❤❛t s♦♠❡ ♦♣t✐♠✐③❛t✐♦♥ ❡✛♦rt ✐s ✇❛rr❛♥t❡❞ ❢♦r t❤✐s ❢✉♥❝t✐♦♥✳ ❚❤❡ ❡①t❡♥❞❡❞ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ar + ms = gcd(a, m) ❢♦r ❣✐✈❡♥ a, m ❜② ✐♥✐t✐❛❧✐s✐♥❣ e0 = 0, f0 = 1, g0 = m ❛♥❞ e1= 1, f1= 0, g1= a, ❛♥❞ ❝♦♠♣✉t✐♥❣ s❡q✉❡♥❝❡s ei, fi ❛♥❞ gi t❤❛t ♠❛✐♥t❛✐♥ aei+ mfi= gi ✭✶✮ ✇❤❡r❡ gcd(a, m) | gi ❛♥❞ t❤❡ gi❛r❡ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✉♥t✐❧ gi = 0✳ ❚❤❡ ♦r✐❣✐♥❛❧
❊✉❝❧✐❞❡❛♥ ❛❧❣♦r✐t❤♠ ✉s❡s gi= gi−2mod gi−1,t❤❛t ✐s✱ ✐♥ ❡❛❝❤ st❡♣ ✇❡ ✇r✐t❡ gi=
gi−2−gi−1⌊gi−2
gi−1⌋ ❛♥❞ ❧✐❦❡✇✐s❡ ei= ei−2−ei−1⌊ gi−2 gi−1⌋ ❛♥❞ fi= fi−2−fi−1⌊ gi−2 gi−1⌋, s♦ t❤❛t ❡q✉❛t✐♦♥ ✭✶✮ ❤♦❧❞s ❢♦r ❡❛❝❤ i. ■❢ n ✐s t❤❡ s♠❛❧❧❡st i s✉❝❤ t❤❛t gi = 0, t❤❡♥ gn−1 = gcd(a, m)✱ s = fn−1, ❛♥❞ r = en−1. ❙✐♥❝❡ ✇❡ ♦♥❧② ✇❛♥t t❤❡ ✈❛❧✉❡ ♦❢ r = en−1, ✇❡ ❞♦♥✬t ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ fi ✈❛❧✉❡s✳ ❲❡ ❝❛♥ ✇r✐t❡ u = ei−1, v = ei, x = gi−1, y = gi ❛♥❞ ❢♦r i = 1 ✐♥✐t✐❛❧✐s❡ u = 0, v = 1, x = m, ❛♥❞ y = a. ❚❤❡♥ ❡❛❝❤ ✐t❡r❛t✐♦♥ i 7→ i + 1 ✐s ❝♦♠♣✉t❡❞ ❜② (u, v, x, y) := (v, u − ⌊x/y⌋v, y, x − ⌊x/y⌋y).
❆t t❤❡ ✜rst ✐t❡r❛t✐♦♥ ✇❤❡r❡ y = 0✱ ✇❡ ❤❛✈❡ r = u ❛♥❞ x = 1 ✐❢ a ❛♥❞ m ✇❡r❡ ✐♥❞❡❡❞ ❝♦♣r✐♠❡✳ ❆ ♣r♦❜❧❡♠ ✇✐t❤ t❤✐s ❛❧❣♦r✐t❤♠ ✐s t❤❡ ❝♦st❧② ❝♦♠♣✉t❛t✐♦♥ ♦❢ ⌊x/y⌋ ❛s ✐♥t❡❣❡r ❞✐✈✐s✐♦♥ ✐s ✉s✉❛❧❧② s❧♦✇✳ ❚❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ ❊✉❝❧✐❞❡❛♥ ❛❧❣♦r✐t❤♠ ❛✈♦✐❞s t❤✐s ♣r♦❜❧❡♠ ❜② ✉s✐♥❣ ♦♥❧② s✉❜tr❛❝t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❜② ♣♦✇❡rs ♦❢ 2. ❖✉r ✐♠♣❧❡♠❡♥✲ t❛t✐♦♥ ✐s ✐♥s♣✐r❡❞ ❜② ❝♦❞❡ ✇r✐tt❡♥ ❜② ❘♦❜❡rt ❍❛r❧❡② ❢♦r t❤❡ ❊❈❈♣✲✾✼ ❝❤❛❧❧❡♥❣❡ ❛♥❞ ✐s s❤♦✇♥ ✐♥ ❆❧❣♦r✐t❤♠ ✸✳ ❚❤❡ ✉♣❞❛t❡s ♠❛✐♥t❛✐♥ ua ≡ −x2t (mod m) ❛♥❞ va ≡ y2t (mod m)s♦ t❤❛t ✇❤❡♥ y = 1, ✇❡ ❤❛✈❡ r = v2−t= a−1 (mod m). ■♥♣✉t✿ ❖❞❞ ♠♦❞✉❧✉s m ❘❡❞✉❝❡❞ r❡s✐❞✉❡ a (mod m) ❖✉t♣✉t✿ ❘❡❞✉❝❡❞ r❡s✐❞✉❡ r (mod m) ✇✐t❤ ar ≡ 1 (mod m)✱ ♦r ❢❛✐❧✉r❡ ✐❢ gcd(a, m) > 1 ✐❢ a = 0 t❤❡♥ r❡t✉r♥ ❢❛✐❧✉r❡❀ t := Val2(a)❀ ✴✯ 2t|| a ✯✴ u := 0; v := 1; x := m; y := a/2t❀ ✇❤✐❧❡ x 6= y ❞♦ ℓ := Val2(x − y)❀ ✴✯ 2ℓ|| x − y ✯✴ ✐❢ x < y t❤❡♥ (u, v, x, y, t) := (u2ℓ, u + v, x, (y − x)/2ℓ, t + ℓ)❀ ❡❧s❡ (u, v, x, y, t) := (u + v, v2ℓ, (x − y)/2ℓ, y, t + ℓ)❀ ✐❢ y 6= 1 t❤❡♥ r❡t✉r♥ ❢❛✐❧✉r❡❀ r := v2−tmod m❀ ❆❧❣♦r✐t❤♠ ✸✿ ❇✐♥❛r② ❡①t❡♥❞❡❞ ●❈❉ ❛❧❣♦r✐t❤♠✳ ■♥ ❡❛❝❤ st❡♣ ✇❡ s✉❜tr❛❝t t❤❡ s♠❛❧❧❡r ♦❢ x, y ❢r♦♠ t❤❡ ❧❛r❣❡r✱ s♦ t❤❡② ❛r❡ ❞❡❝r❡❛s✐♥❣ ❛♥❞ ♥♦♥✲♥❡❣❛t✐✈❡✳ ◆❡✐t❤❡r ❝❛♥ ❜❡❝♦♠❡ ③❡r♦ ❛s t❤❛t ✐♠♣❧✐❡s x = y
✶✹ ❆❧❡①❛♥❞❡r ❑r✉♣♣❛ ✐♥ t❤❡ ♣r❡✈✐♦✉s ✐t❡r❛t✐♦♥✱ ✇❤✐❝❤ t❡r♠✐♥❛t❡s t❤❡ ❧♦♦♣✳ ❙✐♥❝❡ ❜♦t❤ ❛r❡ ♦❞❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ✐t❡r❛t✐♦♥✱ t❤❡✐r ❞✐✛❡r❡♥❝❡ ✐s ❡✈❡♥✱ s♦ ♦♥❡ ✈❛❧✉❡ ❞❡❝r❡❛s❡s ❜② ❛t ❧❡❛st ❛ ❢❛❝t♦r ♦❢ 2, ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✐s ❛t ♠♦st log2(am).■♥ ❡❛❝❤ ✐t❡r❛t✐♦♥✱ uy + vx = m, ❛♥❞ s✐♥❝❡ x ❛♥❞ y ❛r❡ ♣♦s✐t✐✈❡✱ u, v ≤ m s♦ t❤❛t ♥♦ ♦✈❡r✢♦✇ ♦❝❝✉rs ✇✐t❤ ✜①❡❞✲♣r❡❝✐s✐♦♥ ❛r✐t❤♠❡t✐❝✳ ❚♦ ♣❡r❢♦r♠ t❤❡ ♠♦❞✉❧❛r ❞✐✈✐s✐♦♥ r = v/2ti, ✇❡ ❝❛♥ ✉s❡ ❘❊❉❈✳ ❲❤✐❧❡ t ≥ log2(β),✇❡ r❡♣❧❛❝❡ v := ❘❊❉❈(v) ❛♥❞ t := t − log2(β).❚❤❡♥✱ ✐❢ t > 0✱ ✇❡ ♣❡r❢♦r♠ ❛ ✈❛r✐❛❜❧❡✲✇✐❞t❤ ❘❊❉❈ t♦ ❞✐✈✐❞❡ ❜② 2tr❛t❤❡r t❤❛♥ ❜② β ❜② ❝♦♠♣✉t✐♥❣ r = (v + ((vm✐♥✈) mod 2t) m) /2t ✇✐t❤ mm✐♥✈ ≡ −1 (mod β). ❙✐♥❝❡ v < m, ✇❡ ❞♦♥✬t ♥❡❡❞ ❛ ✜♥❛❧ s✉❜tr❛❝t✐♦♥ ✐♥ t❤❡s❡ ❘❊❉❈✳ ■❢ t❤❡ r❡s✐❞✉❡ a ✇❤♦s❡ ✐♥✈❡rs❡ ✇❡ ✇❛♥t ✐s ❣✐✈❡♥ ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥ aβkmod m ✇✐t❤ k✲✇♦r❞ ♠♦❞✉❧✉s m✱ ✇❡ ❝❛♥ ✉s❡ ❘❊❉❈ 2k t✐♠❡s t♦ ❝♦♠♣✉t❡ aβ−kmod m, t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ♠♦❞✉❧❛r ✐♥✈❡rs❡ t♦ ♦❜t❛✐♥ t❤❡ ✐♥✈❡rs❡ ♦❢ a ✐♥ ▼♦♥t❣♦♠❡r② r❡♣r❡s❡♥t❛t✐♦♥✿ a−1βk ≡ aβ−k−1 (mod m).❚❤✐s ❝❛♥ ❜❡ s✐♠♣❧✐✲ ✜❡❞ ❜② ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t t❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ ●❈❉ ❝♦♠♣✉t❡s v = a−12t.■❢ ✇❡ ❦♥♦✇ ❜❡❢♦r❡❤❛♥❞ t❤❛t t ≥ log2β, ✇❡ ❝❛♥ s❦✐♣ ❞✐✈✐s✐♦♥s ❜② β ✈✐❛ ❘❊❉❈ ❜♦t❤ ❜❡❢♦r❡ ❛♥❞ ❛❢t❡r t❤❡ ❜✐♥❛r② ❡①t❡♥❞❡❞ ●❈❉✳ ▲❡t t❤❡ ❢✉♥❝t✐♦♥ t(x, y) ❣✐✈❡ t❤❡ ✈❛❧✉❡ ♦❢ t ❛t t❤❡ ❡♥❞ ♦❢ ❆❧❣♦r✐t❤♠ ✸ ❢♦r ❝♦♣r✐♠❡ ✐♥♣✉ts x, y✳ ■t s❛t✐s✜❡s t(x, y) = 0 ✐❢ x = y (✐♠♣❧✐❡s x = y = 1), t(x/2, y) + 1 ✐❢ x 6= y, 2 | x, t(x − y, y) ✐❢ x > y, 2 ∤ x, t(y, x) ✐❢ x < y, 2 ∤ x. ❆ss✉♠✐♥❣ y ♦❞❞✱ ❝❛s❡ ✸ ✐s ❛❧✇❛②s ❢♦❧❧♦✇❡❞ ❜② ❝❛s❡ ✷✱ ❛♥❞ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ ❝❛s❡ ✸ ❜② t(x, y) = t((x − y)/2, y) + 1. ❲❡ ❝♦♠♣❛r❡ t❤❡ ❞❡❝r❡❛s❡ ♦❢ t❤❡ s✉♠ x + y ❛♥❞ t❤❡ ✐♥❝r❡❛s❡ ♦❢ t✳ ■♥ ❝❛s❡ ✷✱ (x + y) 7→ x/2 + y > (x + y)/2, ❛♥❞ t ✐♥❝r❡❛s❡s ❜② 1. ■♥ t❤❡ s✉❜st✐t✉t❡❞ ❝❛s❡ ✸✱ (x + y) 7→ (x + y)/2, ❛♥❞ t ✐♥❝r❡❛s❡s ❜② 1. ❲❡ s❡❡ t❤❛t ✇❤❡♥❡✈❡r x + y ❞❡❝r❡❛s❡s✱ t ✐♥❝r❡❛s❡s✱ ❛♥❞ ✇❤❡♥❡✈❡r t ✐♥❝r❡❛s❡s ❜② 1✱ x + y ❞r♦♣s ❜② ❛t ♠♦st ❤❛❧❢✱ ✉♥t✐❧ x + y = 2. ❍❡♥❝❡ t(x, y) ≥ log2(x + y) − 1, ❛♥❞ t❤❡r❡❢♦r❡ t(x, y) ≥ log2(y),s✐♥❝❡ x > 0. ❚❤✉s ✐♥ ❝❛s❡ ♦❢ k✲✇♦r❞ ♠♦❞✉❧✐ βk−1 < m < βk✱ ✇❡ ❤❛✈❡ t(x, m) ≥ (k −
1) log2(β) ❢♦r ❛♥② ♣♦s✐t✐✈❡ x✱ s♦ ✉s✐♥❣ aβ−1 (mod m) ❛s ✐♥♣✉t t♦ t❤❡ ❜✐♥❛r②
❡①t❡♥❞❡❞ ●❈❉ ✐s s✉✣❝✐❡♥t t♦ ❡♥s✉r❡ t❤❛t ❛t t❤❡ ❡♥❞ ✇❡ ❣❡t a−1β ≡ v2−t
(mod m)✱ ♦r a−1βk ≡ v2−t+(k−1) log2(β) (mod m)❛♥❞ t❤❡ ❞❡s✐r❡❞ r❡s✉❧t a−1βk
❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ v2−t ✇✐t❤ ❛ ❞✐✈✐s✐♦♥ ❜② 2t−(k−1) log2(β) ✈✐❛ ❘❊❉❈✳
