Cryptography is the study of encryptying and decrypting messages and deciphering encrypted messages when the code is unknown. We consider Λπ(Δx, Δy) which is a count of how many ways a permutation satisfies a certain property. According to Hawkes and O'Connor, the distribution of Λπ(Δx, Δy) tends to a Poisson distribution with parameter ½ as m → ∞ for all Δx,Δy ∈ (Z/qZ)m - 0. We give a proof of this theorem using the Stein-Chen method: As qm approaches infinity, the distribution of Λπ(Δx, Δy) is approximately Poisson with parameter ½. Error bounds for this approximation are provided.
","abstract_html":"<p>Cryptography is the study of encryptying and decrypting messages and deciphering encrypted messages when the code is unknown. We consider Λ<sub>π</sub>(Δ<em>x</em>, Δ<em>y</em>) which is a count of how many ways a permutation satisfies a certain property. According to Hawkes and O'Connor, the distribution of Λ<sub>π</sub>(Δ<em>x</em>, Δ<em>y</em>) tends to a Poisson distribution with parameter ½ as <em>m</em> → ∞ for all Δ<em>x</em>,Δ<em>y</em> ∈ (<b>Z</b>/<em>q</em><b>Z</b>)<sup><em>m</em></sup> - 0. We give a proof of this theorem using the Stein-Chen method: As <em>q<sup>m</sup></em> approaches infinity, the distribution of Λ<sub>π</sub>(Δ<em>x</em>, Δ<em>y</em>) is approximately Poisson with parameter ½. Error bounds for this approximation are provided.</p>","abstract_has_math":false,"creators":["Lynch, Kevin"],"institution":null,"degree_name":"MS (Master of Science)","degree_level":"Thesis - unrestricted","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2005,"date_issued":"2005-08-16T07:00:00Z","date_published":"2005-08-16T07:00:00Z","updated_at":"2026-07-24T02:19:43Z","subjects":["Cryptography","Stein-Chen Method","Poisson Approximation","Cryptanalysis","Error Bound","Coupling Method","Even Word","Multiple of Three Word","Applied Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":["Copyright by the authors."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://dc.etsu.edu/etd/1042","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lynch, Kevin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2005-08-16T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis - unrestricted"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS (Master of Science)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cryptography","Stein-Chen Method","Poisson Approximation","Cryptanalysis","Error Bound","Coupling Method","Even Word","Multiple of Three Word","Applied Mathematics","Physical Sciences and Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Copyright by the authors."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://dc.etsu.edu/context/etd/article/2199/viewcontent/LynchK072505f.pdf","https://dc.etsu.edu/etd/1042"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Cryptography is the study of encryptying and decrypting messages and deciphering encrypted messages when the code is unknown. We consider Λπ(Δx, Δy) which is a count of how many ways a permutation satisfies a certain property. According to Hawkes and O'Connor, the distribution of Λπ(Δx, Δy) tends to a Poisson distribution with parameter ½ as m → ∞ for all Δx,Δy ∈ (Z/qZ)m - 0. We give a proof of this theorem using the Stein-Chen method: As qm approaches infinity, the distribution of Λπ(Δx, Δy) is approximately Poisson with parameter ½. Error bounds for this approximation are provided.
"]},{"key":"dc:title","label":"Title","values":["A Limit Theorem in Cryptography."]}]}],"canonical_facts":{"dc:creator":["Lynch, Kevin"],"dc:date.issued":["2005-08-16T07:00:00Z"],"dc:description.abstract":["Cryptography is the study of encryptying and decrypting messages and deciphering encrypted messages when the code is unknown. We consider Λπ(Δx, Δy) which is a count of how many ways a permutation satisfies a certain property. According to Hawkes and O'Connor, the distribution of Λπ(Δx, Δy) tends to a Poisson distribution with parameter ½ as m → ∞ for all Δx,Δy ∈ (Z/qZ)m - 0. We give a proof of this theorem using the Stein-Chen method: As qm approaches infinity, the distribution of Λπ(Δx, Δy) is approximately Poisson with parameter ½. Error bounds for this approximation are provided.
"],"dc:identifier":["https://dc.etsu.edu/context/etd/article/2199/viewcontent/LynchK072505f.pdf","https://dc.etsu.edu/etd/1042"],"dc:rights":["Copyright by the authors."],"dc:subject":["Cryptography","Stein-Chen Method","Poisson Approximation","Cryptanalysis","Error Bound","Coupling Method","Even Word","Multiple of Three Word","Applied Mathematics","Physical Sciences and Mathematics"],"dc:title":["A Limit Theorem in Cryptography."],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis - unrestricted"],"thesis:degree_name":["MS (Master of Science)"]},"updated_at":"2026-07-24T02:19:43Z"}