Recurrence sequences over residual rings

In this work we are carried out an algebraic study of the congruential lineal generator. The obtained results make possible several combinatorial approaches that improve significantly the period length and their behavior.

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Datum:	2005
Hauptverfasser:	Sanchez, S., Criado, R., Vega, C.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2005
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spelling	irk-123456789-1573382019-06-21T01:26:54Z Recurrence sequences over residual rings Sanchez, S. Criado, R. Vega, C. In this work we are carried out an algebraic study of the congruential lineal generator. The obtained results make possible several combinatorial approaches that improve significantly the period length and their behavior. 2005 Article Recurrence sequences over residual rings / S. Sanchez, R. Criado, C. Vega // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 81–92. — Бібліогр.: 3 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20B35, 11B50; 05A05. http://dspace.nbuv.gov.ua/handle/123456789/157338 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this work we are carried out an algebraic study of the congruential lineal generator. The obtained results make possible several combinatorial approaches that improve significantly the period length and their behavior.
format	Article
author	Sanchez, S. Criado, R. Vega, C.
spellingShingle	Sanchez, S. Criado, R. Vega, C. Recurrence sequences over residual rings Algebra and Discrete Mathematics
author_facet	Sanchez, S. Criado, R. Vega, C.
author_sort	Sanchez, S.
title	Recurrence sequences over residual rings
title_short	Recurrence sequences over residual rings
title_full	Recurrence sequences over residual rings
title_fullStr	Recurrence sequences over residual rings
title_full_unstemmed	Recurrence sequences over residual rings
title_sort	recurrence sequences over residual rings
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/157338
citation_txt	Recurrence sequences over residual rings / S. Sanchez, R. Criado, C. Vega // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 81–92. — Бібліогр.: 3 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT sanchezs recurrencesequencesoverresidualrings AT criador recurrencesequencesoverresidualrings AT vegac recurrencesequencesoverresidualrings
first_indexed	2025-07-14T09:46:54Z
last_indexed	2025-07-14T09:46:54Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2005). pp. 81 – 92 c© Journal “Algebra and Discrete Mathematics” Recurrence sequences over residual rings S. Sánchez, R. Criado, C. Vega Communicated by V. A. Artamonov Abstract. In this work we are carried out an algebraic study of the congruential lineal generator. The obtained results make possible several combinatorial approaches that improve sig- nificantly the period length and their behavior. 1. Introduction In this paper we analyze algebraic structure of the linear congruential generator xn+1 = (a · xn + b) mod m. Our goal is to analyze, for a given m, the influence of the coefficients a and b in the length of the period of generator. The period length is a decisive characteristic for the use of this generator type in many applications. First we analyze the linear congruential generator xn+1 = (a · xn + b) mod m when the parameter m is a prime number, showing that (in that case) we can choose the parameters a, b in such a form that the cyclic permutation is divided in disjoint cycles of the same order in such a manner that the length of the period obtained is close to the theoretical upper bound of m!. For more detailed exposition see [2]. Next we analyze linear congruential generator for m = p · q, when p and q are prime numbers. We show that it is possible also in this case to divide the cyclic permutation in disjoint cycles of (possibly) different order. We determine the order of each permutation and we estimate the length of the period obtained in an certain example. 2000 Mathematics Subject Classification: 20B35, 11B50; 05A05. Key words and phrases: cyclic permutation groups, linear congruences. Jo u rn al A lg eb ra D is cr et e M at h .82 Recurrence sequences over residual rings 2. Basic results and preliminaries Let us recall that if F is a set with a finite number of elements, a generator of F is an algorithm that obtains a sequence of elements of F . A sequence {xn}n≥0 is periodic if there exists k such that xn+k = xn for all n ∈ N. For a periodic sequence, the set of numbers k satisfying the above condition constitutes a subset of N−{0}. If λ is the smallest element of that subset, the subsequence x0, x1, . . . , xλ−1 is called a period of {xn}n≥0 and λ is called the length of that period. We say that {xn}n≥0 is almost periodic if there exists m ∈ N such that the sequence {xn}n≥m is periodic. In this case, the smallest number µ satisfying this condition is called the occurrence index to the period. We shall say that the period of the sequence {xn}n≥0 is the period of the sequence {xn}n≥µ. It is said that {F, f, x0} is of maximum period if the period length is equal to \|F \|. In the sequel, we shall concentrate in single-step generators, that is, those that can be written as xn+1 = f(xn), where f is a mapping f : F → F and F is a finite set. We shall interpret a generator of F as an algorithm that produces the sequence {xn}n≥0 of elements of F and we shall denote it by {F, f, x0}. As the set F is finite, there are h, k, with h < k, such that xh = xk. Applying f , we have that xh+r = xk+r; therefore, k − h is the period of the sequence {xi}i≥h. If xi = xj for j > i, as {xl}l≥i is periodic for i ≥ µ then j − i ≡ 0 (mod λ). We have the following simple result: Proposition 1. Let {F, f, x0} be a generator, where F is a finite set. The sequence {xn}n≥0defined according to the rule xn+1 = f (xn)is almost periodic. In our case F = Zm = {0, 1, ..., m − 1} is a commutative ring with unit. We are interested in knowing under what conditions the affine mapping x 7−→ a · x + b is of cycle length m. The mapping f is bijective if and only if a is an invertible element of Zm. The mapping fk is the mapping x 7−→ akx0 + (1 + a + a2 + ... + ak−1) · b. Using the notation Sk(a) = (1+a+a2+...+ak−1) we may rewrite fk as x 7−→ akx0+b·Sk(a). We now provide some properties related with Sk(a). For that purpose, we define the following polynomial with integer coefficients for k ≥ 1, Sk(x, y) = xk − yk x − y = xk−1 + xk−2y + ... + yk−1. Proposition 2. [3]Let k = p be a prime number and let x, y ∈ Zk. If x ≡ y (mod k), then Sk (x, y) ≡ 0 (mod k). Jo u rn al A lg eb ra D is cr et e M at h .S. Sánchez, R. Criado, C. Vega 83 Proof. Indeed, if x ≡ y (mod k), then Sk(x, y) = xk−1 + xk−1 + ... + xk−1 = kxk−1 and because k = p, according to Fermat’s theorem xk−1 ≡ 1 (mod k). Finally Sk(x, y) = k ≡ 0 (mod k). If y = 1, then x ≡ 1 (mod k) and Sk(x) ≡ 0 (mod k). On the other hand Sk(a) = (1+a+a2 + ...+ak−1) = 1 a−1(a−1)(1+a+a2 + ...+ak−1) = 1 a−1(ak−1). Therefore Sk(a) = (a − 1)−1(ak − 1) and Sk(x) ≡ 0 (mod k), so ak ≡ 1 (mod k). It is important to remark that for f to be of length cycle m, fixed points should not exist. For the sake of simplicity, let b = 1. Then the equation x = a · x + a has a unique solution if a 6= 1 (mod m). Hence, if f is fixed-point free, then a ≡ 1 (mod m). But, in this case, Sk(x) ≡ 0 (mod k) and fm = e, where e is the identity element. Following this reasoning, it is not difficult to prove: Proposition 3. [3]If m is a prime number,a ≡ 1 (mod m) and b is not congruent to 0 (mod m), the mapping fa,b : Zm → Zm defined by fa,b (x) = a · x + b, where a · x + b is the equivalence class modulo m corresponding to the number a · x + b, is a cyclic permutation of order m in Zm (and hence a maximum length generator). The single-step generator that we have seen, cannot generate se- quences with period longer than \|F \|. Our objective is to design a gen- erator whose period length is close to the theoretical boundary m!, the order of a symmetric group with m elements. It is well known that if the order of the cyclic group 〈f〉 generated by f satisfies \|〈f〉\| = s, given x ∈ Zm, we have that either x is a fixed point of certain element of 〈f〉 (in this case, we denote x by xF ), or the set Hs x = {x, f(x), ..., fs−1(x)} con- tains exactly s elements. Moreover, it is also possible to find a sequence x1, x2, ..., xl ∈ Zm such that Zm = H0 ∪ Hs x1 ∪ ... ∪ Hs xl where H0 = {xF }, ∣ ∣Hs x1 ∣ ∣ = ... = ∣ ∣Hs xl ∣ ∣ = s, and l · s = m − 1. 3. Generation of pseudo-random numbers and linear con- gruences Generation based on the expression xn+1 = (a · xn + b) mod m, (3.1) depends on four parameters a, b, m and x0. The parameters in expression (1) can be divided in two categories: Jo u rn al A lg eb ra D is cr et e M at h .84 Recurrence sequences over residual rings 1. Parameters a, b, m, providing a maximum length generator. 2. Parameters a, b, m, which do not provide a maximum length gener- ator. The first case is thoroughly considered in [1]. We shall carry out here the analysis of the second option, concentrating first on the case where m is a prime number and second on the case where m = p · q. To do so, we shall use the following well-known results that summarize part of what was mentioned in the previous section: Theorem 1. If a ≡ 1 (mod m), then for any sequence produced accord- ing to (1) there exists a fixed point xF such that xF = (a · xF + b) (mod m) Remark. The multiplicative group Z ∗ m = {1, 2, ..., m − 1} of the field Zm is cyclic with generating element z. The order of element z is m− 1. Therefore a = zl 6= 1 has the order s = m−1 l , as = 1 and 1 + a + a2 + ... + ak−1 = 0 when a 6= 1. So asx + ( 1 + a + a2 + ... + ak−1 ) b = x, s > 1,∀x. Definitely for a 6= 1 and for ∀b, ( 1 + a + a2 + ... + ak−1 ) b = 0, but 1 + a + a2 + ... + ak−1 = 0 so it is verified for ∀b. Thus if m ≥ 5 is a prime number, then m − 1 is composed, so the period formed by the previous m− 1 numbers is divided into l subgroups of s elements each, so that l · s = m − 1 = φ(m) where φ(m) is the Euler φ-function, which counts the number of integers in {1, ..., n} that are relatively prime to n. The number of possible values of s is τ(φ(m)) − 1 where τ(pα1 1 pα2 2 ...pαn n ) = (α1 + 1)(α2 + 1)...(αn + 1) for (m − 1) = pα1 1 pα2 2 ...pαn n . The number of subsets l is l = m − 1 s Now, for a given value of s, the valid values of parameter a are the solutions of the following equation: as ≡ 1 (mod m) (3.2) The number of solutions of this equation is φ(s), so that, if s=φ(m), we have the Euler theorem. Jo u rn al A lg eb ra D is cr et e M at h .S. Sánchez, R. Criado, C. Vega 85 Remark. As we have seen Z ∗ m is a cyclic permutation divided into l disjoint cosets consisting of s elements, z ∈ Z ∗ m and we can express this in the form Z ∗ m = 〈z〉x0 ∪ 〈z〉x1 ∪ ... ∪ 〈z〉xl−1 (3.3) where xi is a generating element of each coset x0 = min 〈z〉 x1 = min Z ∗ m \ 〈z〉 x2 = min Z ∗ m \ {〈z〉x0 ∪ 〈z〉x1} ..... 4. Combinatorial approach In a computer applications the choice of a modulo m is restricted by computer word size. For a generator of maximum period, the only way to increase the period length is to increase the value of the parameter m, which is bounded, or to use a combination of generators. In our case, the following procedures can be followed in order to achieve a large period length. As we have l subsets we can form one of the l! permutation of the numbers of the set {0, 1, ..., l − 1} as {i0, i1, ..., il−1} . So we have in this way a permutation of the subsets { Hi0 , Hi1 , ..., Hil−1 } or a permutation of the seeds { x0i0 , x0i1 , ..., x0il−1 } . Next we can follow the following basic procedure for the case m = p: • First Step. Start with one permutation obtaining a list of subsets { Hi0 , Hi1 , ..., Hil−1 } or a permutation of the seeds { x0i0 , x0i1 , ..., x0il−1 } . • Second Step. Of each subset Hij , in the list of permutation { Hi0 , Hi1 , ..., Hil−1 } , choose the first element x1, or the last element xs−1, or one at ran- dom xj until all subsets in the list of permutation { Hi0 , Hi1 , ..., Hil−1 } are examining. • Third Step.The obtained element is sent to the output. • Fourth step. The next permutation (list of subsets) is generated [1]. • The steps 3,4 are continued until all of the possible permutation are examined. Jo u rn al A lg eb ra D is cr et e M at h .86 Recurrence sequences over residual rings • Fifth step. Repeat the Second Step choosing the second element x2, or the element xs−2, or one at random xk 6= xj until all subsets in the list of permutation are examined. • Sixth step. 3,4 are continued until all of the possible permutation are examined. • ..... and so forth . The complexity of the presented algorithm doesn’t turn out to be very superior to the complexity of the algorithm LCG, since differs only in necessity of generating the next permutation in the step 4 and needs to evaluate the formula (4.1), slightly more complex then (3.1). xj = ( aj · x0ij + aj − 1 a − 1 · b ) mod m (4.1) When j is chosen at random the procedure is more complex then the algorithm LCG. It is necessary to generate an integer number uniformly distributed between 1 and s − 1. This can be obtaining using the re- lationship (3.1). In the successive steps it is necessary to try that the distribution law is verified, in other words it is necessary to generate an integer number uniformly distributed between 1 and s− 2 , s− 3, .... and so on. We need, also, to evaluate aj mod m.We can use the "power algo- rithm" or "binary method". This algorithm computes aj mod m, using O ( (lg j) (lg m)2 ) bit operations. The statistical properties of the sequences that is generated by the algorithm as well as other statistical aspects are shown in [2]. Therefore, it is possible to produce a series whose generation depends on: 1. The order in which we go through the subsets. The number of possible variations is l!, hence the period length will be l! · m. 2. The choice of seed x0 in each subset. The number of combinations for the choice of initial value is sl, hence the period length will be l! ·m ·sl. 3. The choice of subset can be random, as well as the choice of the number of elements in the chosen subset. This process is carried out so that the uniform law of distribution of x in the interval between 0 and m − 1 holds; in other words, each value x in the interval between 0 and m − 1 should appear only once. In consequence, the period length will be l! · m · (s!)l. In this way, for the chosen parameters a, b, m we can generate not only one serie2 (as in the generator of complete period case) but many Jo u rn al A lg eb ra D is cr et e M at h .S. Sánchez, R. Criado, C. Vega 87 different series. Therefore, it is possible to come closer to the theoretically possible upper bound of m! elements. Following the generation procedures above and if we take initially m = 655339, its closest prime number is 655337, so taking m = 655337, we have that m − 1 = 655336 = (23) · (11)2 · 677. If, for instance, l = 2 · 11 = 22, we determine s = 655337−1 22 = 29778. The period length for each generation from 1 to 3 will be: 1. 22! · 6.55336 · 105 ≈ 1021 · 6.55336 · 105 ≈ 1026; 2. 22! · 6.55336 · 105 · (29778)22 ≈ 1021 · 6.55336 · 105 · 1098 ≈ 10124; 3. 48! · 6.55336 · 105 · (29778!)22 ≈ 1021 · 6.55336 · 105 · 102.647.414 ≈ 102.646.441. To see the degree of approximation, we can apply Stirling’s formula to approximate the value 655337!, ln(m!) ≈ ( m + 1 2 ) · ln(m) − m + ln( √ 2π) from where 6.55336! ≈ 103.527.102 5. Linear congruences and pseudorandom-numbers gen- eration for a composite module We have analyzed a linear congruential generator xn+1 = (a · xn + b) mod m when the parameter m is a prime number, showing that (in that case) we can choose the parameters a, b in such a form that the cyclic permutation is divided in disjoint cycles of the same order in such a manner that the length of the period obtained is close to the theoretical upper bound of m!. In this chapter we analyze the linear congruential generator for m = m1 ·m2, when m1 and m2 are prime numbers. We show that it is possible also in this case to divide the cyclic permutation in disjoint cycles of (possibly) different order. We determine the order of each permutation and we estimate the length of the period obtained in an specific example. In order to do that, we recall that a pseudo-random numbers generator based in a recursive relation xn+1 = (a · xn + b) mod m (5.1) makes an ordered arrangement of the different elements of an specific set. If we consider the following affine function: fa,b : Zm → Zm x 7→ a · x + b (5.2) Jo u rn al A lg eb ra D is cr et e M at h .88 Recurrence sequences over residual rings we have that when the parameter a has an inverse (modulo m) this func- tion is a bijection. Moreover, the set of invertible functions of this form constitutes a twice transitive group with respect to the operation: ∗ : fa,b ∗ fc,d = fac,ad+b (5.3) At this moment it is important to recall the following theorem: Theorem.(Chinese Remainder Theorem) If m1 and m2 are positive integers such that gcd(m1, m2) = 1 then the groups Zm1 ×Zm2 and Zm1m2 are isomorphic. In our context one could say: If m1 and m2 are two prime numbers, then the function Zm1m2 −→ Zm1 × Zm2 xm1m2 −→ (xm1 , xm2 ) (5.4) is an isomorphism of groups (in fact an isomorphism of rings), so we have that Zm1m2 ⋍ Zm1 × Zm2 . 6. Linear congruences over Zm1m2 We denote by {F, f, x0} a generic single-step generator, that is, a genera- tor that can be written as xn+1 = f(xn), where f is a mapping f : F → F and F is a finite set, as a consequence of Chinese Remainder theorem we can combine two generators {F, f, x0} and {G, g, y0} obtaining the carte- sian product generator {F × G, f × g, (x0, y0)}. Lemma. [3] If m is the product of two distinct prime numbers m = m1 · m2 , the generator G : zi+1 = (a · zi + b) mod m is the cartesian product of G1 : xi+1 = (a · xi + b) mod m1 and G2 : yi+1 = (a · yi + b) mod m2. Moreover, the length of G is equal to the least common multiple of the lengths of G1 and G2. Now, if we consider the functions f1, f2 and f defined by f1(x) = a · x + b mod m1,f2(y) = a · y + b mod m2 and f(z) = a · z + b mod m, the following diagram show us how these functions are relating: f Zm → Zm ↓ ↓ f1 × f2 Zm1 × Zm2 → Zm1 × Zm2 (6.1) In this diagram the upright arrows represent the Chinese isomorphism. Let us consider the generator G1 : xi+1 = (a · xi + b) mod m1. Fol- lowing the underlying ideas of previous lemma, the decomposition into independent cycles of G1 is Jo u rn al A lg eb ra D is cr et e M at h .S. Sánchez, R. Criado, C. Vega 89 m1 − 1 = l1 · s1 m1 − 1 = ϕ (m1) l1 = m1−1 s1 = ϕ(m1) s1 and the same for generator G2: m2 − 1 = l2 · s2 m2 − 1 = ϕ (m2) l2 = m2−1 s2 = ϕ( m2) s2 On the other hand, we have that m1 · m2 − 1 = ϕ (m1) + ϕ (m2) + ϕ (m1 · m2) where ϕ(m) is the Euler φ-function. Therefore, m1 · m2 − 1 = ϕ (m1) + ϕ (m2) + ϕ (m1 · m2) = s1 · l1 + s2 · l2 + s3 · l3. G : zi+1 = (a · zi + b) mod (m1 · m2), ϕ(m1 · m2) = l3 · s3. hence (m1 · m2) = s3 · l3, ϕ (m1 · m2) = ϕ (m1) · ϕ (m2) = s3 · l3 and (l1 · s1) · (l2 · s2) = (l3 · s3) so, we can obtain G : zi+1 = (a · zi + b) mod (m1 · m2 ) ϕ(m1 · m2) = l3 · s3 Thus ord(s3) = (a1,a2) , a1, a2 6= 1, when 〈z1〉 = Z ∗ m1 , 〈z2〉 = Z ∗ m2 , (a1,a2) = ( zl1 1 , zl2 2 ) , and (m1 − 1) · (m1 − 1) = l3 · s3. Therefor the condicion for s3 = s1 · s2 is that gcd (s1, s2) = 1. Consequently, we have obtained the following decomposition of the cyclic permutation associated to generator G: l1 − groups of s1 elements l2 − groups of s2 elements l3 = l1 · l2 s3 = s1 · s2 l3 − groups of s3 elements Jo u rn al A lg eb ra D is cr et e M at h .90 Recurrence sequences over residual rings For example, if we want a generator with m of the order 6 · 105, we can choose m = 644773, so m = m1 · m2 = 797 · 809, m1 − 1 = 796 = (2)2 · 199 m2 − 1 = 808 = (2)3 · 101 If, for example, l1 = 22, l2 = 22, we have that s =797−1 22 = 199 and s2= 809−1 22 = 202. Then, for the composite module generator m = m1 · m2 = 797 · 809, we have the following decomposition: l3 = l1 · l2 = 22 · 22 = 24 = 16 groups of s3 = s1 ·s2 = 199 ·202 = 40198 elements, l3 = l1 = 4 groups of s4 = s1 = 199 elements, and l4 = l2 = 4 groups of s5 = s2 = 202 elements. The lenghts of each period are the following: 1. [(4!) · 199] · [(4!) · 202] · [(16!) · 40198] ≈ 1027 2. [ (4!) · 4 · 1994 ] · [ (4!) · 4 · 2024 ] · [ (16!) · 16 · 4019816 ] ≈ 10110 3. [ (4!) · 4 · (199!)4 ] · [ (4!) · 4 · (202!)4 ] · [ (16!) · 16 · (40198!)16 ] ≈ 102.685.022 To see the degree of approximation, we can apply Stirling’s formula to approximate the value 644773!, ln (m!) ≈ ( m + 1 2 ) · ln (m!) − m + ln (√ 2 · π ) , from where 644773! ≈ 103.465.730. We can resume the results of this example in the following table: Jo u rn al A lg eb ra D is cr et e M at h .S. Sánchez, R. Criado, C. Vega 91 G1 G2 G3 a=797 a=797 a=797 b=73 b=73 b=73 m=797 m=809 m=644773 l1 = 4 l2 = 4 l3 = 16 xF = 199 xF = 455 xF = 413045 x0 [i] s1 x0 [i] s2 x0 [i] s3 [i] 4 199 2 202 2 40198 6 199 4 202 3 40198 11 199 5 202 4 40198 40 199 7 202 5 40198 6 40198 7 40198 8 40198 11 40198 12 40198 15 40198 18 40198 26 40198 32 40198 37 40198 49 40198 174 202 385 199 971 202 1194 199 1768 202 3621 199 5239 199 7347 202 References [1] D. Knuth, The Art of Computing Programming, Vol.2, Addison-Wesley, 1997. [2] S. Sanchez, R. Criado and C. Vega, A generator of pseudo-random numbers se- quences with maximum period, Proc. ICMMSE (2003), World Scientific Publishing Co. Pte. Ltd., 561-566 (2003). [3] P.Naudin, C.Quitte, Algoritmique Algebrique, Masson, Paris, Milan, Barcelona, Bonn, 1992 Jo u rn al A lg eb ra D is cr et e M at h .92 Recurrence sequences over residual rings Contact information S. Sánchez Dpto. de Matemáticas y Fisica Aplicadas y CC. de la Naturaleza, Universidad Rey Juan Carlos, E-28933 Móstoles, Madrid, Spain E-Mail: sergio.sanchez@urjc.es R. Criado Dpto. de Matemáticas y Fisica Aplicadas y CC. de la Naturaleza, Universidad Rey Juan Carlos, E-28933 Móstoles, Madrid, Spain E-Mail: regino.criado@urjc.es C. Vega Dpto.de Matemática Aplicada a las Tec- nologias de la Información, Universidad Politécnica de Madrid, E-28040, Madrid, Spain E-Mail: cvega@mat.upm.es Received by the editors: 17.05.2005 and final form in 17.10.2005.

Recurrence sequences over residual rings

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