Number Theoretic Methods in Cryptography: Complexity lower bounds
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1999
|
| Schriftenreihe: | Progress in Computer Science and Applied Logic
17 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | The book introduces new techniques which imply rigorous lower bounds on the complexity of some number theoretic and cryptographic problems. These methods and techniques are based on bounds of character sums and numbers of solutions of some polynomial equations over finite fields and residue rings. It also contains a number of open problems and proposals for further research. We obtain several lower bounds, exponential in terms of logp, on the de grees and orders of • polynomials; • algebraic functions; • Boolean functions; • linear recurring sequences; coinciding with values of the discrete logarithm modulo a prime p at suf ficiently many points (the number of points can be as small as pI/He). These functions are considered over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1. The case of d = 2 is of special interest since it corresponds to the representation of the right most bit of the discrete logarithm and defines whether the argument is a quadratic residue. We also obtain non-trivial upper bounds on the de gree, sensitivity and Fourier coefficients of Boolean functions on bits of x deciding whether x is a quadratic residue. These results are used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm. For example, we prove that any unbounded fan-in Boolean circuit. of sublogarithmic depth computing the discrete logarithm modulo p must be of superpolynomial size |
| Beschreibung: | 1 Online-Ressource (IX, 182 p) |
| ISBN: | 9783034886642 9783034897235 |
| DOI: | 10.1007/978-3-0348-8664-2 |
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Datensatz im Suchindex
| _version_ | 1804153096294629376 |
|---|---|
| any_adam_object | |
| author | Shparlinski, Igor |
| author_facet | Shparlinski, Igor |
| author_role | aut |
| author_sort | Shparlinski, Igor |
| author_variant | i s is |
| building | Verbundindex |
| bvnumber | BV042422201 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)1184262488 (DE-599)BVBBV042422201 |
| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-0348-8664-2 |
| format | Electronic eBook |
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| id | DE-604.BV042422201 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:10Z |
| institution | BVB |
| isbn | 9783034886642 9783034897235 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027857618 |
| oclc_num | 1184262488 |
| open_access_boolean | |
| owner | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| owner_facet | DE-384 DE-703 DE-91 DE-BY-TUM DE-634 |
| physical | 1 Online-Ressource (IX, 182 p) |
| psigel | ZDB-2-SMA ZDB-2-BAE ZDB-2-SMA_Archive |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Birkhäuser Basel |
| record_format | marc |
| series2 | Progress in Computer Science and Applied Logic |
| spelling | Shparlinski, Igor Verfasser aut Number Theoretic Methods in Cryptography Complexity lower bounds by Igor Shparlinski Basel Birkhäuser Basel 1999 1 Online-Ressource (IX, 182 p) txt rdacontent c rdamedia cr rdacarrier Progress in Computer Science and Applied Logic 17 The book introduces new techniques which imply rigorous lower bounds on the complexity of some number theoretic and cryptographic problems. These methods and techniques are based on bounds of character sums and numbers of solutions of some polynomial equations over finite fields and residue rings. It also contains a number of open problems and proposals for further research. We obtain several lower bounds, exponential in terms of logp, on the de grees and orders of • polynomials; • algebraic functions; • Boolean functions; • linear recurring sequences; coinciding with values of the discrete logarithm modulo a prime p at suf ficiently many points (the number of points can be as small as pI/He). These functions are considered over the residue ring modulo p and over the residue ring modulo an arbitrary divisor d of p - 1. The case of d = 2 is of special interest since it corresponds to the representation of the right most bit of the discrete logarithm and defines whether the argument is a quadratic residue. We also obtain non-trivial upper bounds on the de gree, sensitivity and Fourier coefficients of Boolean functions on bits of x deciding whether x is a quadratic residue. These results are used to obtain lower bounds on the parallel arithmetic and Boolean complexity of computing the discrete logarithm. For example, we prove that any unbounded fan-in Boolean circuit. of sublogarithmic depth computing the discrete logarithm modulo p must be of superpolynomial size Mathematics Data structures (Computer science) Data encryption (Computer science) Information theory Number theory Number Theory Data Encryption Theory of Computation Data Structures, Cryptology and Information Theory Mathematik Kryptosystem (DE-588)4209132-9 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Kryptologie (DE-588)4033329-2 gnd rswk-swf Boolesche Funktion (DE-588)4146281-6 gnd rswk-swf Faktorisierung (DE-588)4128927-4 gnd rswk-swf Komplexität (DE-588)4135369-9 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Modularithmetik (DE-588)4325008-7 gnd rswk-swf Polynom (DE-588)4046711-9 s Faktorisierung (DE-588)4128927-4 s Komplexität (DE-588)4135369-9 s Kryptosystem (DE-588)4209132-9 s 1\p DE-604 Algebraische Zahlentheorie (DE-588)4001170-7 s 2\p DE-604 Boolesche Funktion (DE-588)4146281-6 s 3\p DE-604 Kryptologie (DE-588)4033329-2 s Zahlentheorie (DE-588)4067277-3 s 4\p DE-604 Modularithmetik (DE-588)4325008-7 s 5\p DE-604 https://doi.org/10.1007/978-3-0348-8664-2 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Shparlinski, Igor Number Theoretic Methods in Cryptography Complexity lower bounds Mathematics Data structures (Computer science) Data encryption (Computer science) Information theory Number theory Number Theory Data Encryption Theory of Computation Data Structures, Cryptology and Information Theory Mathematik Kryptosystem (DE-588)4209132-9 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd Boolesche Funktion (DE-588)4146281-6 gnd Faktorisierung (DE-588)4128927-4 gnd Komplexität (DE-588)4135369-9 gnd Polynom (DE-588)4046711-9 gnd Modularithmetik (DE-588)4325008-7 gnd |
| subject_GND | (DE-588)4209132-9 (DE-588)4001170-7 (DE-588)4067277-3 (DE-588)4033329-2 (DE-588)4146281-6 (DE-588)4128927-4 (DE-588)4135369-9 (DE-588)4046711-9 (DE-588)4325008-7 |
| title | Number Theoretic Methods in Cryptography Complexity lower bounds |
| title_auth | Number Theoretic Methods in Cryptography Complexity lower bounds |
| title_exact_search | Number Theoretic Methods in Cryptography Complexity lower bounds |
| title_full | Number Theoretic Methods in Cryptography Complexity lower bounds by Igor Shparlinski |
| title_fullStr | Number Theoretic Methods in Cryptography Complexity lower bounds by Igor Shparlinski |
| title_full_unstemmed | Number Theoretic Methods in Cryptography Complexity lower bounds by Igor Shparlinski |
| title_short | Number Theoretic Methods in Cryptography |
| title_sort | number theoretic methods in cryptography complexity lower bounds |
| title_sub | Complexity lower bounds |
| topic | Mathematics Data structures (Computer science) Data encryption (Computer science) Information theory Number theory Number Theory Data Encryption Theory of Computation Data Structures, Cryptology and Information Theory Mathematik Kryptosystem (DE-588)4209132-9 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Kryptologie (DE-588)4033329-2 gnd Boolesche Funktion (DE-588)4146281-6 gnd Faktorisierung (DE-588)4128927-4 gnd Komplexität (DE-588)4135369-9 gnd Polynom (DE-588)4046711-9 gnd Modularithmetik (DE-588)4325008-7 gnd |
| topic_facet | Mathematics Data structures (Computer science) Data encryption (Computer science) Information theory Number theory Number Theory Data Encryption Theory of Computation Data Structures, Cryptology and Information Theory Mathematik Kryptosystem Algebraische Zahlentheorie Zahlentheorie Kryptologie Boolesche Funktion Faktorisierung Komplexität Polynom Modularithmetik |
| url | https://doi.org/10.1007/978-3-0348-8664-2 |
| work_keys_str_mv | AT shparlinskiigor numbertheoreticmethodsincryptographycomplexitylowerbounds |