Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime
Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, an...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Holden, Joshua [verfasserIn] |
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| Format: |
E-Artikel |
|---|---|
| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Anmerkung: |
© Pleiades Publishing, Ltd. 2020 |
|---|
| Übergeordnetes Werk: |
Enthalten in: P-adic numbers, ultrametric analysis, and applications - Moscow : MAIK Nauka, Interperiodica Publ., 2009, 12(2020), 1 vom: Jan., Seite 12-28 |
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| Übergeordnetes Werk: |
volume:12 ; year:2020 ; number:1 ; month:01 ; pages:12-28 |
| Links: |
|---|
| DOI / URN: |
10.1134/S2070046620010021 |
|---|
| Katalog-ID: |
SPR026329557 |
|---|
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| 100 | 1 | |a Holden, Joshua |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime |
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| 520 | |a Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. | ||
| 700 | 1 | |a Richardson, Pamela A. |4 aut | |
| 700 | 1 | |a Robinson, Margaret M. |4 aut | |
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10.1134/S2070046620010021 doi (DE-627)SPR026329557 (SPR)S2070046620010021-e DE-627 ger DE-627 rakwb eng Holden, Joshua verfasserin aut Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2020 Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. Richardson, Pamela A. aut Robinson, Margaret M. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 12(2020), 1 vom: Jan., Seite 12-28 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:12 year:2020 number:1 month:01 pages:12-28 https://dx.doi.org/10.1134/S2070046620010021 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2020 1 01 12-28 |
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10.1134/S2070046620010021 doi (DE-627)SPR026329557 (SPR)S2070046620010021-e DE-627 ger DE-627 rakwb eng Holden, Joshua verfasserin aut Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2020 Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. Richardson, Pamela A. aut Robinson, Margaret M. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 12(2020), 1 vom: Jan., Seite 12-28 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:12 year:2020 number:1 month:01 pages:12-28 https://dx.doi.org/10.1134/S2070046620010021 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2020 1 01 12-28 |
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10.1134/S2070046620010021 doi (DE-627)SPR026329557 (SPR)S2070046620010021-e DE-627 ger DE-627 rakwb eng Holden, Joshua verfasserin aut Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2020 Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. Richardson, Pamela A. aut Robinson, Margaret M. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 12(2020), 1 vom: Jan., Seite 12-28 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:12 year:2020 number:1 month:01 pages:12-28 https://dx.doi.org/10.1134/S2070046620010021 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2020 1 01 12-28 |
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10.1134/S2070046620010021 doi (DE-627)SPR026329557 (SPR)S2070046620010021-e DE-627 ger DE-627 rakwb eng Holden, Joshua verfasserin aut Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Pleiades Publishing, Ltd. 2020 Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. Richardson, Pamela A. aut Robinson, Margaret M. aut Enthalten in P-adic numbers, ultrametric analysis, and applications Moscow : MAIK Nauka, Interperiodica Publ., 2009 12(2020), 1 vom: Jan., Seite 12-28 (DE-627)59571546X (DE-600)2487796-7 2070-0474 nnns volume:12 year:2020 number:1 month:01 pages:12-28 https://dx.doi.org/10.1134/S2070046620010021 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 12 2020 1 01 12-28 |
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Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime |
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counting fixed points and rooted closed walks of the singular map %$x \mapsto {x^{{x^n}}}%$ modulo powers of a prime |
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Counting Fixed Points and Rooted Closed Walks of the Singular Map %$x \mapsto {x^{{x^n}}}%$ Modulo Powers of a Prime |
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Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. © Pleiades Publishing, Ltd. 2020 |
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Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. © Pleiades Publishing, Ltd. 2020 |
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Abstract The “self-power” map x ⟼ xx modulo m and its generalized form %$x \mapsto {x^{{x^n}}}%$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix. © Pleiades Publishing, Ltd. 2020 |
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