Jacobi Sums and Correlations of Sidelnikov Sequences
aa r X i v : . [ m a t h . C O ] F e b JACOBI SUMS AND CORRELATIONS OF SIDELNIKOVSEQUENCES
AYS¸E ALACA AND GOLDWYN MILLAR
Abstract.
We consider the problem of determining the cross-correlation valuesof the sequences in the families comprised of constant multiples of M -ary Sidel-nikov sequences over F q , where q is a power of an odd prime p . We show thatthe cross-correlation values of pairs of sequences from such a family can be ex-pressed in terms of certain Jacobi sums. This insight facilitates the computationof the cross-correlation values of these sequence pairs so long as φ ( M ) φ ( M ) ≤ q. We are also able to use our Jacobi sum expression to deduce explicit formulaefor the cross-correlation distribution of a family of this type in the special casethat there exists an integer x such that p x ≡ − M ) . Introduction
A sequence a = a a a . . . of elements from the ring Z /M Z , where M is apositive integer, is called an M -ary sequence. A sequence a is said to be periodic if there is an integer v > a i = a v + i for all integers i ≥
0. If v is thesmallest such integer, then we say that a has period v . For the rest of this section,we assume that a is a periodic M -ary sequence of period v .Let b = b b b . . . be another periodic M -ary sequence of period v. The (peri-odic) correlation C a , b of a and b is defined as follows: for each nonnegative integer τ , C a , b ( τ ) := v − X t =0 exp (cid:16) πi ( a t − b t + τ ) M (cid:17) , (1.1)where the terms of a and b appearing in the exponents are interpreted as integers.The function C a , a is called the autocorrelation of a , and the values C a , a ( τ ) for1 ≤ τ ≤ v − out-of-phase autocorrelation values of a . We say that a has low out-of-phase autocorrelation if its out-of-phase autocorrelation valuesare small compared to v. Low out-of-phase autocorrelation is one of the criteriafor a periodic sequence to be suitable for use as a key sequence in a stream-cipher cryptosystem (see the discussion of Golomb’s postulate R3 in [6, Section5.1] and [15]). Indeed, for a sequence to be useful for this purpose, its out-of-phaseautocorrelation values should be close to zero.If there exists an integer ℓ such that for each positive integer i, a i = b i + ℓ , thenwe say that a and b are shift-equivalent and that a and b are shifts of one another; n this case, C a , b can be obtained from C a , a by C a , b ( τ ) = ( C a , a ( τ − ℓ ) if τ ≥ ℓ, C a , a ( τ + v − ℓ ) if τ < ℓ. If a and b are shift-inequivalent, then we say that C a , b is the cross-correlation of a and b . Furthermore, we consider a family F of shift-inequivalent sequencesto have low cross-correlation if for any pair of sequences c , d in F and for any τ, C c , d ( τ ) is small compared to v. The following result, which is due to Welch[16], gives a lower bound on the cross-correlation and out-of-phase autocorrelationvalues of the sequences in a family of a given size.
Theorem 1.1. [16, Theorem (periodic correlation)]
Let S be a family of s shift-inequivalent M -ary sequences of period ℓ. Let C max denote the maximum of thecross-correlation and out-of-phase autocorrelation values of the sequences in S. Then C max ≥ r ℓ ( s − sℓ − . Remark 1.1.
Following [6] , we say that F has low cross-correlation if for any pairof sequences c , d from F , C c , d outputs only values less than or equal to δ √ v + ǫ, for some small integers δ and ǫ . Families of shift-inequivalent sequences with low cross-correlation (and such thateach member has low out-of-phase autocorrelation) have found use in the designof code division multiple access (CDMA) radio communications systems (see, forinstance, [6] or [7]). For CDMA applications, one would like such families to beas large as possible (i.e. to include as many sequences as possible). One may alsodesire that sequences in these families have certain additional properties (such ascryptographic strength).One indicator of cryptographic strength (besides low out-of-phase autocorrela-tion) is the balance property . We say that the sequence a is balanced if in a givenperiod of a (i.e. in a given list of v consecutive elements of a ) each element of Z /M Z appears either ⌊ v/M ⌋ or ⌈ v/M ⌉ times. Definition 1.1.
Let p be an odd prime, let d be a positive integer, and let q = p d . Let α be a primitive element of F q , and let M | q − . Following [8] , we set D k = { α Mi + k − | ≤ i < ( q − /M } for ≤ k ≤ M − . An M-ary Sidelnikov sequence s = s s s . . . is a sequence of period q − whosefirst q − elements are defined as follows: for ≤ j < q − s j = ( if α j = − ,k if α j ∈ D k . his class of sequences was originally discovered by Sidelnikov in 1969 [12] and,in the binary case (i.e. the case in which M = 2) rediscovered independently byLempel, Cohn, and Eastman in 1977 [4].It is known that the Sidelnikov sequences have low out-of-phase autocorrelation.Indeed, their out-of-phase autocorrelation values have magnitude at most 4 (see,for instance, [3]). It is clear from the definition of these sequences that they havethe balance property. Remark 1.2. If c ∈ Z /M Z , then we stipulate that c a is the sequence whose i thentry is ca i and we say that c a is a constant multiple of a . The authors of [11] consider the family S of sequences consisting of all nonzeroconstant multiples of an M -ary Sidelnikov sequence. They are able to prove that S has rather remarkable correlation properties. Indeed, they use the Weil bound toprove the following upper bound on the cross-correlation of a two distinct constantmultiples of a Sidelnikov sequence. Theorem 1.2. [11, Theorem 3]
Let q be a power of an odd prime, and let M | q − . Let s be an M-ary Sidelnikov sequence over F q . Let c , c ∈ Z /M Z , c = c and c , c = 0 . Let a := c s , and let b := c s . Then for each τ = 0 , . . . , q − , |C a , b ( τ ) | ≤ √ q + 3 . As the authors of [11] note, the upper bound in Theorem 1.2 implies that S is afamily of M − C max denotes the maximum of the cross-correlation and out-of-phase auto-correlation values of the sequences in S , then by Theorem 1.1, C max ≥ s ( q − ( M − q − M − − , and this expression is asymptotically equivalent to √ q. But the suitability of a sequence family for CDMA depends not just on the max-imum cross-correlation value of its sequence pairs (although the maximum valueis important). It is also important to determine the rest of the cross-correlationvalues of the pairs from the sequence family. Furthermore, we note that for fixedvalues of M and q, there is some distance between the lower bound given in The-orem 1.1 and the upper bound given in Theorem 1.2. For instance, if q = 5 and M = 4 , then s ( q − ( M − q − M − − ≈ .
63 and √ q + 3 ≈ . . So, the known bounds given do not completely specify the maximum cross-correlationvalues. Thus, it is of interest to determine the precise cross-correlation values ofthe sequences from S . n this paper, we show that the cross-correlation values of two sequences from S can be represented in terms of certain Jacobi sums. We then use known evaluationsof Jacobi sums to determine the precise cross-correlation distribution of S in certaincases. 2. Preliminaries
Let q be a prime power, and let F q be a finite field with q elements. A homo-morphism χ : F ∗ q → C is called a character on F ∗ q . The trivial character on F ∗ q isdefined by the rule that χ ( x ) = 1 for all x ∈ F ∗ q . Any character on F ∗ q which isdifferent from the trivial character is called a nontrivial character .We note that if χ is a nontrivial character on F ∗ q , it is common to extend χ toa map on F q by setting χ (0) = 0. However, it is sometimes useful to define χ (0)to be equal to something else. In this paper, we consider both characters χ on F q for which χ (0) = 0 and characters χ on F q for which χ (0) = 1.It is possible to define a logarithm over F q . Let α be a primitive element of F q . For x ∈ F q , we stipulate thatlog α ( x ) = ( i if x = α i , ≤ i ≤ q − , x = 0 . As Gong and Yu note ([8] and [9]) the M -ary Sidelnikov sequence s defined over F q using α is completely determined by the congruences(2.1) s i ≡ log α ( α i + 1) (mod M ) , ≤ i ≤ q − . Still following Gong and Yu, we define a multiplicative character ψ M of order M on F q by the rule that for x ∈ F q ,(2.2) ψ M ( x ) = exp (cid:16) πi log α ( x ) M (cid:17) . Note that ψ M (0) = 1. Gong and Yu remark that (2.1) and (2.2) imply the identity(2.3) exp (cid:16) πis j M (cid:17) = ψ M ( α j + 1) , ≤ j ≤ q − . Let χ M denote the character on F q defined by χ M ( x ) = ( ψ M ( x ) if x = 0 , x = 0 . We shall need to make use of certain types of character sums called
Gauss sums and
Jacobi sums . These sums were originally studied by C. F. Gauss and C. G. I.Jacobi, hence the names. efinition 2.1. Let q = p k be a power of a prime p , and let χ and ψ be nontrivialcharacters on F q ( which map to . Then the sum J ( χ, ψ ) = X x ∈ F q χ ( x ) ψ (1 − x ) is called a Jacobi sum , and the sum G ( χ ) = X x ∈ F q χ ( x ) exp (cid:16) πi Tr ( x ) p (cid:17) ( where Tr denotes the field trace from F q to F p ) is called a Gauss sum . The following identity relates Gauss and Jacobi sums, see [10, Theorem 2.1.3].If χψ is not the trivial character, then(2.4) J ( χ, ψ ) = G ( χ ) G ( ψ ) G ( χψ ) . We note the following basic properties of Gauss and Jacobi sums (see [10]). G ( χ ) = q − χ is trivial . | G ( χ ) | = √ q if χ is nontrivial .J ( χ, ψ ) = − χ = ψ. | J ( χ, ψ ) | = √ q if χ = ψ. It is not known, in general, how to produce exact, simple, closed form evaluationsof J ( χ, ψ ) and G ( χ ) . That being said, the sums J ( χ, ψ ) and G ( χ ) have beenevaluated in certain special cases. One case in which there are known evaluationsof Gauss and Jacobi sums is the small order case (when the order of the charactersinvolved in the sums is small relative to q ). Explicit formulae for Gauss and Jacobisums over characters of orders 3 , , , . . . etc have been given in the literature (see[1, Chapters 3 and 4]). However, these formulae are rather lengthy and difficult toapply. Alternatively, there are algorithms that can be used to compute Jacobi sumsover characters of small order (see, for instance, [2] and [14]). Let χ and ψ be twocharacters on F q , and let e = gcd( | χ | , | ψ | ) . The algorithm given in [14] computesthe Jacobi sum J ( χ, ψ ) faster than just naively summing the series so long as φ ( e ) φ ( e ) ≤ q , where φ is the Euler totient function. There are other situations inwhich it is possible to give explicit evaluations of Gauss and Jacobi sums. We willmake use of known evaluations for the so-called pure Gauss and Jacobi sums. AGauss or Jacobi sum is called pure if some positive integral power of it is real. Thefollowing theorem completely classifies pure Gauss sums (see [1, Theorems 11.6.3and 11.6.4]). Theorem 2.1.
Let m ∈ N , let p be an odd prime, and let q = p m . Let k | q − ,and let χ be a character of order k . Then G ( χ ) is pure if and only if there existsa positive integer x such that p x ≡ − k ) . Furthermore, if there exist such ntegers and t is the least such integer, then there exists a positive integer s suchthat m = 2 ts, and G ( χ ) = ( − s − p t +1) s/k p m/ . We can use Theorem 2.1 to evaluate a large class of pure Jacobi sums.
Corollary 2.1.
Let m ∈ N , let p be an odd prime, and let q = p m . Let k | q − , andlet χ be a character of order k . Suppose that there exists a positive integer x suchthat p x ≡ − k ) . Let a and b be integers such that a + b k ) . Thenthere exist positive integers w , y , and z such that p w ≡ − k/ gcd ( k, a )) , p y ≡ − k/ gcd ( k, b )) , and p z ≡ − k/ gcd ( k, a + b )) . Let t a , t b , and t a + b be the least such integers. Then there exist positive integers s a , s b , and s a + b such that m = 2 t a s a = 2 t b s b = 2 t a + b s a + b . Furthermore, J ( χ a , χ b ) = ( − s a + s b + s a + b +1+(( p ta +1) s a gcd ( k,a )+( p tb +1) s b gcd ( k,b )+( p ta + b +1) s a + b gcd ( k,a + b )) /k p m/ . Proof.
For each positive integer c, | χ c | = k/ gcd( k, c ) divides k. Hence, thereexists a positive integer x such that p x ≡ − k/ gcd( k, c )) . Let t c be theleast such integer. Then by Theorem 2.1, there exists an integer s c such that m = 2 t c s c and G ( χ c ) = ( − s c − p tc +1) s c gcd( k,c ) /k p m/ . (2.5)By (2.4) and (2.5) we have J ( χ a , χ b ) = G ( χ a ) G ( χ b ) G ( χ a + b )= ( − s a − p ta +1) s a gcd( k,a ) /k p m/ ( − s b − p tb +1) s b gcd( k,b ) /k p m/ ( − s a + b − p ta + b +1) s a + b gcd( k,a + b ) /k p m/ = ( − s a + s b + s a + b +1+(( p ta +1) s a gcd( k,a )+( p tb +1) s b gcd( k,b )+( p ta + b +1) s a + b gcd( k,a + b )) /k p m/ . (cid:3) The Jacobi sum evaluation given in Corollary 2.1 is a bit cumbersome, but insome cases it simplifies nicely (as in the following corollary).
Corollary 2.2.
Let m ∈ N be such that m ≡ , let p be an odd prime,and let q = p m . Let k | q − , and let χ be a character of order k . Suppose thatthere exists a positive integer x such that p x ≡ − k ) . Furthermore, supposethat the least positive integer satisfying this equation is odd. Let a and b be integerssuch that a + b k ) . Then J ( χ a , χ b ) = − p m/ . Proof.
Let c be a positive integer, and suppose that t is the least positive integersatisfying p x ≡ − k ) . As above, let t c be the least positive solution of p y ≡− k/ gcd( k, c )) . Note that, in this case, the order of p in the multiplicative roup ( Z / gcd( k, c ) Z ) ∗ is 2 t c . By the Division Algorithm, there exist unique integers y and r such that t = t c y + r and 0 ≤ r < t c . Thus, − ≡ p t ≡ p t c y + r ≡ ( − y p r (mod k/ gcd( k, c )) . (2.6)If y is even, then p r ≡ − k/ gcd( k, c )) . But since r < t c , this contradictsthe fact that t c is the least positive integer satisfying p y ≡ − k/ gcd( k, c )).Hence y must be odd. Then by (2.6) we have p r ≡ k/ gcd( k, c )). Thus,unless r = 0 , since r < t c < t c , this contradicts the fact that 2 t c is the order of p in( Z / gcd( k, c ) Z ) ∗ . It follows that r = 0 . Hence, t c | t. Thus, if t is odd, then the terms t a , t b , and t a + b in Corollary 2.1 must also be odd. Hence, if m ≡ , then s a , s b , and s a + b must all be even. The assertion now follows from Corollary 2.1. (cid:3) We also need to make use of cyclotomic numbers.
Definition 2.2.
Let q = p m be an odd prime power, let k | q − , and let α be aprimitive element in F q . Then for ≤ u ≤ k − , we set C u = (cid:26) α kℓ + u | ≤ ℓ ≤ q − k (cid:27) . The sets C u are called cyclotomic classes . For ≤ u, v ≤ k − , the cyclotomicnumber ( u, v ) k is defined to be the number of elements x ∈ C u such that x ∈ C v . Cyclotomic numbers were first studied by C. F. Gauss as part of his work onthe problem of finding straight edge and compass constructions for regular n -gons[13]. There is a connection between cyclotomic numbers and Jacobi sums. For aproof of the next result, see [1, Theorem 2.5.1]. Theorem 2.2.
Let q = p m be an odd prime power, let k | q − , and let α be aprimitive element in F q . Let χ be a character of order k . Then for ≤ u ≤ k − , k ( u, v ) k = X w,x (mod k ) χ u ( − J ( χ u , χ v ) − ℓ ) ζ − uw − vx , where ζ is a certain primitive k th root of unity and ℓ is the number of trivialcharacters in the set { χ u , χ v } . A number of authors have made use of Theorem 2.2 to compute formulae forcyclotomic numbers of “small order.” See [13] for a summary of many such results.By making use of the evaluations of Jacobi sums in the pure case, one can deducethe following corollary to Theorem 2.2 (see [1, Section 11.6] and [5]).
Corollary 2.3.
Let m ∈ N , let p be an odd prime, and let q = p m . Let k | q − ,and suppose that there exists a positive integer x such that p x ≡ − k ) . Let t be the least such integer. Then there exists a positive integer s such that m = 2 ts .Furthermore, k (0 , k = q + 1 − k − ( k − k − − s p m/ . f b k ) , then k (0 , b ) k = k ( b, k = k ( k − b, k − b ) k = q + 1 − k + ( − s ( k − p m/ . If a, b, a − b k ) , then k ( a, b ) k = q + 1 − − s p m/ . Cross-correlation distributions
Theorem 3.1.
Let q be an odd prime power, let α be a primitive element of F q ,and let M | q − . Let c , c ∈ { , . . . , M − } with c = c . Let s be the M -arySidelnikov sequence over F q defined using α. Then C c s ,c s (0) = − and for τ = 1 , . . . , q − , C c s ,c s ( τ ) = χ c M (1 − α − τ ) χ − c M (1 − α τ )( J ( χ c M , χ − c M ) + 2) − . Proof.
For any τ , by (1.1) and (2.3) we have C c s ,c s ( τ ) = q − X t =0 exp (cid:16) πi ( c s t − c s t + τ ) M (cid:17) = q − X t =0 ψ c M ( α t + 1) ψ − c M ( α t + τ + 1)As t ranges from 0 to q − α t + 1 ranges over all of the elements of F q except for1. So, C c s ,c s (0) = q − X t =0 ψ c − c M ( α t + 1) = − . Now, assume that τ = 1 , ..., q − . Note that C c s ,c s ( τ ) = ψ − c M ( α τ ) q − X t =0 ψ c M ( α t + 1) ψ − c M ( α t + 1 + ( α − τ − . Again, using the fact that as t ranges from 0 to q − α t + 1 ranges over all of theelements of F q except for 1 , we get that C c s ,c s ( τ ) = ψ − c M ( α τ ) (cid:0) X x ∈ F q ψ c M ( x ) ψ − c M ( x + ( α − τ − − ψ − c M ( α − τ ) (cid:1) = ψ − c M ( α τ ) X x ∈ F q ψ c M ( x ) ψ − c M ( x + ( α − τ − − . inally, as x runs over the elements of F q , − ( α − τ − x also runs over the elementsof F q . So, C c s ,c s ( τ ) = ψ − c M ( α τ ) X x ∈ F q ψ c M ( − ( α − τ − x ) ψ − c M ( − ( α − τ − x + ( α − τ − − ψ − c M ( α τ ) ψ c M ( − ( α − τ − ψ − c M ( α − τ − X x ∈ F q ψ c M ( x ) ψ − c M (1 − x ) − χ c M (1 − α − τ ) χ − c M (1 − α τ )( J ( χ c M , χ − c M ) + 2) − . (cid:3) So long as φ ( M ) φ ( M ) ≤ q, Theorem 3.1 can be used in concert with the algorithmfrom [14] to facilitate computations of cross-correlations of constant multiples ofM-ary Sidelnikov sequences. Indeed, we consider this to be the main applicationof Theorem 3.1.We will also use Theorem 3.1 together with known evaluations of pure Jacobisums to deduce explicit formulae for the cross-correlation values of constant mul-tiples of M-ary Sidelnikov sequences in certain special cases. Note that(3.1) J ( χ c M , χ − c M ) = − c + c ≡ M ) . The next corollary follows directly from (3.1), Theorem 3.1, Corollary 2.1, andCorollary 2.2.
Corollary 3.1.
Let m ∈ N , p be an odd prime, and q = p m . Let α be a primitiveelement of F q , and let M | q − . Let c , c ∈ { , . . . , M − } with c = c . Let s be the M -ary Sidelnikov sequence over F q defined using α . Let τ ∈ { , . . . , q − } .If c + c ≡ M ) , then C c s ,c s ( τ ) = χ c M (1 − α − τ ) χ − c M (1 − α τ ) − . Suppose that c + c M ) . Suppose also that there exists a positive integer x such that p x ≡ − M ) . Then there exist positive integers w, y, and z suchthat p w ≡ − M/ gcd ( M, c )) , p y ≡ − M/ gcd ( M, c )) , and p z ≡ − M/ gcd ( M, c + c )) . Let t c , t c , and t c + c be the least such integers. Thenthere exist positive integers s c , s c , and s c + c such that m = 2 t c s c = 2 t c s c = 2 t c + c s c + c . Furthermore, C c s ,c s ( τ ) = χ c M (1 − α − τ ) χ − c M (1 − α τ )(( − ǫ p m/ + 2) − , where ǫ = s c + s c + s c + c + 1 + (( p t c + 1) s c gcd ( M, c ) + ( p t c + 1) s c gcd ( M, c )+( p t c c + 1) s c + c gcd ( M, c + c )) /M. f m ≡ and the least positive integer satisfying the equation p x ≡ − M ) is odd, then C c s ,c s ( τ ) = χ c M (1 − α − τ ) χ − c M (1 − α τ )( − p m/ + 2) − . By borrowing some ideas from the authors of [3], we can use Corollary 3.1 to de-termine the cross-correlation distributions of certain families of constant multiplesof Sidelnikov sequences.By Corollary 3.1, for fixed c and c , the cross-correlation C c s ,c s ( τ ) depends on χ c M (1 − α − τ ) χ − c M (1 − α τ ) . For τ ∈ { , , . . . , q − } , let y = α τ . Note that y = 0 , ω M = exp (cid:0) πiM (cid:1) . Let 0 ≤ u, v ≤ M − χ c M (cid:16) y − y (cid:17) = ω uM and χ c M (cid:16) − y (cid:17) = ω vM . From (3.3) we have χ c M (1 − α − τ ) χ − c M (1 − α τ ) = χ c M ( α − τ ( α τ − χ − c M (1 − α τ )(3.4) = χ c M (cid:16) y − y (cid:17) χ c M (cid:16) − y (cid:17) = ω c u + c vM . Hence, the values taken on by χ c M (1 − α − τ ) χ − c M (1 − α τ ) are completely determinedby the cardinalities of the sets S u,v = (cid:26) y ∈ F p m \ { , } | χ M (cid:16) y − y (cid:17) = ω uM and χ M (cid:16) − y (cid:17) = ω vM (cid:27) , where 0 ≤ u, v ≤ M − . The authors of [3] characterized the cardinalities of thesesets in terms of cyclotomic numbers.
Theorem 3.2. [3, Theorem 11]
Let ≤ u, v ≤ M − . Then | S u,v | = ( u + v, v ) M . In the following example we use Corollary 3.1, Theorem 3.2, and Corollary 2.3to illustrate how to determine the cross-correlation distribution of the family ofconstant multiples of an M -ary Sidelnikov sequence. Example 3.1.
Let p = 3 , m = 4 , q = p m = 81 , and M = 4 . By (3.2) , ω M = i .Let α be a primitive element of F , and let s be the M -ary Sidelnikov sequencedefined using α . Note that M | q − . Furthermore, p ≡ − M ) , so that t = 1 . We also have that m = 2 t · , so that s = 2 . Thus, we can apply Corollary3.1 to determine the cross-correlation values of any two constant multiples of s .We will explicitly show how this is done for c = 1 and c = 2 . ince c + c M ) , for τ ∈ { , , . . . , q − } , by Corollary 3.1 we have (3.5) C s , s ( τ ) = χ c M (1 − α − τ ) χ − c M (1 − α τ )( − − . For ≤ u, v ≤ , by (3.4) and Theorem 3.2 we have (3.6) χ c M (1 − α − τ ) χ − c M (1 − α τ ) = ω u +2 vM = i u +2 v for exactly ( u + v, v ) M values of τ . Note that u + 2 v ≡ if and only if u = v = 0 , or u = 0 and v = 2 , or u = 2 and v = 1 , or u = 2 and v = 3 . By Corollary 2.3 , we obtain (3.7) (0 , + (2 , + (3 , + (1 , = 15 . Thus by (3.6) and (3.7) , we have χ c M (1 − α − τ ) χ − c M (1 − α τ ) = 1 for exactly values of τ .Note that u + 2 v ≡ if and only if u = 1 and v = 0 , or u = 1 and v = 2 , or u = 3 and v = 1 , or u = 3 and v = 3 . By Corollary 2.3 , we obtain (1 , + (3 , + (0 , + (2 , = 20 . (3.8) Thus by (3.6) and (3.8) , we have χ c M (1 − α − τ ) χ − c M (1 − α τ ) = i for exactly values of τ .Note that u + 2 v ≡ if and only if u = 0 and v = 1 , or u = 0 and v = 3 , or u = 2 and v = 0 , or u = 2 and v = 2 . By Corollary 2.3 , we obtain (1 , + (3 , + (2 , + (0 , = 24 . (3.9) hus by (3.6) and (3.9) , we have χ c M (1 − α − τ ) χ − c M (1 − α τ ) = − for exactly values of τ .Note that u + 2 v ≡ if and only if u = v = 1 , or u = 1 and v = 3 , or u = 3 and v = 0 , or u = 3 and v = 2 . By Corollary 2.3 , we obtain (2 , + (0 , + (3 , + (1 , = 20 . (3.10) Thus by (3.6) and (3.10) , we have χ c M (1 − α − τ ) χ − c M (1 − α τ ) = − i for exactly values of τ .Thus, as τ ranges over { , , . . . , } , by (3.5) we have C s , s ( τ ) = − exactly times − i − exactly times exactly times i − exactly times . We also note that C , s ( τ ) = C s , s ( τ ) and C s , s ( τ ) = exactly times i − exactly times − exactly times − i − exactly times . Acknowledgements
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