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41-580: [REDACTED] Look up qpp in Wiktionary, the free dictionary. QPP may refer to: Quadratic permutation polynomial Quebec Pension Plan (QPP) Queensland People's Party Queerplatonic partnership Sûreté du Québec , sometimes referred to in English as "Quebec Provincial Police" Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

82-419: A 0 + ∑ 0 < i ≤ M a i x i {\textstyle g(x)=a_{0}+\sum _{0<i\leq M}a_{i}x^{i}} defines a permutation for the ring Z / n Z if and only if all the polynomials g p t ( x ) = a 0 , p t + ∑ 0 < i ≤ M

123-414: A i , p t x i {\textstyle g_{p_{t}}(x)=a_{0,p_{t}}+\sum _{0<i\leq M}a_{i,p_{t}}x^{i}} defines the permutations for all rings Z / p t k t Z {\displaystyle Z/p_{t}^{k_{t}}Z} , where a j , p t {\displaystyle a_{j,p_{t}}} are remainders of

164-547: A j {\displaystyle a_{j}} modulo p t k t {\displaystyle p_{t}^{k_{t}}} . As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider n = p 1 k 1 p 2 k 2 … p l k l {\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}\dots p_{l}^{k_{l}}} , assume that k 1 >1. Consider

205-697: A x 2 + b x {\displaystyle ax^{2}+bx} , such that a = 0 mod p 1 {\displaystyle a=0{\bmod {p}}_{1}} , but a ≠ 0 mod p 1 k 1 {\displaystyle a\neq 0{\bmod {p}}_{1}^{k_{1}}} ; assume that a = 0 mod p i k i {\displaystyle a=0{\bmod {p}}_{i}^{k_{i}}} , i > 1. And assume that b ≠ 0 mod p i {\displaystyle b\neq 0{\bmod {p}}_{i}} for all i = 1, ..., l . (For example, one can take

246-394: A x 2 + b x + c {\displaystyle g(x)=ax^{2}+bx+c} for the ring Z / p Z . Lemma: for k =1 (i.e. Z / p Z ) such polynomial defines a permutation only in the case a =0 and b not equal to zero. So the polynomial is not quadratic, but linear. Lemma: for k >1, p >2 ( Z / p Z ) such polynomial defines a permutation if and only if

287-555: A ≡ 0 ( mod p ) {\displaystyle a\equiv 0{\pmod {p}}} and b ≢ 0 ( mod p ) {\displaystyle b\not \equiv 0{\pmod {p}}} . Consider n = p 1 k 1 p 2 k 2 . . . p l k l {\displaystyle n=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{l}^{k_{l}}} , where p t are prime numbers. Lemma: any polynomial g ( x ) =

328-535: A ) = x {\displaystyle D_{1}(x,a)=x} . The first few Dickson polynomials are: If a ≠ 0 and n > 1 then D n ( x , a ) permutes GF( q ) if and only if ( n , q − 1) = 1 . If a = 0 then D n ( x , 0) = x and the previous result holds. The linearized polynomials that are permutation polynomials over GF( q ) form a group under the operation of composition modulo x q r − x {\displaystyle x^{q^{r}}-x} , which

369-419: A = p 1 p 2 k 2 . . . p l k l {\displaystyle a=p_{1}p_{2}^{k_{2}}...p_{l}^{k_{l}}} and b = 1 {\displaystyle b=1} ). Then such polynomial defines a permutation. To see this we observe that for all primes p i , i > 1, the reduction of this quadratic polynomial modulo p i

410-419: A , b and c are chosen so that g ( x ) is monic , g (0) = 0 and (provided the characteristic p does not divide the degree n of the polynomial) the coefficient of x is 0. There are many open questions concerning permutation polynomials defined over finite fields. Hermite's criterion is computationally intensive and can be difficult to use in making theoretical conclusions. However, Dickson

451-568: A Ph.D. student, and Dickson initially accepted Harvard's offer, but chose to attend Chicago instead. In 1896, when he was only 22 years of age, he was awarded Chicago's first doctorate in mathematics, for a dissertation titled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group , supervised by E. H. Moore . Dickson then went to Leipzig and Paris to study under Sophus Lie and Camille Jordan , respectively. On returning to

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492-628: A Texan by virtue of having grown up in Cleburne , where his father was a banker, merchant, and real estate investor. He attended the University of Texas at Austin , where George Bruce Halsted encouraged his study of mathematics. Dickson earned a B.S. in 1893 and an M.S. in 1894, under Halsted's supervision. Dickson first specialised in Halsted's own specialty, geometry . Both the University of Chicago and Harvard University welcomed Dickson as

533-597: A finite projective plane , PG(2, q ) with q a power of 2, can be coordinatized in such a way that the relationship between the coordinates is given by an o-polynomial , which is a special type of permutation polynomial over the finite field GF( q ) . The problem of testing whether a given polynomial over a finite field is a permutation polynomial can be solved in polynomial time . A polynomial f ∈ F q [ x 1 , … , x n ] {\displaystyle f\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]}

574-410: A polynomial with integer coefficients (i.e., in ℤ[ x ] ) is a permutation polynomial over GF( p ) for infinitely many primes p , then it is the composition of linear and Dickson polynomials. (See Schur's conjecture below). In finite geometry coordinate descriptions of certain point sets can provide examples of permutation polynomials of higher degree. In particular, the points forming an oval in

615-438: A proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. She concluded that Dickson should be credited with the first correct proof. Dickson's search for a counterexample to Wedderburn's theorem led him to investigate nonassociative algebras , and in

656-626: A series of papers he found all possible three and four-dimensional (nonassociative) division algebras over a field . In 1919 Dickson constructed Cayley numbers by a doubling process starting with quaternions H {\displaystyle \mathbb {H} } . His method was extended to a doubling of R {\displaystyle \mathbb {R} } to produce C {\displaystyle \mathbb {C} } , and of C {\displaystyle \mathbb {C} } to produce H {\displaystyle \mathbb {H} } by A. A. Albert in 1922, and

697-406: A theorem stating that all finite division algebras were commutative , now known as Wedderburn's theorem . The proofs all made clever use of the interplay between the additive group of a finite division algebra A , and the multiplicative group A * =  A  − {0}. Karen Parshall noted that the first of these three proofs had a gap not noticed at the time. Dickson also found

738-399: Is a permutation polynomial of F q if the function from F q to itself defined by c ↦ f ( c ) {\displaystyle c\mapsto f(c)} is a permutation of F q . Due to the finiteness of F q , this definition can be expressed in several equivalent ways: A characterization of which polynomials are permutation polynomials

779-638: Is a permutation polynomial in n variables over F q {\displaystyle \mathbb {F} _{q}} if the equation f ( x 1 , … , x n ) = α {\displaystyle f(x_{1},\ldots ,x_{n})=\alpha } has exactly q n − 1 {\displaystyle q^{n-1}} solutions in F q n {\displaystyle \mathbb {F} _{q}^{n}} for each α ∈ F q {\displaystyle \alpha \in \mathbb {F} _{q}} . For

820-778: Is actually linear polynomial and hence is permutation by trivial reason. For the first prime number we should use the lemma discussed previously to see that it defines the permutation. For example, consider Z /12 Z and polynomial 6 x 2 + x {\displaystyle 6x^{2}+x} . It defines a permutation ( 0 1 2 3 4 5 6 7 8 ⋯ 0 7 2 9 4 11 6 1 8 ⋯ ) . {\begin{pmatrix}0&1&2&3&4&5&6&7&8&\cdots \\0&7&2&9&4&11&6&1&8&\cdots \end{pmatrix}}. A polynomial g ( x ) for

861-450: Is given by ( Hermite 's Criterion ) f ∈ F q [ x ] is a permutation polynomial of F q if and only if the following two conditions hold: If f ( x ) is a permutation polynomial defined over the finite field GF( q ) , then so is g ( x ) = a f ( x + b ) + c for all a ≠ 0, b and c in GF( q ) . The permutation polynomial g ( x ) is in normalized form if

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902-499: Is known as the Betti-Mathieu group, isomorphic to the general linear group GL( r , F q ) . An exceptional polynomial over GF( q ) is a polynomial in F q [ x ] which is a permutation polynomial on GF( q ) for infinitely many m . A permutation polynomial over GF( q ) of degree at most q is exceptional over GF( q ) . Every permutation of GF( q ) is induced by an exceptional polynomial. If

943-794: The University of California in 1914, 1918, and 1922. In 1939, he returned to Texas to retire. Dickson married Susan McLeod Davis in 1902; they had two children, Campbell and Eleanor. Dickson was elected to the National Academy of Sciences in 1913, and was also a member of the American Philosophical Society, the American Academy of Arts and Sciences , the London Mathematical Society , the French Academy of Sciences and

984-626: The finite ring Z / n Z one can construct quadratic permutation polynomials. Actually it is possible if and only if n is divisible by p for some prime number p . The construction is surprisingly simple, nevertheless it can produce permutations with certain good properties. That is why it has been used in the interleaver component of turbo codes in 3GPP Long Term Evolution mobile telecommunication standard (see 3GPP technical specification 36.212 e.g. page 14 in version 8.8.0). Consider g ( x ) = 2 x 2 + x {\displaystyle g(x)=2x^{2}+x} for

1025-469: The recursion D n ( x , a ) = x D n − 1 ( x , a ) − a D n − 2 ( x , a ) , {\displaystyle D_{n}(x,a)=xD_{n-1}(x,a)-aD_{n-2}(x,a),} with the initial conditions D 0 ( x , a ) = 2 {\displaystyle D_{0}(x,a)=2} and D 1 ( x ,

1066-416: The 56 having order less than 1 million. The remaining three were found in 1960, 1965, and 1967. Dickson worked on finite fields and extended the theory of linear associative algebras initiated by Joseph Wedderburn and Cartan . He started the study of modular invariants of a group . In 1905, Wedderburn, then at Chicago on a Carnegie Fellowship, published a paper that included three claimed proofs of

1107-584: The Galois field theory , a revision and expansion of his Ph.D. thesis. Teubner in Leipzig published the book, as there was no well-established American scientific publisher at the time. Dickson had already published 43 research papers in the preceding five years; all but seven on finite linear groups . Parshall (1991) described the book as follows: An appendix in this book lists the non-abelian simple groups then known having order less than 1 billion. He listed 53 of

1148-534: The Theory of Numbers (1919–23) is still much consulted today, covering divisibility and primality, Diophantine analysis , and quadratic and higher forms. The work contains little interpretation and makes no attempt to contextualize the results being described, yet it contains essentially every significant number theoretic idea from the dawn of mathematics up to the 1920s except for quadratic reciprocity and higher reciprocity laws. A planned fourth volume on these topics

1189-528: The US, he became an instructor at the University of California . In 1899 and at the extraordinarily young age of 25, Dickson was appointed associate professor at the University of Texas. Chicago countered by offering him a position in 1900, and he spent the balance of his career there. At Chicago, he supervised 53 Ph.D. theses; his most accomplished student was probably A. A. Albert . He was a visiting professor at

1230-803: The Union of Czech Mathematicians and Physicists. Dickson was the first recipient of a prize created in 1924 by The American Association for the Advancement of Science, for his work on the arithmetics of algebras. Harvard (1936) and Princeton (1941) awarded him honorary doctorates. Dickson presided over the American Mathematical Society in 1917–1918. His December 1918 presidential address, titled "Mathematics in War Perspective", criticized American mathematics for falling short of those of Britain, France, and Germany: In 1928, he

1271-577: The assertion that, if a polynomial f defined over K is a permutation polynomial on R / P for infinitely many prime ideals P , then f is the composition of Dickson polynomials, degree-one polynomials, and polynomials of the form x . In fact, Schur did not make any conjecture in this direction. The notion that he did is due to Fried, who gave a flawed proof of a false version of the result. Correct proofs have been given by Turnwald and Müller. Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954)

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1312-422: The case of finite rings Z / n Z , such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms. Let F q = GF( q ) be the finite field of characteristic p , that is, the field having q elements where q = p for some prime p . A polynomial f with coefficients in F q (symbolically written as f ∈ F q [ x ] )

1353-419: The polynomial defines the permutation ( 0 1 2 3 4 5 6 7 0 3 2 5 4 7 6 1 ) . {\displaystyle {\begin{pmatrix}0&1&2&3&4&5&6&7\\0&3&2&5&4&7&6&1\end{pmatrix}}.} Consider g ( x ) =

1394-708: The procedure is known now as the Cayley–Dickson construction of composition algebras . Dickson proved many interesting results in number theory , using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory . He proved the Waring's problem for k ≥ 7 {\displaystyle k\geq 7} under the further condition of independently of Subbayya Sivasankaranarayana Pillai who proved it for k ≥ 6 {\displaystyle k\geq 6} ahead of him. The three-volume History of

1435-451: The ring Z / p Z is a permutation polynomial if and only if it permutes the finite field Z / p Z and g ′ ( x ) ≠ 0 mod p {\displaystyle g'(x)\neq 0{\bmod {p}}} for all x in Z / p Z , where g ′( x ) is the formal derivative of g ( x ). Let K be an algebraic number field with R the ring of integers . The term "Schur's conjecture" refers to

1476-610: The ring Z /4 Z . One sees: g ( 0 ) = 0 {\displaystyle g(0)=0} ; g ( 1 ) = 3 {\displaystyle g(1)=3} ; g ( 2 ) = 2 {\displaystyle g(2)=2} ; g ( 3 ) = 1 {\displaystyle g(3)=1} , so the polynomial defines the permutation ( 0 1 2 3 0 3 2 1 ) . {\displaystyle {\begin{pmatrix}0&1&2&3\\0&3&2&1\end{pmatrix}}.} Consider

1517-798: The same polynomial g ( x ) = 2 x 2 + x {\displaystyle g(x)=2x^{2}+x} for the other ring Z / 8 Z . One sees: g ( 0 ) = 0 {\displaystyle g(0)=0} ; g ( 1 ) = 3 {\displaystyle g(1)=3} ; g ( 2 ) = 2 {\displaystyle g(2)=2} ; g ( 3 ) = 5 {\displaystyle g(3)=5} ; g ( 4 ) = 4 {\displaystyle g(4)=4} ; g ( 5 ) = 7 {\displaystyle g(5)=7} ; g ( 6 ) = 6 {\displaystyle g(6)=6} ; g ( 7 ) = 1 {\displaystyle g(7)=1} , so

1558-508: The title QPP . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=QPP&oldid=1197552671 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Permutation polynomial#Quadratic permutation polynomials (QPP) over finite rings In

1599-504: Was able to use it to find all permutation polynomials of degree at most five over all finite fields. These results are: A list of all monic permutation polynomials of degree six in normalized form can be found in Shallue & Wanless (2013) . Beyond the above examples, the following list, while not exhaustive, contains almost all of the known major classes of permutation polynomials over finite fields. These can also be obtained from

1640-610: Was also the first recipient of the Cole Prize for algebra, awarded annually by the AMS, for his book Algebren und ihre Zahlentheorie . It appears that Dickson was a hard man: Dickson had a major impact on American mathematics, especially abstract algebra . His mathematical output consists of 18 books and more than 250 papers. The Collected Mathematical Papers of Leonard Eugene Dickson fill six large volumes. In 1901, Dickson published his first book Linear groups with an exposition of

1681-417: Was an American mathematician . He was one of the first American researchers in abstract algebra , in particular the theory of finite fields and classical groups , and is also remembered for a three-volume history of number theory , History of the Theory of Numbers . The L. E. Dickson instructorships at the University of Chicago Department of Mathematics are named after him. Dickson considered himself

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