The 21 references with contexts in paper N. Bezverkhniy V., O. Chernisheva A., Н. Безверхний В., О. Чернышева А. (2016) “Односторонние функции, основанные на проблеме дискретного логарифмирования в группах с условиями C(3)-T(6) // One-way functions based on the discrete logarithm problem in the groups meeting conditions C(3)-T (6)” / spz:neicon:technomag:y:2014:i:0:p:70-101

1
Magnus W., Karras A., Solitar D.Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. John Wiley and Sons, Inc., New York-London-Sydney, 1966. 444 p. (Russ. ed.: Magnus W., Karras A., Solitar D.Kombinatornaja teorija grupp. Moscow, Nauka Publ., 1974. 456 p.).
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  1. In-text reference with the coordinate start=2119
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    –ÂÁÛÎ ̧Ú‡Ú ̊ ‰‡ÌÌÓ„Ó ËÒÒΉӂ‡Ìˡ Û͇Á ̊‚‡ ̨Ú Ì‡ Ì‚ÓÁÏÓÊÌÓÒÚ ̧ ÔÓÒÚÓÂÌˡ ÍËÔÚÓÒÚÓÈÍÓÈ ÒËÒÚÂÏ ̊ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Û͇Á‡ÌÌ ̊ı ‡Î„ÓËÚÏÓ‚ ‚ ‰‡ÌÌÓÏ Í·ÒÒ „ÛÔÔ. »ÒÒΉӂ‡Ìˡ ÔÓ‚Ó‰ˇÚÒˇ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ „ÂÓÏÂÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚ ÍÓÏ·Ë̇ÚÓÌÓÈ ÚÂÓËË „ÛÔÔ, ‡ ËÏÂÌÌÓ, ÏÂÚÓ‰‡ ‰Ë‡„‡ÏÏ Ì‡‰ „ÛÔÔ‡ÏË, ·‡ÁËÛ ̨ ̆„ÓÒˇ ̇ ÒÎÂ‰Û ̨ ̆Ëı ‰‚Ûı ÛÚ‚ÂʉÂÌˡı
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    . œÂ‚Ó | ÎÂÏχ ¬‡Ì ‡ÏÔÂ̇, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚Ó ‡‚ÌÓ Â‰ËÌˈ ‚ „ÛÔÔ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ Ó‰ÌÓÒ‚ˇÁ̇ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃ÌÓÈ ÏÂÚÍÓÈ, ‡‚ÌÓÈ ̋ÚÓÏÛ ÒÎÓ‚Û. ¬ÚÓÓ | ÎÂÏχ Ó ÒÓÔˇÊÂÌÌ ̊ı ̋ÎÂÏÂÌÚ‡ı „ÛÔÔ ̊, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚‡uËvÔ‰ÒÚ‡‚Ρ ̨Ú ÒÓÔˇÊÂÌÌ ̊ ̋ÎÂÏÂÌÚ ̊ ‰‡ÌÌÓÈ „ÛÔÔ ̊ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ ÍÓÎ ̧ˆÂ‚‡ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃Ì ̊ÏË ÏÂÚ͇Ï

2
Lyndon R., Schupp P.Combinatorial group theory. Springer-Verlag, Berlin, 1977. (Russ. ed.: Lyndon R., Schupp P.Kombinatornaja teorija grupp. Moscow, Mir Publ., 1980. 448 p.).
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    –ÂÁÛÎ ̧Ú‡Ú ̊ ‰‡ÌÌÓ„Ó ËÒÒΉӂ‡Ìˡ Û͇Á ̊‚‡ ̨Ú Ì‡ Ì‚ÓÁÏÓÊÌÓÒÚ ̧ ÔÓÒÚÓÂÌˡ ÍËÔÚÓÒÚÓÈÍÓÈ ÒËÒÚÂÏ ̊ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Û͇Á‡ÌÌ ̊ı ‡Î„ÓËÚÏÓ‚ ‚ ‰‡ÌÌÓÏ Í·ÒÒ „ÛÔÔ. »ÒÒΉӂ‡Ìˡ ÔÓ‚Ó‰ˇÚÒˇ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ „ÂÓÏÂÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚ ÍÓÏ·Ë̇ÚÓÌÓÈ ÚÂÓËË „ÛÔÔ, ‡ ËÏÂÌÌÓ, ÏÂÚÓ‰‡ ‰Ë‡„‡ÏÏ Ì‡‰ „ÛÔÔ‡ÏË, ·‡ÁËÛ ̨ ̆„ÓÒˇ ̇ ÒÎÂ‰Û ̨ ̆Ëı ‰‚Ûı ÛÚ‚ÂʉÂÌˡı
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    . œÂ‚Ó | ÎÂÏχ ¬‡Ì ‡ÏÔÂ̇, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚Ó ‡‚ÌÓ Â‰ËÌˈ ‚ „ÛÔÔ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ Ó‰ÌÓÒ‚ˇÁ̇ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃ÌÓÈ ÏÂÚÍÓÈ, ‡‚ÌÓÈ ̋ÚÓÏÛ ÒÎÓ‚Û. ¬ÚÓÓ | ÎÂÏχ Ó ÒÓÔˇÊÂÌÌ ̊ı ̋ÎÂÏÂÌÚ‡ı „ÛÔÔ ̊, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚‡uËvÔ‰ÒÚ‡‚Ρ ̨Ú ÒÓÔˇÊÂÌÌ ̊ ̋ÎÂÏÂÌÚ ̊ ‰‡ÌÌÓÈ „ÛÔÔ ̊ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ ÍÓÎ ̧ˆÂ‚‡ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃Ì ̊ÏË ÏÂÚ͇Ï

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    ̨ ̆ËÏ ÛÒÎӂˡÏ: 1) ÂÒÎËe| ·Ó ËÁM, ÚÓφ(e)6= 1; 2)MÒ‚ˇÁ̇ Ë Ó‰ÌÓÒ‚ˇÁ̇ Ò ‚ ̊‰ÂÎÂÌÌÓÈ ‚Â ̄ËÌÓÈȮ∂M, ÔË ̋ÚÓÏ ÒÛ ̆ÂÒÚ‚ÛÂÚ „‡ÌË ̃Ì ̊È ˆËÍÎe1. . . et͇Ú ̊M, ̇ ̃Ë̇ ̨ ̆ËÈÒˇ ‚OË Ú‡ÍÓÈ, ̃ÚÓφ(e1). . . φ(et) =c1. . . cn; 3) ÂÒÎËD| ÔÓËÁ‚ÓÎ ̧̇ˇ ӷ·ÒÚ ̧ ËÁMËe1. . . ej| ÔÓËÁ‚ÓÎ ̧Ì ̊È „‡ÌË ̃Ì ̊È ˆËÍÎ ̋ÚÓÈ Ó·Î‡ÒÚË, ÚÓφ(e1). . . φ(ej)| ˆËÍÎË ̃ÂÒÍË Ô˂‰ÂÌ̇ˇ ÔÂÂÒÚ‡Ìӂ͇ ÌÂÍÓÚÓÓ„Óci. “ÂÓÂχ 1 (
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    ).≈ÒÎËF| ҂ӷӉ̇ˇ „ÛÔÔ‡, ÚÓ ‰Îˇ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚË c1, . . . ,cn,n≥0, ÌÂÚ˂ˇΠ̧Ì ̊ı ̋ÎÂÏÂÌÚÓ‚ ËÁFÒÛ ̆ÂÒÚ‚ÛÂÚ ‰Ë‡„‡ÏχM(c1, . . . , cn), Û‰Ó‚ÎÂÚ‚Óˇ ̨ ̆‡ˇ ÛÒÎӂˡÏ1{3. “ÂÓÂχ 2 ([2]).œÛÒÚ ̧M| Ò‚ˇÁ̇ˇ Ó‰ÌÓÒ‚ˇÁ̇ˇ ‰Ë‡„‡Ïχ Ò Ó·Î‡ÒÚˇÏËD1, .

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    “ÂÓÂχ 1 ([2]).≈ÒÎËF| ҂ӷӉ̇ˇ „ÛÔÔ‡, ÚÓ ‰Îˇ ÔÓËÁ‚ÓÎ ̧ÌÓÈ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚË c1, . . . ,cn,n≥0, ÌÂÚ˂ˇΠ̧Ì ̊ı ̋ÎÂÏÂÌÚÓ‚ ËÁFÒÛ ̆ÂÒÚ‚ÛÂÚ ‰Ë‡„‡ÏχM(c1, . . . , cn), Û‰Ó‚ÎÂÚ‚Óˇ ̨ ̆‡ˇ ÛÒÎӂˡÏ1{3. “ÂÓÂχ 2 (
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    ).œÛÒÚ ̧M| Ò‚ˇÁ̇ˇ Ó‰ÌÓÒ‚ˇÁ̇ˇ ‰Ë‡„‡Ïχ Ò Ó·Î‡ÒÚˇÏËD1, . . . ,Dm. ƒÓÔÛÒÚËÏ, ̃ÚÓα| „‡ÌË ̃Ì ̊È ˆËÍÎ ‰ÎˇM, ̇ ̃Ë̇ ̨ ̆ËÈÒˇ ‚ ‚Â ̄ËÌÂv0∈∂M, Ë ÔÛÒÚ ̧ w=φ(α). “Ó„‰‡ ÒÛ ̆ÂÒÚ‚Û ̨Ú ÏÂÚÍËriӷ·ÒÚÂÈDiË ̋ÎÂÏÂÌÚ ̊uiËÁF,1≤i≤m, Ú‡ÍËÂ, ̃ÚÓ w= (u1r1u−11)· · ·(umrmu−1m). œÓ‰ÏÌÓÊÂÒÚ‚ÓRÒ‚Ó·Ó‰ÌÓÈ „ÛÔÔ ̊ḞÁ ̊‚‡ÂÚÒˇ ÒËÏÏÂÚËÁÓ‚‡ÌÌ ̊Ï, ÂÒÎË ‚Ò ̋ÎÂÏÂÌÚ ̊ ËÁRÔ˂‰ÂÌ ̊ Ë ËÁr∈RÒΉÛÂÚ, ̃ÚÓ ‚Ò ˆËÍÎË ̃ÂÒÍË ÔÂÂÒÚ

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    R-ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧ ̨, ÂÒÎË ÔÓËÁ‚‰ÂÌËÂw=c1· · ·cnÌ ÏÓÊÂÚ · ̊Ú ̧ Á‡ÔËÒ‡ÌÓ Í‡Í ÔÓËÁ‚‰ÂÌË ÏÂÌ ̃ÂÏnÒÓÔˇÊÂÌÌ ̊ı Ò ̋ÎÂÏÂÌÚ‡ÏË ËÁR. œÛÒÚ ̧M| ÔÓËÁ‚ÓÎ ̧̇ˇ ‰Ë‡„‡Ïχ ̇‰F. ƒÓÔÛÒÚËÏ, ̃ÚÓD1ËD2{ ӷ·ÒÚË ËÁMÒ Ó· ̆ËÏ ·ÓÏe⊆∂D1∩∂D2. œÛÒÚ ̧eδ1Ëδ2e−1| „‡ÌË ̃Ì ̊ ˆËÍÎ ̊ ӷ·ÒÚÂÈD1ËD2 ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ. œÓÎÓÊËÏφ(δ1) =f1Ëφ(δ2) =f2. ƒË‡„‡ÏχṀÁ ̊‚‡ÂÚÒˇ Ô˂‰ÂÌÌÓÈ, ÂÒÎË ‚Ò„‰‡f26=f−11. “ÂÓÂχ 3.
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    ƒË‡„‡ÏχMÏËÌËχΠ̧ÌÓÈR-ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚË Ô˂‰Â̇. œÛÒÚ ̧M| ÌÂÍÓÚÓ‡ˇ ͇Ú‡. √‡ÌË ̃ÌÓÈ ‚Â ̄ËÌÓÈ (ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ „‡ÌË ̃Ì ̊Ï ·ÓÏ) Ï ̊ ·Û‰ÂÏ Ì‡Á ̊‚‡Ú ̧ ‚Â ̄ËÌÛ (·Ó) ËÁ∂M. √‡ÌË ̃ÌÓÈ Ó·Î‡ÒÚ ̧ ̨ ͇Ú ̊ṀÁ ̊‚‡ÂÚÒˇ ڇ͇ˇ ӷ·ÒÚ ̧DËÁM, ̃ÚÓ∂D∪∂M6=∅.

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    —ËÏ‚ÓÎi(D)Ó·ÓÁ̇ ̃‡ÂÚ ̃ËÒÎÓ ‚ÌÛÚÂÌÌËı ·Â ËÁD, ÔË ̃ÂÏ ÒÌÓ‚‡ ·Ó, ‚ÒÚ ̃‡ ̨ ̆ÂÂÒˇ ‚ „‡ÌË ̃ÌÓÏ ˆËÍΠ‰ÎˇD‰‚‡Ê‰ ̊, Ò ̃ËÚ‡ÂÚÒˇ ‰‚‡ ‡Á‡. —ÎÂ‰Û ̨ ̆‡ˇ ÎÂÏχ ‰‡ÂÚ „ÂÓÏÂÚË ̃ÂÒÍÛ ̨ ËÌÚÂÔÂÚ‡ˆË ̨ ÛÒÎÓ‚ËÈC(p)ËT(q). “ÂÓÂχ 4 (
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    ).œÛÒÚ ̧R| ÒËÏÏÂÚËÁÓ‚‡ÌÌÓ ÏÌÓÊÂÒÚ‚Ó ̋ÎÂÏÂÌÚÓ‚ Ò‚Ó·Ó‰ÌÓÈ „ÛÔÔ ̊ FËM| Ô˂‰ÂÌ̇ˇR-‰Ë‡„‡Ïχ. “Ó„‰‡ ‚ÂÌ ̊ ÒÎÂ‰Û ̨ ̆Ë ÛÚ‚ÂʉÂÌˡ: 1) ÂÒÎËRÛ‰Ó‚ÎÂÚ‚ÓˇÂÚ ÛÒÎÓ‚Ë ̨C(k), ÚÓ Í‡Ê‰‡ˇ ӷ·ÒÚ ̧DËÁM, ڇ͇ˇ, ̃ÚÓ∂D∪∂M Ì ÒÓ‰ÂÊËÚ ·Â, ËÏÂÂÚ ÒÚÂÔÂÌ ̧d(D)≥k; 2) ÂÒÎËRÛ‰Ó‚ÎÂÚ‚ÓˇÂÚ ÛÒÎÓ‚Ë ̨T(m), ÚÓ Í‡Ê‰‡ˇ ‚ÌÛÚÂÌÌˇˇ ‚Â ̄Ë̇v͇Ú ̊MËÏÂÂÚ ÒÚÂÔÂÌ ̧d(v)≥m. œÛÒÚ ̧pËq| Ú‡ÍË ̇ÚÛ‡Î ̧Ì ̊ ̃ËÒ·, ̃ÚÓ1/p+ 1/q=

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    ◊ÚÓ· ̊ ÔÓÎÛ ̃ËÚ ̧E•‚ Ó· ̆ÂÏ ÒÎÛ ̃‡Â, ÌÛÊÌÓ ÒÎÓÊËÚ ̧ ̃ËÒ· ·Â ‚ ˆËÍ·ı, ÌÂÓ·ıÓ‰ËÏ ̊ı ‰Îˇ ÓÔËÒ‡Ìˡ „‡Ìˈ ̊ ͇Ú ̊M. œË ËÒÒΉӂ‡ÌËË ‰Ë‡„‡ÏÏ Ì‡‰ „ÛÔÔ‡ÏË Ò ÛÒÎӂˡÏËC(p)-T(q)ÔÓÎÂÁÌÓÈ Ó͇Á ̊‚‡ÂÚÒˇ ÒÎÂ‰Û ̨ ̆‡ˇ ÙÓÏÛ· ÍË‚ËÁÌ ̊. œÓ·„‡ÂÏ, ̃ÚÓ Ì‡ÚÛ‡Î ̧Ì ̊ ̃ËÒ·pËqÒ‚ˇÁ‡Ì ̊ ‡‚ÂÌÒÚ‚ÓÏ 1/p+ 1/q= 1/2. “ÂÓÂχ 5 (
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    ).≈ÒÎËM| Ó‰ÌÓÒ‚ˇÁ̇ˇ[p, q]-͇Ú‡, ÒÓ‰Âʇ ̆‡ˇ Ì ÏÂÌ ‰‚Ûı ‚Â ̄ËÌ, ÚÓ ∑• M [ p q + 2−d(v) ] ≥p. ≈ÒÎËM{ Ó‰ÌÓÒ‚ˇÁ̇ˇ(q, p)-͇Ú‡, ÒÓ‰Âʇ ̆‡ˇ ·ÓΠ̃ÂÏ Ó‰ÌÛ Ó·Î‡ÒÚ ̧, ÚÓ ∑• M [ p q + 2−i(D) ] ≥p. “ÂÓÂχ 6 ([2]).

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    “ÂÓÂχ 5 ([2]).≈ÒÎËM| Ó‰ÌÓÒ‚ˇÁ̇ˇ[p, q]-͇Ú‡, ÒÓ‰Âʇ ̆‡ˇ Ì ÏÂÌ ‰‚Ûı ‚Â ̄ËÌ, ÚÓ ∑• M [ p q + 2−d(v) ] ≥p. ≈ÒÎËM{ Ó‰ÌÓÒ‚ˇÁ̇ˇ(q, p)-͇Ú‡, ÒÓ‰Âʇ ̆‡ˇ ·ÓΠ̃ÂÏ Ó‰ÌÛ Ó·Î‡ÒÚ ̧, ÚÓ ∑• M [ p q + 2−i(D) ] ≥p. “ÂÓÂχ 6 (
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    ).≈ÒÎËM|[p, q]-͇Ú‡, ͇ʉ‡ˇ ÍÓÏÔÓÌÂÌÚ‡ ÍÓÚÓÓÈ ˇ‚ΡÂÚÒˇ ÎË·Ó Ó‰ÌÓÒ‚ˇÁÌÓÈ, ÎË·Ó ÍÓÎ ̧ˆÂ‚ÓÈ, ÚÓ E•M≤ q p ∑• M[p−d(v)](1) Ë V•M≤ q p ∑• M [p−d(v)](2) Œ·ÓÁ̇ ̃ËÏ ̃ÂÂÁM1ÔӉ͇ÚÛ Í‡Ú ̊M, ÔÓÎÛ ̃‡ ̨ ̆Û ̨Òˇ ‚ ̊ ̃ÂÍË‚‡ÌËÂÏ ‚ÒÂı ËÁÓÎËÓ‚‡ÌÌ ̊ı ‚Â ̄ËÌ. œÛÒÚ ̧M| ÔÓËÁ‚ÓÎ ̧̇ˇ ͇Ú‡.

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    √‡ÌË ̃Ì ̊È ÒÎÓÈ Í‡Ú ̊MÒÓÒÚÓËÚ ËÁ ‚ÒÂı „‡ÌË ̃Ì ̊ı ‚Â ̄ËÌ, ·Â, ÒÓ‰Âʇ ̆Ëı „‡ÌË ̃Ì ̊ ‚Â ̄ËÌ ̊, Ë „‡ÌË ̃Ì ̊ı ӷ·ÒÚÂÈ Í‡Ú ̊M. ƒÎˇ[p, q]-͇Ú ̊M‚‚‰ÂÏ Ó·ÓÁ̇ ̃ÂÌË σ(M) = ∑• M [p−d(v)]. “ÂÓÂχ 7 (Ó ÔÎÓ ̆‡‰Ë
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    ).≈ÒÎËM1| Ó‰ÌÓÒ‚ˇÁ̇ˇ[p, q]-͇Ú‡, ÚÓ VM≤ q p2 σ(M)2. ≈ÒÎËM|- Ó‰ÌÓÒ‚ˇÁ̇ˇ(q, p)-͇Ú‡, ÚÓ FM≤ q p2 (∑• M [p−i(D)] )2 . “ÂÓÂχ 8 ([2]).œÛÒÚ ̧F| ÍÓÌ ̃ÌÓ ÔÓÓʉÂÌ̇ˇ ҂ӷӉ̇ˇ „ÛÔÔ‡,R| ÍÓÌ ̃ÌÓ ÒËÏÏÂÚËÁÓ‚‡ÌÌÓ ÏÌÓÊÂÒÚ‚Ó ̋ÎÂÏÂÌÚÓ‚ „ÛÔÔ ̊FËN| ÌÓχΠ̧ÌÓ Á‡Ï ͇̊ÌË ÏÌÓÊÂÒÚ‚‡ R.

  9. In-text reference with the coordinate start=27740
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    √‡ÌË ̃Ì ̊È ÒÎÓÈ Í‡Ú ̊MÒÓÒÚÓËÚ ËÁ ‚ÒÂı „‡ÌË ̃Ì ̊ı ‚Â ̄ËÌ, ·Â, ÒÓ‰Âʇ ̆Ëı „‡ÌË ̃Ì ̊ ‚Â ̄ËÌ ̊, Ë „‡ÌË ̃Ì ̊ı ӷ·ÒÚÂÈ Í‡Ú ̊M. ƒÎˇ[p, q]-͇Ú ̊M‚‚‰ÂÏ Ó·ÓÁ̇ ̃ÂÌË σ(M) = ∑• M [p−d(v)]. “ÂÓÂχ 7 (Ó ÔÎÓ ̆‡‰Ë [2]).≈ÒÎËM1| Ó‰ÌÓÒ‚ˇÁ̇ˇ[p, q]-͇Ú‡, ÚÓ VM≤ q p2 σ(M)2. ≈ÒÎËM|- Ó‰ÌÓÒ‚ˇÁ̇ˇ(q, p)-͇Ú‡, ÚÓ FM≤ q p2 (∑• M [p−i(D)] )2 . “ÂÓÂχ 8 (
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    [2]
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    ).œÛÒÚ ̧F| ÍÓÌ ̃ÌÓ ÔÓÓʉÂÌ̇ˇ ҂ӷӉ̇ˇ „ÛÔÔ‡,R| ÍÓÌ ̃ÌÓ ÒËÏÏÂÚËÁÓ‚‡ÌÌÓ ÏÌÓÊÂÒÚ‚Ó ̋ÎÂÏÂÌÚÓ‚ „ÛÔÔ ̊FËN| ÌÓχΠ̧ÌÓ Á‡Ï ͇̊ÌË ÏÌÓÊÂÒÚ‚‡ R. ≈ÒÎËRÛ‰Ó‚ÎÂÚ‚ÓˇÂÚ ÎË·Ó ÛÒÎÓ‚Ë ̨C(6), ÎË·Ó ÛÒÎӂˡÏC(4)ËT(4), ÎË·Ó ÛÒÎÓ‚ËˇÏ C(3)ËT(6), ÚÓ ÔÓ·ÎÂχ ‡‚ÂÌÒÚ‚‡ ÒÎÓ‚ ‚ „ÛÔÔÂF/N‡Î„ÓËÚÏË ̃ÂÒÍË ‡Á ̄Ëχ.

  10. In-text reference with the coordinate start=28280
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    √Ó‚ÓˇÚ, ̃ÚÓ ‚ „ÛÔÔÂG=〈X;R〉‡Á ̄Ëχ ÔÓ·ÎÂχ ‚ıÓʉÂÌˡ ‚ ˆËÍÎË ̃ÂÒÍÛ ̨ ÔÓ‰„ÛÔÔÛ, ÂÒÎË ‰Îˇ Î ̨· ̊ı ÒÎÓ‚wËv‚ ‡ÎÙ‡‚ËÚÂXÏÓÊÌÓ ‚ ̊ˇÒÌËÚ ̧, ÒÛ ̆ÂÒÚ‚ÛÂÚ ÎË Ú‡ÍÓ ̃ËÒÎÓn∈Z, ̃ÚÓ ÒÎÓ‚‡wnËvÔ‰ÒÚ‡‚Ρ ̨Ú Ó‰ËÌ Ë ÚÓÚ Ê ̋ÎÂÏÂÌÚ „ÛÔÔ ̊G. “ÂÓÂχ 9 (
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    ).œÛÒÚ ̧F| ÍÓÌ ̃ÌÓ ÔÓÓʉÂÌ̇ˇ ҂ӷӉ̇ˇ „ÛÔÔ‡,R| ÍÓÌ ̃ÌÓ ÒËÏÏÂÚËÁÓ‚‡ÌÌÓ ÏÌÓÊÂÒÚ‚Ó ̋ÎÂÏÂÌÚÓ‚ „ÛÔÔ ̊FËN| ÌÓχΠ̧ÌÓ Á‡Ï ͇̊ÌË ÏÌÓÊÂÒÚ‚‡R. ≈ÒÎËRÛ‰Ó‚ÎÂÚ‚ÓˇÂÚ ÎË·Ó ÛÒÎÓ‚Ë ̨C(6), ÎË·Ó ÛÒÎӂˡÏC(4)ËT(4), ÎË·Ó ÛÒÎӂˡÏC(3)Ë T(6), ÚÓ ÔÓ·ÎÂχ ‚ıÓʉÂÌˡ ‚ ˆËÍÎË ̃ÂÒÍÛ ̨ ÔÓ‰„ÛÔÔÛ ‚ „ÛÔÔÂG=F/N‡Î„ÓËÚÏË ̃ÂÒÍË ‡Á ̄Ëχ. œË ̋ÚÓÏ ˆÂÎÓ ̃ËÒÎÓn, ‰Îˇ ÍÓÚÓÓ„Ó ‚ ̊ÔÓÎÌˇÂÚÒˇ ‡‚ÂÌÒÚ‚Ó ÒÎÓ‚wnËv‚ „ÛÔÔÂ

3
Ol'shanskii A.Iu.Geometriia opredeliaiushchikh sootnoshenii v gruppakh[The geometry of dening relations in groups]. Moscow, Nauka Publ., 1989. 448 p. (in Russian).
Total in-text references: 1
  1. In-text reference with the coordinate start=2119
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    –ÂÁÛÎ ̧Ú‡Ú ̊ ‰‡ÌÌÓ„Ó ËÒÒΉӂ‡Ìˡ Û͇Á ̊‚‡ ̨Ú Ì‡ Ì‚ÓÁÏÓÊÌÓÒÚ ̧ ÔÓÒÚÓÂÌˡ ÍËÔÚÓÒÚÓÈÍÓÈ ÒËÒÚÂÏ ̊ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ Û͇Á‡ÌÌ ̊ı ‡Î„ÓËÚÏÓ‚ ‚ ‰‡ÌÌÓÏ Í·ÒÒ „ÛÔÔ. »ÒÒΉӂ‡Ìˡ ÔÓ‚Ó‰ˇÚÒˇ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ „ÂÓÏÂÚË ̃ÂÒÍËı ÏÂÚÓ‰Ó‚ ÍÓÏ·Ë̇ÚÓÌÓÈ ÚÂÓËË „ÛÔÔ, ‡ ËÏÂÌÌÓ, ÏÂÚÓ‰‡ ‰Ë‡„‡ÏÏ Ì‡‰ „ÛÔÔ‡ÏË, ·‡ÁËÛ ̨ ̆„ÓÒˇ ̇ ÒÎÂ‰Û ̨ ̆Ëı ‰‚Ûı ÛÚ‚ÂʉÂÌˡı
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    . œÂ‚Ó | ÎÂÏχ ¬‡Ì ‡ÏÔÂ̇, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚Ó ‡‚ÌÓ Â‰ËÌˈ ‚ „ÛÔÔ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ Ó‰ÌÓÒ‚ˇÁ̇ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃ÌÓÈ ÏÂÚÍÓÈ, ‡‚ÌÓÈ ̋ÚÓÏÛ ÒÎÓ‚Û. ¬ÚÓÓ | ÎÂÏχ Ó ÒÓÔˇÊÂÌÌ ̊ı ̋ÎÂÏÂÌÚ‡ı „ÛÔÔ ̊, ÛÚ‚Âʉ‡ ̨ ̆‡ˇ, ̃ÚÓ ÒÎÓ‚‡uËvÔ‰ÒÚ‡‚Ρ ̨Ú ÒÓÔˇÊÂÌÌ ̊ ̋ÎÂÏÂÌÚ ̊ ‰‡ÌÌÓÈ „ÛÔÔ ̊ ÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡, ÍÓ„‰‡ ÒÛ ̆ÂÒÚ‚ÛÂÚ ÍÓÎ ̧ˆÂ‚‡ˇ ‰Ë‡„‡Ïχ Ò „‡ÌË ̃Ì ̊ÏË ÏÂÚ͇Ï

4
Novikov P.S. On the algorithmic unsolvability of word identity problem in group theory.Tr. Matematicheskogo in-ta AN SSSR[Proc. of Mathematical Institute of the USSR Academy of Sciences], 1955, vol. 44, pp. 1{144. (in Russian).
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    Ì ÒÛ ̆ÂÒÚ‚ÛÂÚ ‡Î„ÓËÚχ, ‡ÒÔÓÁ̇ ̨ ̆Â„Ó Á‡ ÍÓÌ ̃ÌÓ ̃ËÒÎÓ ̄‡„Ó‚ ÔÓ ÔÓËÁ‚ÓÎ ̧ÌÓÏÛ ÒÎÓ‚ÛW ̋ÚÓ„Ó Ô‰ÒÚ‡‚ÎÂÌˡ, ÓÔ‰ÂΡÂÚ ÎËW‰ËÌË ̃Ì ̊È ̋ÎÂÏÂÌÚ. “‡ÍËÏ Ó·‡ÁÓÏ, ÌÂÚ Ôӈ‰Û ̊ ‰Îˇ  ̄ÂÌˡ ÔÓ·ÎÂÏ ̊ ÒÎÓ‚, ÍÓÚÓ‡ˇ ‡·Óڇ· · ̊ ‰Îˇ Í‡Ê‰Ó„Ó Ô‰ÒÚ‡‚ÎÂÌˡ, ËÎË ‚ Í·ÒÒ ‚ÒÂı „ÛÔÔ
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    . œÓ·ÎÂχ ÒÓÔˇÊÂÌÌÓÒÚË Â ̆ ·ÓΠÚۉ̇, ̃ÂÏ ÔÓ·ÎÂχ ÒÎÓ‚. ¬ Ò‡ÏÓÏ ‰ÂÎÂ, ÒÓÔˇÊÂÌÌÓÒÚ ̧ ÒÎÓ‚‡W2ÔÛÒÚÓÏÛ ÒÎÓ‚ÛW1ÂÒÚ ̧ Ì ̃ÚÓ ËÌÓÂ, Í‡Í ‡‚ÂÌÒÚ‚Ó ÒÎÓ‚‡W2ÔÛÒÚÓÏÛ ÒÎÓ‚ÛW1, ÔÓ ̋ÚÓÏÛ  ̄ÂÌË ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‰‡ÂÚ, ‚ ̃‡ÒÚÌÓÒÚË,  ̄ÂÌË ÔÓ·ÎÂÏ ̊ ÒÎÓ‚.

5
Bezverkhnij N.V. On the solvability of the general word problem for a cyclic subgroup of a group with conditionC(6).Fundamental'naia i prikladnaia matematika, 1999, vol. 5, no. 1, pp. 39{46. (in Russian).
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  1. In-text reference with the coordinate start=29724
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    ‰ÂΡ ̨ ̆Ë ÒÓÓÚÌÓ ̄ÂÌˡ, „‰Âa,b,r1,r2| ÌÂÔÛÒÚ ̊ ÒÎÓ‚‡ ‚ ‡ÎÙ‡‚ËÚÂX, Ë ‡ÒÒÏÓÚËÏ ÒÎÓ‚‡ ËÁR, Ó·‡ÚÌ ̊ Ír1,r2, ‡ Ú‡ÍÊ Ëı ˆËÍÎË ̃ÂÒÍË ÔÂÂÒÚ‡ÌÓ‚ÍË:u1≡br′1a,u2≡a−1(r′2)−1b−1, u3≡br′1a,u4≡a−1(r′2)−1b−1. œÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧u1,u2,u3,u4,u1ÔÓÚË‚Ó ̃ËÚ ÛÒÎÓ‚Ë ̨ T(6). «Ì‡ ̃ËÚ, Ó· ̆ ̇ ̃‡ÎÓab‰‚Ûı ÓÔ‰ÂΡ ̨ ̆Ëı ÒÓÓÚÌÓ ̄ÂÌËÈ ËÁRËÏÂÂÚ Â‰ËÌË ̃ÌÛ ̨ ‰ÎËÌÛ. ¡Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚ ÒÎÓ‚ÂwÂÒÚ ̧R-ÒÓÍ‡ ̆ÂÌËÂ
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    , ÂÒÎË ÒÛ ̆ÂÒÚ‚ÛÂÚ ̋ÎÂÏÂÌÚ r∈R, Ú‡ÍÓÈ, ̃ÚÓ: 1)r≡r1r2, 2)w≡w1w2w3, 3)r1≡w2, 4) ÒÎÓ‚Ór2ÎË·Ó ÔÛÒÚÓ, ÎË·Ó ˇ‚ΡÂÚÒˇ ÍÛÒÍÓÏ, 5) ÒÎÓ‚‡w1r−12,r−12w3ÌÂÒÓÍ‡ÚËÏ ̊ ‚ Ò‚Ó·Ó‰ÌÓÈ „ÛÔÔÂ. ¬ ÒÎÛ ̃‡Â Á‡ÏÂÌ ̊ ÒÎÓ‚‡w‡‚Ì ̊Ï ÂÏÛ ‚ „ÛÔÔÂGÒÎÓ‚ÓÏw1r−12w3·Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚w‚ ̊ÔÓÎÌÂÌÓR-ÒÓÍ‡ ̆ÂÌËÂ.

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    . œËÏÂ ÔÓÚÓÍÓ· ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ.¬ ÒÚ‡Ú ̧ [10] ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÍËÔÚÓÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚ÌÓÈ „ÛÔÔÂ Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚  ˆËÍÎË ̃ÂÒÍÓÈ ÔÓ‰„ÛÔÔÂ. –‡Á ̄ËÏÓÒÚ ̧ ÌÂÍÓÚÓ ̊ı ËÁ ̋ÚËı ÔÓ·ÎÂÏ ‚ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(p)-T(q)‰Ó͇Á‡Ì‡ ‚ ‡·ÓÚ‡ı
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    ¬ Ô‰·„‡ÂÏ ̊ı ÒËÒÚÂχı ‰Îˇ ÔÓÒÚÓÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‡·ÓÌÂÌÚ ̊AËB‚ ̊·Ë‡ ̨Ú Ó· ̆Û ̨ ̇·ÂÎÂ‚Û „ÛÔÔÛGË Â ̋ÎÂÏÂÌÚg‰ÓÒÚ‡ÚÓ ̃ÌÓ ·ÓÎ ̧ ̄Ó„Ó ÔÓÒÚÓ„Ó ÔÓˇ‰Í‡. œÓÚÓÍÓÎ ÔÓÎÛ ̃ÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‚ ̊„Ρ‰ËÚ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. 1.

6
Bezverhnij N.V. Normal forms for elements of infinite order in group withC(3)-T(6)condition. Izvestija TulGU. Estestvennye nauki, 2010, iss. 1, pp. 6{25. (in Russian).
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  1. In-text reference with the coordinate start=29724
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    ‰ÂΡ ̨ ̆Ë ÒÓÓÚÌÓ ̄ÂÌˡ, „‰Âa,b,r1,r2| ÌÂÔÛÒÚ ̊ ÒÎÓ‚‡ ‚ ‡ÎÙ‡‚ËÚÂX, Ë ‡ÒÒÏÓÚËÏ ÒÎÓ‚‡ ËÁR, Ó·‡ÚÌ ̊ Ír1,r2, ‡ Ú‡ÍÊ Ëı ˆËÍÎË ̃ÂÒÍË ÔÂÂÒÚ‡ÌÓ‚ÍË:u1≡br′1a,u2≡a−1(r′2)−1b−1, u3≡br′1a,u4≡a−1(r′2)−1b−1. œÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧u1,u2,u3,u4,u1ÔÓÚË‚Ó ̃ËÚ ÛÒÎÓ‚Ë ̨ T(6). «Ì‡ ̃ËÚ, Ó· ̆ ̇ ̃‡ÎÓab‰‚Ûı ÓÔ‰ÂΡ ̨ ̆Ëı ÒÓÓÚÌÓ ̄ÂÌËÈ ËÁRËÏÂÂÚ Â‰ËÌË ̃ÌÛ ̨ ‰ÎËÌÛ. ¡Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚ ÒÎÓ‚ÂwÂÒÚ ̧R-ÒÓÍ‡ ̆ÂÌËÂ
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    , ÂÒÎË ÒÛ ̆ÂÒÚ‚ÛÂÚ ̋ÎÂÏÂÌÚ r∈R, Ú‡ÍÓÈ, ̃ÚÓ: 1)r≡r1r2, 2)w≡w1w2w3, 3)r1≡w2, 4) ÒÎÓ‚Ór2ÎË·Ó ÔÛÒÚÓ, ÎË·Ó ˇ‚ΡÂÚÒˇ ÍÛÒÍÓÏ, 5) ÒÎÓ‚‡w1r−12,r−12w3ÌÂÒÓÍ‡ÚËÏ ̊ ‚ Ò‚Ó·Ó‰ÌÓÈ „ÛÔÔÂ. ¬ ÒÎÛ ̃‡Â Á‡ÏÂÌ ̊ ÒÎÓ‚‡w‡‚Ì ̊Ï ÂÏÛ ‚ „ÛÔÔÂGÒÎÓ‚ÓÏw1r−12w3·Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚w‚ ̊ÔÓÎÌÂÌÓR-ÒÓÍ‡ ̆ÂÌËÂ.

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    . œËÏÂ ÔÓÚÓÍÓ· ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ.¬ ÒÚ‡Ú ̧ [10] ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÍËÔÚÓÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚ÌÓÈ „ÛÔÔÂ Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚  ˆËÍÎË ̃ÂÒÍÓÈ ÔÓ‰„ÛÔÔÂ. –‡Á ̄ËÏÓÒÚ ̧ ÌÂÍÓÚÓ ̊ı ËÁ ̋ÚËı ÔÓ·ÎÂÏ ‚ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(p)-T(q)‰Ó͇Á‡Ì‡ ‚ ‡·ÓÚ‡ı
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    ¬ Ô‰·„‡ÂÏ ̊ı ÒËÒÚÂχı ‰Îˇ ÔÓÒÚÓÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‡·ÓÌÂÌÚ ̊AËB‚ ̊·Ë‡ ̨Ú Ó· ̆Û ̨ ̇·ÂÎÂ‚Û „ÛÔÔÛGË Â ̋ÎÂÏÂÌÚg‰ÓÒÚ‡ÚÓ ̃ÌÓ ·ÓÎ ̧ ̄Ó„Ó ÔÓÒÚÓ„Ó ÔÓˇ‰Í‡. œÓÚÓÍÓÎ ÔÓÎÛ ̃ÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‚ ̊„Ρ‰ËÚ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. 1.

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    ¬ÓÒÔÓÎ ̧ÁÛÂÏÒˇ ÒÎÂ‰Û ̨ ̆ËÏ ÂÁÛÎ ̧Ú‡ÚÓÏ ËÁ [7]. “ÂÓÂχ 10.≈ÒÎË ‚ Ô˂‰ÂÌÌÓÈ Ó‰ÌÓÒ‚ˇÁÌÓÈR-‰Ë‡„‡ÏÏÂMÒ „‡ÌˈÂÈ∂M=σ∪τ ÒÎÓ‚‡φ(σ)Ëφ(τ)R-, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊, ÚÓ ÒÛ ̆ÂÒÚ‚ÛÂÚ ÍÓÌÒÚ‡ÌÚ‡C >0, ‰Îˇ ÍÓÚÓÓÈ |φ(σ)| ≤C|φ(τ)|,|φ(τ)| ≤C|φ(σ)|. ¬ ÒÚ‡Ú ̧Â
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    [6]
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    ÔË‚Ó‰ËÚÒˇ ‡Î„ÓËÚÏ ÔÓÒÚÓÂÌˡ ÌÓχΠ̧ÌÓÈ ÙÓÏ ̊w0ÒÎÓ‚‡w, ÒÓÔˇÊÂÌÌÓÈ ‚ „ÛÔÔÂGÒwε, ε∈ {1,2}, Ú‡ÍÓÈ, ̃ÚÓ ‚Ò ÒÚÂÔÂÌË ÒÎÓ‚‡w0R-, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊ÏË. —‡‚ÌË‚‡ˇ ‰ÎËÌ ̊ ÒÎÓ‚wnËwn0, Á‡Ï ̃‡ÂÏ, ̃ÚÓ ËÁ ÌÂÒÓÍ‡ÚËÏÓÒÚË ÔÓÒΉÌÂ„Ó ÒΉÛÂÚ, ̃ÚÓ|wn| ≥ |wn0|, ÔÓÒÍÓÎ ̧ÍÛ, Í‡Í „Ó‚ÓËÎÓÒ ̧ ‚ ̊ ̄Â, ÔË ‚ ̊ÔÓÎÌÂÌËËR-, ̄R-ÒÓÍ‡ ̆ÂÌËÈ ‰ÎË̇ ÒÎÓ‚‡ ÒÚÓ„Ó ÛÏÂÌ ̧ ̄‡ÂÚÒˇ.

7
Bezverhnij N.V. The power conjugacy search problem in a cyclic subgroup in groups with the conditionC(3)-T(6).Diskretnaja matematika, 2012, vol. 24, iss. 4, pp. 27{46. (English Translation:Discrete Mathematics and Applications, 2012, vol. 22, iss. 5{6, pp. 521{544. DOI:10.1515/dma-2012-036).
Total in-text references: 4
  1. In-text reference with the coordinate start=29724
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    ‰ÂΡ ̨ ̆Ë ÒÓÓÚÌÓ ̄ÂÌˡ, „‰Âa,b,r1,r2| ÌÂÔÛÒÚ ̊ ÒÎÓ‚‡ ‚ ‡ÎÙ‡‚ËÚÂX, Ë ‡ÒÒÏÓÚËÏ ÒÎÓ‚‡ ËÁR, Ó·‡ÚÌ ̊ Ír1,r2, ‡ Ú‡ÍÊ Ëı ˆËÍÎË ̃ÂÒÍË ÔÂÂÒÚ‡ÌÓ‚ÍË:u1≡br′1a,u2≡a−1(r′2)−1b−1, u3≡br′1a,u4≡a−1(r′2)−1b−1. œÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧u1,u2,u3,u4,u1ÔÓÚË‚Ó ̃ËÚ ÛÒÎÓ‚Ë ̨ T(6). «Ì‡ ̃ËÚ, Ó· ̆ ̇ ̃‡ÎÓab‰‚Ûı ÓÔ‰ÂΡ ̨ ̆Ëı ÒÓÓÚÌÓ ̄ÂÌËÈ ËÁRËÏÂÂÚ Â‰ËÌË ̃ÌÛ ̨ ‰ÎËÌÛ. ¡Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚ ÒÎÓ‚ÂwÂÒÚ ̧R-ÒÓÍ‡ ̆ÂÌËÂ
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    , ÂÒÎË ÒÛ ̆ÂÒÚ‚ÛÂÚ ̋ÎÂÏÂÌÚ r∈R, Ú‡ÍÓÈ, ̃ÚÓ: 1)r≡r1r2, 2)w≡w1w2w3, 3)r1≡w2, 4) ÒÎÓ‚Ór2ÎË·Ó ÔÛÒÚÓ, ÎË·Ó ˇ‚ΡÂÚÒˇ ÍÛÒÍÓÏ, 5) ÒÎÓ‚‡w1r−12,r−12w3ÌÂÒÓÍ‡ÚËÏ ̊ ‚ Ò‚Ó·Ó‰ÌÓÈ „ÛÔÔÂ. ¬ ÒÎÛ ̃‡Â Á‡ÏÂÌ ̊ ÒÎÓ‚‡w‡‚Ì ̊Ï ÂÏÛ ‚ „ÛÔÔÂGÒÎÓ‚ÓÏw1r−12w3·Û‰ÂÏ „Ó‚ÓËÚ ̧, ̃ÚÓ ‚w‚ ̊ÔÓÎÌÂÌÓR-ÒÓÍ‡ ̆ÂÌËÂ.

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    . œËÏÂ ÔÓÚÓÍÓ· ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ.¬ ÒÚ‡Ú ̧ [10] ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÍËÔÚÓÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚ÌÓÈ „ÛÔÔÂ Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚  ˆËÍÎË ̃ÂÒÍÓÈ ÔÓ‰„ÛÔÔÂ. –‡Á ̄ËÏÓÒÚ ̧ ÌÂÍÓÚÓ ̊ı ËÁ ̋ÚËı ÔÓ·ÎÂÏ ‚ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(p)-T(q)‰Ó͇Á‡Ì‡ ‚ ‡·ÓÚ‡ı
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    ¬ Ô‰·„‡ÂÏ ̊ı ÒËÒÚÂχı ‰Îˇ ÔÓÒÚÓÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‡·ÓÌÂÌÚ ̊AËB‚ ̊·Ë‡ ̨Ú Ó· ̆Û ̨ ̇·ÂÎÂ‚Û „ÛÔÔÛGË Â ̋ÎÂÏÂÌÚg‰ÓÒÚ‡ÚÓ ̃ÌÓ ·ÓÎ ̧ ̄Ó„Ó ÔÓÒÚÓ„Ó ÔÓˇ‰Í‡. œÓÚÓÍÓÎ ÔÓÎÛ ̃ÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‚ ̊„Ρ‰ËÚ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. 1.

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    ÕÓ ÓÚÎÓÊËÏ  ̄ÂÌË ̋ÚÓÈ Á‡‰‡ ̃Ë ‰Ó ÒÎÂ‰Û ̨ ̆Â„Ó ÔÛÌÍÚ‡, ‡ ÔÓ͇ ÔÓÔ ̊Ú‡ÂÏÒˇ ÔÓÎÛ ̃ËÚ ̧ ÓˆÂÌÍË ̃ËÒ·n‰Û„ËÏ ÒÔÓÒÓ·ÓÏ. ¡Û‰ÂÏ ‡Ì‡ÎËÁËÓ‚‡Ú ̧ ‰Ë‡„‡ÏÏ ̊ ‡‚ÂÌÒÚ‚‡ ÒÎÓ‚ ‚ „ÛÔÔÂGÒ ÛÒÎӂˡÏËC(3)-T(6). ¬ÓÒÔÓÎ ̧ÁÛÂÏÒˇ ÒÎÂ‰Û ̨ ̆ËÏ ÂÁÛÎ ̧Ú‡ÚÓÏ ËÁ
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    . “ÂÓÂχ 10.≈ÒÎË ‚ Ô˂‰ÂÌÌÓÈ Ó‰ÌÓÒ‚ˇÁÌÓÈR-‰Ë‡„‡ÏÏÂMÒ „‡ÌˈÂÈ∂M=σ∪τ ÒÎÓ‚‡φ(σ)Ëφ(τ)R-, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊, ÚÓ ÒÛ ̆ÂÒÚ‚ÛÂÚ ÍÓÌÒÚ‡ÌÚ‡C >0, ‰Îˇ ÍÓÚÓÓÈ |φ(σ)| ≤C|φ(τ)|,|φ(τ)| ≤C|φ(σ)|. ¬ ÒÚ‡Ú ̧ [6] ÔË‚Ó‰ËÚÒˇ ‡Î„ÓËÚÏ ÔÓÒÚÓÂÌˡ ÌÓχΠ̧ÌÓÈ ÙÓÏ ̊w0ÒÎÓ‚‡w, ÒÓÔˇÊÂÌÌÓÈ ‚ „ÛÔÔÂGÒwε, ε∈ {1,2}, Ú‡ÍÓÈ, ̃ÚÓ ‚Ò ÒÚÂÔÂÌË ÒÎÓ‚‡w0R-, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊ÏË.

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    —‡‚ÌË‚‡ˇ ‰ÎËÌ ̊ ÒÎÓ‚wnËwn0, Á‡Ï ̃‡ÂÏ, ̃ÚÓ ËÁ ÌÂÒÓÍ‡ÚËÏÓÒÚË ÔÓÒΉÌÂ„Ó ÒΉÛÂÚ, ̃ÚÓ|wn| ≥ |wn0|, ÔÓÒÍÓÎ ̧ÍÛ, Í‡Í „Ó‚ÓËÎÓÒ ̧ ‚ ̊ ̄Â, ÔË ‚ ̊ÔÓÎÌÂÌËËR-, ̄R-ÒÓÍ‡ ̆ÂÌËÈ ‰ÎË̇ ÒÎÓ‚‡ ÒÚÓ„Ó ÛÏÂÌ ̧ ̄‡ÂÚÒˇ. ≈ÒÎË Ê Ó͇ÊÂÚÒˇ, ̃ÚÓ ÒÎÓ‚ÓwnÌÂÒÓÍ‡ÚËÏÓ, ÚÓ Í ÌÂÏÛ ÔËÏÂÌËÏ ̊ ÌÂ‡‚ÂÌÒÚ‚‡, ‡Ì‡ÎÓ„Ë ̃Ì ̊ ÔË‚Ó‰ËÏ ̊Ï ÌËÊ ‰Îˇwn0. ƒÂÈÒÚ‚ËÚÂÎ ̧ÌÓ, ‚ ÒÚ‡Ú ̧Â
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    ‰Ó͇Á‡ÌÓ, ̃ÚÓ ÂÒÎËR-, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊ ÒÎÓ‚‡u,vÒÓÔˇÊÂÌ ̊ ‚ (C(3)-T(6))-„ÛÔÔÂG, ÚÓ |u| 9 ≤ |v| ≤9|u|. »Ú‡Í, ‚ „ÛÔÔÂG‚ ̊ÔÓÎÌÂÌ ̊ ÒÓÓÚÌÓ ̄ÂÌˡwn0∼wn=v, „‰ÂvÔÓÎÛ ̃ÂÌÓ ÔˇÏ ̊Ï ‚ ̊ ̃ËÒÎÂÌËÂÏ Ó‰ÌÓÒÚÓÓÌÌÂÈ ÙÛÌ͈ËË.

8
Bezverhnij V.N. On normalizers of elements inC(p)-T(q)-groups.Algoritmicheskie problemy teorii grupp i polugrupp: mezhvuz. sb. nauch. trudov[Algorithmic problems of the theory of groups and semigroups: interuniversity collection of scientific papers]. Tula, Tolstoi TSPU Publ., 1994, pp. 4{58. (in Russian).
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  1. In-text reference with the coordinate start=31280
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    ŒÔ‰ÂÎËÏ ÔÓÌˇÚË ̄R-ÒÓÍ‡ ̆ÂÌˡ Ò ËÒÔÓÎ ̧ÁÓ‚‡ÌËÂÏ ‰Ë‡„‡ÏÏ. “‡ÍÊ ‰‡‰ËÏ „ÂÓÏÂÚË ̃ÂÒÍÓ ÓÔ‰ÂÎÂÌËÂR-ÒÓÍ‡ ̆ÂÌˡ. ƒÎˇ ̋ÚÓ„Ó ‡ÒÒÏÓÚËÏ ÒÎÂ‰Û ̨ ̆Ë ÔÓÌˇÚˡ. –‡ÒÒÏÓÚËÏ ‰Ë‡„‡ÏÏÛM. Œ·Î‡ÒÚ ̧D⊂ṀÁ ̊‚‡ÂÚÒˇ ‰ ̋ÌÓ‚ÒÍÓÈ
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    , ÂÒÎË: 1)∂D∩∂M| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M(Ú.Â.∂D∩∂M=p| ÔÓ‰ÔÛÚ ̧ ‚ „‡ÌË ̃Ì ̊ı ˆËÍ·ı ӷ·ÒÚËDË ‰Ë‡„‡ÏÏ ̊M); 2)i(D)∈ {0,1}. œÓÎÓÒÓÈ [8] ‚ ‰Ë‡„‡ÏÏÂṀÁ ̊‚‡ÂÚÒˇ ÔÓ‰‰Ë‡„‡ÏÏ‡Π = ⋃k i=1 DiÒÓ Ò‚ÓÈÒÚ‚‡ÏË: 1)∂Di∩∂M=p| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M; 2)∂Π∩∂M=p| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M; 3) ÂÒÎËk= 3, ÚÓi(D1) =i(D2) =i(D3) = 2, ÔË ̃ÂÏ ÒÓÒ‰ÌË ӷ·ÒÚË ËÏ ̨Ú Ó· ̆ ·Ó, ‡ ‚

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    –‡ÒÒÏÓÚËÏ ‰Ë‡„‡ÏÏÛM. Œ·Î‡ÒÚ ̧D⊂ṀÁ ̊‚‡ÂÚÒˇ ‰ ̋ÌÓ‚ÒÍÓÈ [8], ÂÒÎË: 1)∂D∩∂M| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M(Ú.Â.∂D∩∂M=p| ÔÓ‰ÔÛÚ ̧ ‚ „‡ÌË ̃Ì ̊ı ˆËÍ·ı ӷ·ÒÚËDË ‰Ë‡„‡ÏÏ ̊M); 2)i(D)∈ {0,1}. œÓÎÓÒÓÈ
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    ‚ ‰Ë‡„‡ÏÏÂṀÁ ̊‚‡ÂÚÒˇ ÔÓ‰‰Ë‡„‡ÏÏ‡Π = ⋃k i=1 DiÒÓ Ò‚ÓÈÒÚ‚‡ÏË: 1)∂Di∩∂M=p| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M; 2)∂Π∩∂M=p| ÔÓÒΉӂ‡ÚÂÎ ̧̇ˇ ̃‡ÒÚ ̧ „‡Ìˈ ̊∂M; 3) ÂÒÎËk= 3, ÚÓi(D1) =i(D2) =i(D3) = 2, ÔË ̃ÂÏ ÒÓÒ‰ÌË ӷ·ÒÚË ËÏ ̨Ú Ó· ̆ ·Ó, ‡ ‚Ò ÚË Ó·Î‡ÒÚË ÔÓÎÓÒ ̊ ËÏ ̨Ú Ó· ̆Û ̨ ‚Â ̄ËÌÛ; 4) ÂÒÎËk >3,k= 2l+ 1, ÚÓi(D1) =i(D2) =i(D2l) =i(D2l+1) = 2,i(D3) =i(D5) = . . .=i(D2l−3) =i(D2l−1)

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    ,wnv−1= 1‚ „ÛÔÔÂG. ƒÎˇ ̋ÚÓ„Ó ÔÓÚË‚ÌËÍ ‰ÓÎÊÂÌ ÔÓÒΉӂ‡ÚÂÎ ̧ÌÓ ‚ÓÁ‚Ó‰ËÚ ̧w‚ ÒÚÂÔÂÌ ̧k, „‰Âk=1, n(·Û‰ÂÏ Ò ̃ËÚ‡Ú ̧, ̃ÚÓn >0) Ë  ̄‡Ú ̧ ÔÓ·ÎÂÏÛ ‡‚ÂÌÒÚ‚‡ ÒÎÓ‚ ‚ „ÛÔÔÂG. ƒÎˇ  ̄ÂÌˡ ÔÓ·ÎÂÏ ̊ ‡‚ÂÌÒÚ‚‡ ÒÎÓ‚ ‚ „ÛÔÔ‡ı Ò ÛÒÎÓ‚ËÂÏC(3)-T(6)ËÒÔÓÎ ̧ÁÛ ̨ÚÒˇ Ôӈ‰Û ̊ ˆËÍÎË ̃ÂÒÍËıR, ̄R-ÒÓÍ‡ ̆ÂÌËÈ: ‰ËÌÒÚ‚ÂÌÌ ̊Ï ˆËÍÎË ̃ÂÒÍËR, ̄R-ÌÂÒÓÍ‡ÚËÏ ̊Ï ÒÎÓ‚ÓÏ, ‡‚Ì ̊Ï Â‰ËÌˈ ‚ „ÛÔÔÂG, ˇ‚ΡÂÚÒˇ ÔÛÒÚÓ ÒÎÓ‚Ó
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    . “‡ÍËÏ Ó·‡ÁÓÏ, ÔÓÒΠ‚ ̊ÔÓÎÌÂÌˡ ‚Ó ‚ÒÂı ˆËÍÎË ̃ÂÒÍËı ÔÂÂÒÚ‡Ìӂ͇ı ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÒÎÓ‚‡x‚ÒÂıR, ̄R-ÒÓÍ‡ ̆ÂÌËÈ ÔÓÎÛ ̃‡ÂÚÒˇ ÎË·Ó ˆËÍÎË ̃ÂÒÍËR, ̄R-ÌÂÒÓÍ‡ÚËÏÓ ÒÎÓ‚Óx1(Á‰ÂÒ ̧|x1| ≥1) ÔËx6= 1‚G, ÎË·Ó ÔÛÒÚÓ ÒÎÓ‚Ó, ÂÒÎËx= 1‚ „ÛÔÔÂG.

9
Bezverhnij V.N., Parshikova E.V. The solution of problems of integration in a cyclic subgroup of a group with conditionC(4)-T(4).Algoritmicheskie problemy teorii grupp i polugrupp: mezhvuz. sb. nauch. trudov[Algorithmic problems of the theory of groups and semigroups: interuniversity collection of scientific papers]. Tula, Tolstoi TSPU Publ., 2001, pp. 120{139. (in Russian).
Total in-text references: 1
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    . œËÏÂ ÔÓÚÓÍÓ· ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ.¬ ÒÚ‡Ú ̧ [10] ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÍËÔÚÓÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚ÌÓÈ „ÛÔÔÂ Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚  ˆËÍÎË ̃ÂÒÍÓÈ ÔÓ‰„ÛÔÔÂ. –‡Á ̄ËÏÓÒÚ ̧ ÌÂÍÓÚÓ ̊ı ËÁ ̋ÚËı ÔÓ·ÎÂÏ ‚ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(p)-T(q)‰Ó͇Á‡Ì‡ ‚ ‡·ÓÚ‡ı
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    ¬ Ô‰·„‡ÂÏ ̊ı ÒËÒÚÂχı ‰Îˇ ÔÓÒÚÓÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‡·ÓÌÂÌÚ ̊AËB‚ ̊·Ë‡ ̨Ú Ó· ̆Û ̨ ̇·ÂÎÂ‚Û „ÛÔÔÛGË Â ̋ÎÂÏÂÌÚg‰ÓÒÚ‡ÚÓ ̃ÌÓ ·ÓÎ ̧ ̄Ó„Ó ÔÓÒÚÓ„Ó ÔÓˇ‰Í‡. œÓÚÓÍÓÎ ÔÓÎÛ ̃ÂÌˡ Ó· ̆Â„Ó ÍÎ ̨ ̃‡ ‚ ̊„Ρ‰ËÚ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. 1.

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Glukhov M.M. An analysis of some key distribution public systems based on non-abelian groups.Matematicheskie voprosy kriptografii, 2010, vol. 1, no. 4, pp. 5{22. (in Russian).
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    ‚ ‡·ÓÚ [22] · ̊· Ô‰ÎÓÊÂ̇ ÒıÂχ, ÓÒÌÓ‚‡Ì̇ˇ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ. ¡ÓΠÔÓ‰Ó·ÌÛ ̨ ËÌÙÓχˆË ̨ Ë ‡Ì‡ÎËÁ ÍËÔÚÓÒËÒÚÂÏ, Ô‰ÎÓÊÂÌÌ ̊ı ‚ ÔÂ ̃ËÒÎÂÌÌ ̊ı ‡·ÓÚ‡ı, ÏÓÊÌÓ Ì‡ÈÚË ‚ ÒÚ‡Ú ̧Â
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    . ¬ ‰‡Î ̧ÌÂÈ ̄ÂÏ Ï ̊ ·Û‰ÂÏ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ÒÎÂ‰Û ̨ ̆Ë ÓÒÌÓ‚Ì ̊ ӷÓÁ̇ ̃ÂÌˡ. w1≡w2| ÒÎÓ‚Ów1„‡ÙË ̃ÂÒÍË ‡‚ÌÓ ÒÎÓ‚Ûw2; w1=w2| ÒÎÓ‚Ów1‡‚ÌÓ ÒÎÓ‚Ûw2; |w|| ‰ÎË̇ ÒÎÓ‚‡w; ∂D| „‡Ìˈ‡ ӷ·ÒÚËD; ∂M| „‡Ìˈ‡ ͇Ú ̊M; d(v)| ÒÚÂÔÂÌ ̧ ‚Â ̄ËÌ ̊v; d(D)| ÒÚÂÔÂÌ ̧ ӷ·ÒÚËD; i(D)| ̃ËÒÎÓ ‚ÌÛÚÂÌÌËı ·Â ӷ·ÒÚËD; ∑ Md(v)| ÒÛÏχ ÒÚÂÔÂÌÂÈ ‚ÒÂı ‚Â ̄ËÌ Í‡Ú ̊M; ∑ Md(D)| ÒÛÏχ ÒÚÂÔÂÌÂÈ ‚ÒÂı ӷ·ÒÚÂÈ Í‡Ú ̊M; ∑• M|

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    ‰ËÒÍÂÚÌ ̊ı ÎÓ„‡ËÙÏÓ‚, ̋ÍÒÔÓÌÂ̈ˇΠ̧̇. «‡‰‡ ̃Ë, Ò‚ˇÁ‡ÌÌ ̊Â Ò œƒÀ (‰‡Ì‡ ÍÓÌ ̃̇ˇ ‡·Â΂‡ „ÛÔÔ‡GË Â ̋ÎÂÏÂÌÚA): •Á‡‰‡ ̃‡ ƒËÙÙË | ’ÂÎÎχ̇ | ‰Îˇ Á‡‰‡ÌÌ ̊ı ̋ÎÂÏÂÌÚÓ‚A∈G,B=AxËC=Ay ‚ ̊ ̃ËÒÎËÚ ̧D=Axy; •ÔÓ·ÎÂχ ‚ ̊·Ó‡ ƒËÙÙË | ’ÂÎÎχ̇ | ‰Îˇ Á‡‰‡ÌÌ ̊ı ̋ÎÂÏÂÌÚÓ‚A∈G,B=Ax, C=AyËD=AzÚ·ÛÂÚÒˇ ÓÔ‰ÂÎËÚ ̧, ˇ‚ΡÂÚÒˇ ÎËzÔÓËÁ‚‰ÂÌËÂÏx·y. œËÏÂ ÔÓÚÓÍÓ· ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ.¬ ÒÚ‡Ú ̧Â
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    ‡ÒÒχÚË‚‡ ̨ÚÒˇ ÍËÔÚÓÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚ÌÓÈ „ÛÔÔÂ Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚  ˆËÍÎË ̃ÂÒÍÓÈ ÔÓ‰„ÛÔÔÂ.

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Bezverhnij N.V.O kruchenii i o razreshimosti problemy vhozhdenija v ciklicheskuju podgruppu v gruppah s usloviemC(6)[On torsion and solvability of the general word problem for a cyclic subgroup of a group with conditionC(6)]. Moscow, 1995. Dep. VINITI no. 2033{V95. (in Russian).
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    ¬ÚÓÓ Ú·ӂ‡ÌË ڇÍÊ ˇ‚ΡÂÚÒˇ ÌÂÓ·ıÓ‰ËÏ ̊Ï, ÔÓÒÍÓÎ ̧ÍÛ ÂÒÎËwÔ‰ÒÚ‡‚ΡÂÚ ̋ÎÂÏÂÌÚ ÍÓÌ ̃ÌÓ„Ó ÔÓˇ‰Í‡, ÚÓwp= 1‰Îˇ ÌÂÍÓÚÓÓ„Óp. —Ήӂ‡ÚÂÎ ̧ÌÓ, ÔÓÚË‚ÌËÍÛ ÌÛÊÌÓ ·Û‰ÂÚ Ì‡ÈÚË ÌÂn, ‡ ÓÒÚ‡ÚÓÍ ÓÚ ‰ÂÎÂÌˡṅp, ̃ÚÓ ˇ‚ΡÂÚÒˇ „Ó‡Á‰Ó ·ÓΠÔÓÒÚÓÈ Á‡‰‡ ̃ÂÈ, ÂÒÎË Û ̃ÂÒÚ ̧, ̃ÚÓ Ò‡ÏÓpÏÓÊÂÚ · ̊Ú ̧ ̇ȉÂÌÓ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. ¬ ÒÚ‡Ú ̧ˇı
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    ‰Ó͇Á‡ÌÓ, ̃ÚÓ ÒÎÓ‚Ów‚ ‡ÎÙ‡‚ËÚÂXÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡ Ô‰ÒÚ‡‚ΡÂÚ ̋ÎÂÏÂÌÚ ÍÓÌ ̃ÌÓ„Ó ÔÓˇ‰Í‡ ‚ „ÛÔÔÂGÒ ÛÒÎӂˡÏËC(3)-T(6), ÍÓ„‰‡w≡vsË ‚ ÏÌÓÊÂÒÚ‚Â RÒÛ ̆ÂÒÚ‚ÛÂÚ ÓÔ‰ÂΡ ̨ ̆ ÒÓÓÚÌÓ ̄ÂÌË ‚ˉ‡r≡ut, ÔË ̃ÂÏ ÒÎÓ‚‡u,vÒÓÔˇÊÂÌ ̊ ‚G.

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Bogley W.A., Pride S.J. Aspherical relative presentations.Proc. of Edinburg Math. Soc. Ser. 2. 1992, vol. 35, no. 1, pp. 1{39. DOI:10.1017/S0013091500005290
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    ¬ÚÓÓ Ú·ӂ‡ÌË ڇÍÊ ˇ‚ΡÂÚÒˇ ÌÂÓ·ıÓ‰ËÏ ̊Ï, ÔÓÒÍÓÎ ̧ÍÛ ÂÒÎËwÔ‰ÒÚ‡‚ΡÂÚ ̋ÎÂÏÂÌÚ ÍÓÌ ̃ÌÓ„Ó ÔÓˇ‰Í‡, ÚÓwp= 1‰Îˇ ÌÂÍÓÚÓÓ„Óp. —Ήӂ‡ÚÂÎ ̧ÌÓ, ÔÓÚË‚ÌËÍÛ ÌÛÊÌÓ ·Û‰ÂÚ Ì‡ÈÚË ÌÂn, ‡ ÓÒÚ‡ÚÓÍ ÓÚ ‰ÂÎÂÌˡṅp, ̃ÚÓ ˇ‚ΡÂÚÒˇ „Ó‡Á‰Ó ·ÓΠÔÓÒÚÓÈ Á‡‰‡ ̃ÂÈ, ÂÒÎË Û ̃ÂÒÚ ̧, ̃ÚÓ Ò‡ÏÓpÏÓÊÂÚ · ̊Ú ̧ ̇ȉÂÌÓ ÒÎÂ‰Û ̨ ̆ËÏ Ó·‡ÁÓÏ. ¬ ÒÚ‡Ú ̧ˇı
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    ‰Ó͇Á‡ÌÓ, ̃ÚÓ ÒÎÓ‚Ów‚ ‡ÎÙ‡‚ËÚÂXÚÓ„‰‡ Ë ÚÓÎ ̧ÍÓ ÚÓ„‰‡ Ô‰ÒÚ‡‚ΡÂÚ ̋ÎÂÏÂÌÚ ÍÓÌ ̃ÌÓ„Ó ÔÓˇ‰Í‡ ‚ „ÛÔÔÂGÒ ÛÒÎӂˡÏËC(3)-T(6), ÍÓ„‰‡w≡vsË ‚ ÏÌÓÊÂÒÚ‚Â RÒÛ ̆ÂÒÚ‚ÛÂÚ ÓÔ‰ÂΡ ̨ ̆ ÒÓÓÚÌÓ ̄ÂÌË ‚ˉ‡r≡ut, ÔË ̃ÂÏ ÒÎÓ‚‡u,vÒÓÔˇÊÂÌ ̊ ‚G.

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Gersten Sh., Short H.B. Small cancellation theory and automatic groups.Inventiones Mathematicae, 1990, vol. 102, iss. 1, pp. 305{334. DOI:10.1007/BF01233430
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    ›ÚÓ Ó· ̇ˇÒÌˇÂÚÒˇ ÚÂÏ, ̃ÚÓ Î ̨·ÓÈ ÍÛÒÓÍ ‚ Ú‡ÍÓÏ Í·ÒÒ ËÏÂÂÚ Â‰ËÌË ̃ÌÛ ̨ ‰ÎËÌÛ, ̃ÚÓ Ë ‰Ó͇Á‡ÌÓ ÌËÊÂ. CÓÍ‡ ̆ÂÌˡ ‚ (C(3)-T(6))-„ÛÔÔ‡ı.¬ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(p)-T(q)‰ÎË̇ ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó ÍÛÒ͇ ÏÓÊÂÚ · ̊Ú ̧ ÓÚÎË ̃̇ ÓÚ Â‰ËÌˈ ̊. ÕÓ ÂÒÎËq >4, ÚÓ ‚Ò ÍÛÒÍË ËÏ ̨Ú Â‰ËÌË ̃ÌÛ ̨ ‰ÎËÌÛ
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    . ƒÂÈÒÚ‚ËÚÂÎ ̧ÌÓ, Ô‰ÔÓÎÓÊËÏ, ̃ÚÓr1≡abr′1,r2≡abr′2| ‡ÁÎË ̃Ì ̊ ÓÔ‰ÂΡ ̨ ̆Ë ÒÓÓÚÌÓ ̄ÂÌˡ, „‰Âa,b,r1,r2| ÌÂÔÛÒÚ ̊ ÒÎÓ‚‡ ‚ ‡ÎÙ‡‚ËÚÂX, Ë ‡ÒÒÏÓÚËÏ ÒÎÓ‚‡ ËÁR, Ó·‡ÚÌ ̊ Ír1,r2, ‡ Ú‡ÍÊ Ëı ˆËÍÎË ̃ÂÒÍË ÔÂÂÒÚ‡ÌÓ‚ÍË:u1≡br′1a,u2≡a−1(r′2)−1b−1, u3≡br′1a,u4≡a−1(r′2)−1b−1. œÓÒΉӂ‡ÚÂÎ ̧ÌÓÒÚ ̧u1,u2,u3,u4,u1ÔÓÚË‚Ó ̃ËÚ ÛÒÎÓ‚Ë ̨ T(6).

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Shor P.W. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer.SIAM Journal on Computing, 1997, vol. 26, no. 5, pp. 1484{1509. DOI: 10.1137/S0097539795293172
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    ‡Í ÛÊ „Ó‚ÓËÎÓÒ ̧, ‚ ‰‡ÌÌÓÈ ÒÚ‡Ú ̧ ÔÓÒÚÓÂÌË ӉÌÓÒÚÓÓÌÌÂÈ ÙÛÌ͈ËË ·‡ÁËÛÂÚÒˇ ̇ ÔÓ·ÎÂÏ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı Ò ÛÒÎӂˡÏËC(3)-T(6). Õ‡‰Ó ÓÚÏÂÚËÚ ̧, ̃ÚÓ ÔÓÒΠ‰Ó͇Á‡ÚÂÎ ̧ÒÚ‚‡ ‚ 1997 „. ‚ ‡·ÓÚÂ
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    ÔÓÎËÌÓÏˇΠ̧ÌÓÈ ÒÎÓÊÌÓÒÚË Á‡‰‡ ̃Ë ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ ÏÛÎ ̧ÚËÔÎË͇ÚË‚Ì ̊ı „ÛÔÔ‡ı ÍÓÌ ̃Ì ̊ı ÔÓÎÂÈ Ë ÍÓΈ ‚ ̊ ̃ÂÚÓ‚ ÔË Â  ̄ÂÌËË Ì‡ Í‚‡ÌÚÓ‚ÓÏ ÍÓÏÔ ̧ ̨ÚÂ ÔÓˇ‚ËÎ‡Ò ̧ ÒÂˡ ËÒÒΉӂ‡ÌËÈ ÔÓ ÓÚ ̊Ò͇ÌË ̨ ‰Û„Ëı ÒÎÓÊÌ ̊ı χÚÂχÚË ̃ÂÒÍËı ÔÓ·ÎÂÏ, ÍÓÚÓ ̊ ÏÓÊÌÓ · ̊ÎÓ · ̊ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‰Îˇ ÔÓÒÚÓÂÌˡ ‡ÒËÏÏÂÚË ̃Ì ̊ı ÍËÔÚÓÒËÒÚÂÏ. œË ̋ÚÓÏ Ï ̊ Ì ‚ÒÚ ̃‡ÂÏ ‡·ÓÚ, ËÒÔÓÎ ̧ÁÛ ̨ ̆Ëı „ÛÔÔ ̊

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Anshel I., Anshtl M., Goldfeld D. An algebraic method for public key cryptography.Mathematical Research Letters, 1999, vol. 6, no. 3, pp. 287{291. DOI:10.4310/MRL.1999.v6.n3.a3
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    ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı ÍÓÌ ̃Ì ̊ı ÔÓÎÂÈ Ë ÍÓΈ ‚ ̊ ̃ÂÚÓ‚ ÔË Â  ̄ÂÌËË Ì‡ Í‚‡ÌÚÓ‚ÓÏ ÍÓÏÔ ̧ ̨ÚÂ ÔÓˇ‚ËÎ‡Ò ̧ ÒÂˡ ËÒÒΉӂ‡ÌËÈ ÔÓ ÓÚ ̊Ò͇ÌË ̨ ‰Û„Ëı ÒÎÓÊÌ ̊ı χÚÂχÚË ̃ÂÒÍËı ÔÓ·ÎÂÏ, ÍÓÚÓ ̊ ÏÓÊÌÓ · ̊ÎÓ · ̊ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‰Îˇ ÔÓÒÚÓÂÌˡ ‡ÒËÏÏÂÚË ̃Ì ̊ı ÍËÔÚÓÒËÒÚÂÏ. œË ̋ÚÓÏ Ï ̊ Ì ‚ÒÚ ̃‡ÂÏ ‡·ÓÚ, ËÒÔÓÎ ̧ÁÛ ̨ ̆Ëı „ÛÔÔ ̊ Ò ÛÒÎӂˡÏË Ï‡ÎÓ„Ó ÒÓÍ‡ ̆ÂÌˡC(p)-T(q). “‡Í ‚ 1998{2001 „„. ‚ ‡·ÓÚ‡ı
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    · ̊ÎË ‡ÒÒÏÓÚÂÌ ̊ ÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÔÓ·ÎÂÏ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı: χÚË ̃Ì ̊ı „ÛÔÔ‡ı Ë „ÛÔÔ‡ı ÍÓÒ. ¬ 2001 „. ‚ ̊ ̄ÎË ‡·ÓÚ ̊ [20,21], ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „.

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Ko K.H., Lee S.J., Cheon J.H., Han J.W., Kang J., Park C. New Public-Key Cryptosystem Using Braid Groups. In:Advances in Cryptology | CRYPTO 2000. Springer Berlin Heidelberg, 2000, pp. 166{183. (Ser.Lecture Notes in Computer Science; vol. 1880). DOI: 10.1007/3-540-44598-610
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    ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı ÍÓÌ ̃Ì ̊ı ÔÓÎÂÈ Ë ÍÓΈ ‚ ̊ ̃ÂÚÓ‚ ÔË Â  ̄ÂÌËË Ì‡ Í‚‡ÌÚÓ‚ÓÏ ÍÓÏÔ ̧ ̨ÚÂ ÔÓˇ‚ËÎ‡Ò ̧ ÒÂˡ ËÒÒΉӂ‡ÌËÈ ÔÓ ÓÚ ̊Ò͇ÌË ̨ ‰Û„Ëı ÒÎÓÊÌ ̊ı χÚÂχÚË ̃ÂÒÍËı ÔÓ·ÎÂÏ, ÍÓÚÓ ̊ ÏÓÊÌÓ · ̊ÎÓ · ̊ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‰Îˇ ÔÓÒÚÓÂÌˡ ‡ÒËÏÏÂÚË ̃Ì ̊ı ÍËÔÚÓÒËÒÚÂÏ. œË ̋ÚÓÏ Ï ̊ Ì ‚ÒÚ ̃‡ÂÏ ‡·ÓÚ, ËÒÔÓÎ ̧ÁÛ ̨ ̆Ëı „ÛÔÔ ̊ Ò ÛÒÎӂˡÏË Ï‡ÎÓ„Ó ÒÓÍ‡ ̆ÂÌˡC(p)-T(q). “‡Í ‚ 1998{2001 „„. ‚ ‡·ÓÚ‡ı
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    · ̊ÎË ‡ÒÒÏÓÚÂÌ ̊ ÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÔÓ·ÎÂÏ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı: χÚË ̃Ì ̊ı „ÛÔÔ‡ı Ë „ÛÔÔ‡ı ÍÓÒ. ¬ 2001 „. ‚ ̊ ̄ÎË ‡·ÓÚ ̊ [20,21], ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „.

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Yamamura A. Public-key cryptosystems using the modular group. In:Public Key Cryptography. Springer Berlin Heidelberg, 1998, pp. 203{216. (Ser.Lecture Notes in Computer Science; vol. 1431). DOI:10.1007/BFb0054026
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    ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı ÍÓÌ ̃Ì ̊ı ÔÓÎÂÈ Ë ÍÓΈ ‚ ̊ ̃ÂÚÓ‚ ÔË Â  ̄ÂÌËË Ì‡ Í‚‡ÌÚÓ‚ÓÏ ÍÓÏÔ ̧ ̨ÚÂ ÔÓˇ‚ËÎ‡Ò ̧ ÒÂˡ ËÒÒΉӂ‡ÌËÈ ÔÓ ÓÚ ̊Ò͇ÌË ̨ ‰Û„Ëı ÒÎÓÊÌ ̊ı χÚÂχÚË ̃ÂÒÍËı ÔÓ·ÎÂÏ, ÍÓÚÓ ̊ ÏÓÊÌÓ · ̊ÎÓ · ̊ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‰Îˇ ÔÓÒÚÓÂÌˡ ‡ÒËÏÏÂÚË ̃Ì ̊ı ÍËÔÚÓÒËÒÚÂÏ. œË ̋ÚÓÏ Ï ̊ Ì ‚ÒÚ ̃‡ÂÏ ‡·ÓÚ, ËÒÔÓÎ ̧ÁÛ ̨ ̆Ëı „ÛÔÔ ̊ Ò ÛÒÎӂˡÏË Ï‡ÎÓ„Ó ÒÓÍ‡ ̆ÂÌˡC(p)-T(q). “‡Í ‚ 1998{2001 „„. ‚ ‡·ÓÚ‡ı
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    · ̊ÎË ‡ÒÒÏÓÚÂÌ ̊ ÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÔÓ·ÎÂÏ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı: χÚË ̃Ì ̊ı „ÛÔÔ‡ı Ë „ÛÔÔ‡ı ÍÓÒ. ¬ 2001 „. ‚ ̊ ̄ÎË ‡·ÓÚ ̊ [20,21], ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „.

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    ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı ÍÓÌ ̃Ì ̊ı ÔÓÎÂÈ Ë ÍÓΈ ‚ ̊ ̃ÂÚÓ‚ ÔË Â  ̄ÂÌËË Ì‡ Í‚‡ÌÚÓ‚ÓÏ ÍÓÏÔ ̧ ̨ÚÂ ÔÓˇ‚ËÎ‡Ò ̧ ÒÂˡ ËÒÒΉӂ‡ÌËÈ ÔÓ ÓÚ ̊Ò͇ÌË ̨ ‰Û„Ëı ÒÎÓÊÌ ̊ı χÚÂχÚË ̃ÂÒÍËı ÔÓ·ÎÂÏ, ÍÓÚÓ ̊ ÏÓÊÌÓ · ̊ÎÓ · ̊ ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ ‰Îˇ ÔÓÒÚÓÂÌˡ ‡ÒËÏÏÂÚË ̃Ì ̊ı ÍËÔÚÓÒËÒÚÂÏ. œË ̋ÚÓÏ Ï ̊ Ì ‚ÒÚ ̃‡ÂÏ ‡·ÓÚ, ËÒÔÓÎ ̧ÁÛ ̨ ̆Ëı „ÛÔÔ ̊ Ò ÛÒÎӂˡÏË Ï‡ÎÓ„Ó ÒÓÍ‡ ̆ÂÌˡC(p)-T(q). “‡Í ‚ 1998{2001 „„. ‚ ‡·ÓÚ‡ı
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    · ̊ÎË ‡ÒÒÏÓÚÂÌ ̊ ÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÔÓ·ÎÂÏ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı: χÚË ̃Ì ̊ı „ÛÔÔ‡ı Ë „ÛÔÔ‡ı ÍÓÒ. ¬ 2001 „. ‚ ̊ ̄ÎË ‡·ÓÚ ̊ [20,21], ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „.

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    , ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „. ‚ ‡·ÓÚ [22] · ̊· Ô‰ÎÓÊÂ̇ ÒıÂχ, ÓÒÌÓ‚‡Ì̇ˇ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ.

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    ÓÚ‡ı [16,17,18,19] · ̊ÎË ‡ÒÒÏÓÚÂÌ ̊ ÒËÒÚÂÏ ̊ ÓÚÍ ̊ÚÓ„Ó ‡ÒÔ‰ÂÎÂÌˡ ÍÎ ̨ ̃ÂÈ, ÓÒÌÓ‚‡ÌÌ ̊ ̇ ÔÓ·ÎÂÏ ÒÓÔˇÊÂÌÌÓÒÚË ‚ ÌÂÍÓÏÏÛÚ‡ÚË‚Ì ̊ı „ÛÔÔ‡ı: χÚË ̃Ì ̊ı „ÛÔÔ‡ı Ë „ÛÔÔ‡ı ÍÓÒ. ¬ 2001 „. ‚ ̊ ̄ÎË ‡·ÓÚ ̊ [20,21], ‚ ÍÓÚÓ ̊ı Á‡ ÓÒÌÓ‚Û · ̊· ‚ÁˇÚ‡ ÔÓ·ÎÂχ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ ‚ „ÛÔÔ‡ı ‚ÌÛÚÂÌÌËı ‡‚ÚÓÏÓÙËÁÏÓ‚ ÔÓÎÛÔˇÏ ̊ı ÔÓËÁ‚‰ÂÌËÈ „ÛÔÔ SL(2,Zp)ËZp,GL(2,Zp)ËZp. ¬ 2007 „. ‚ ‡·ÓÚÂ
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    · ̊· Ô‰ÎÓÊÂ̇ ÒıÂχ, ÓÒÌÓ‚‡Ì̇ˇ ̇ ÍÓÏÔÓÁˈËË ‰‚Ûı ÔÓ·ÎÂÏ ÚÂÓËË „ÛÔÔ: ÔÓ·ÎÂÏ ̊ ÒÓÔˇÊÂÌÌÓÒÚË Ë ÔÓ·ÎÂÏ ̊ ‰ËÒÍÂÚÌÓ„Ó ÎÓ„‡ËÙÏËÓ‚‡Ìˡ. ¡ÓΠÔÓ‰Ó·ÌÛ ̨ ËÌÙÓχˆË ̨ Ë ‡Ì‡ÎËÁ ÍËÔÚÓÒËÒÚÂÏ, Ô‰ÎÓÊÂÌÌ ̊ı ‚ ÔÂ ̃ËÒÎÂÌÌ ̊ı ‡·ÓÚ‡ı, ÏÓÊÌÓ Ì‡ÈÚË ‚ ÒÚ‡Ú ̧ [10].