Injectivity on distribution of elements in the compressed sequences derived from primitive sequences over $\mathbb {Z}_{p^{e}}$

Abstract

Let p ≥ 3 be a prime, e ≥ 2 an integer and $\mathbb {Z}_{p^{e}}$ the residue ring modulo p^e. Let σ(x) be a primitive polynomial of degree n over $\mathbb {Z}_{p^{e}}$ and let G^′(σ(x), p^e) be the set of primitive linear recurring sequences over $\mathbb {Z}_{p^{e}}$ generated by σ(x). A compressing mapping $\varphi :\mathbb {Z}_{p^{e}}\rightarrow \mathscr {A}$ naturally induces a mapping $\widehat {\varphi }$ on G ′(σ(x), p^e), i.e., $\widehat {\varphi }$ maps a sequence (…,s_i− 1,s_i,s_i+ 1,… ) to (…,φ(s_i− 1),φ(s_i),φ(s_i+ 1),… ). For any pair of sequences in $\{\widehat {\varphi }(\underline {s}):\underline {s}\in G^{\prime }_{~}(\sigma (x),p^{e})\}$, it is desirable to determine whether (at least) one element of $\mathscr {A}$ is distributed differently in them. For $\emptyset \neq D\subseteq \mathscr {A}$, $\widehat {\varphi }$ is said to be injective on G ′(σ(x), p^e) w.r.t. D-uniformity if for any two distinct sequences $\underline {u},\underline {v}\in G^{\prime }_{~}(\sigma (x),p^{e})$, the distribution of at least one element of D in $\widehat {\varphi }(\underline {u})$ differs from that in $\widehat {\varphi }(\underline {v})$. A sufficient condition on φ is given to ensure that $\widehat {\varphi }$ is injective on G ′(σ(x), p^e) w.r.t. D-uniformity. If $\left (\left ((x^{p^{n}-1}-1)^{2}\bmod \sigma (x)\right ) \bmod p^{3}\right ) \notin p^{2}\mathbb {Z}_{p}$, then an equivalent condition on φ is obtained to decide whether $\widehat {\varphi }$ is injective on G ′(σ(x), p^e) w.r.t. D-uniformity. Furthermore, quantitative estimation suggests that almost all mappings on $\mathbb {Z}_{p^{e}}$ induce injective mappings on G ′(σ(x), p^e) as p and e increase.

Appendices

Appendix A: Proof of Lemma 8

Proof

(Proof of Lemma 8) First, we prove the following claim.

Claim. If $\underline {u}\not \sim {h_{\sigma }(x)}\underline {v}$,then for any $a\in {\mathbb {Z}_{p^{e}}^{*}}$, (3) is satisfiable.

Let $a\in {\mathbb {Z}_{p^{e}}^{*}}$. Consider the following four cases.

• (i) $\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {v}\bmod p$and $\underline {v}\bmod p$are linearly independent over $\mathbb {Z}_{p}$.By Lemma 3, (3) is satisfiable.

In Cases (ii)–(iv) below, $\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {v}\bmod p$and$\underline {v}\bmod p$are linearlydependent over $\mathbb {Z}_{p}$.

• (ii) $\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {u}\bmod p$and${h_{\sigma }(x)}\underline {v}\bmod p$are linearlyindependent over $\mathbb {Z}_{p}$.Then $\underline {v} \equiv d_0\underline {u} + d_1 {h_{\sigma }(x)}\underline {u} + d_2{h_{\sigma }(x)}\underline {v}\bmod p$for some$d_0,d_1,d_2\in \mathbb {Z}$. By Statement (iii) of Lemma 4, $\underline {v}\not \sim {h_{\sigma }(x)}\underline {v}$, implying that at least one of d₀ and d₁is relatively prime to p. Then we only have to consider the following two subcases.

  • Subcase p∣d₁.Then p ∤ d₀.By Lemma 3, there exists $t_0\in \mathbb {Z}$satisfying

    $$\left\{ \begin{array}{lcc} \underline{u}(t_{0}) & \equiv & {a} \mod p,\\ {h_{\sigma}(x)}\underline{u}(t_{0}) & \not\equiv & 0 \mod p, \\ {h_{\sigma}(x)}\underline{v}(t_{0}) & \equiv & 0 \mod p. \end{array}\right. $$

    Then$\underline {v}(t_0)\equiv d_0\underline {u}(t_0) + d_1 {h_{\sigma }(x)}\underline {u}(t_0) + d_2{h_{\sigma }(x)}\underline {v}(t_0) \equiv d_0 a \not \equiv 0 \bmod p$.Therefore, (3) is satisfiable.

  • Subcase p ∤ d₁. Sincep ≥ 3, we can choose$b\in \{i\in \mathbb {Z}:p\nmid i,i\not \equiv -d_0a/d_1\bmod p\}$. By Lemma3, there exists $t_0\in \mathbb {Z}$satisfying

    $$\left\{ \begin{array}{lclc} \underline{u}(t_{0}) & \equiv & {a} &\mod p,\\ {h_{\sigma}(x)}\underline{u}(t_{0}) & \equiv & b\not\equiv0 &\mod p, \\ {h_{\sigma}(x)}\underline{v}(t_{0}) & \equiv & 0 &\mod p. \end{array}\right. $$

    Then$\underline {v}(t_0)\equiv d_0\underline {u}(t_0) + d_1 {h_{\sigma }(x)}\underline {u}(t_0) + d_2{h_{\sigma }(x)}\underline {v}(t_0) \equiv d_0 a + d_1 b \not \equiv 0\bmod p$.Therefore, (3) is satisfiable.

In Cases (iii)–(iv) below, $\underline {u}\bmod p$,${h_{\sigma }(x)}\underline {u}\bmod p$and${h_{\sigma }(x)}\underline {v}\bmod p$are linearly dependentover $\mathbb {Z}_{p}$. By Statement(iii) of Lemma 4, $\underline {u}\not \sim {h_{\sigma }(x)}\underline {u}$. Thenthere exist $r_0,r_1\in \mathbb {Z}$satisfying${h_{\sigma }(x)}\underline {v} \equiv r_0\underline {u} + r_1{{h_{\sigma }(x)}}\underline {u}\bmod p$. By Statement(iv) of Lemma 4, ${h_{\sigma }(x)}\underline {u}\not \sim {h_{\sigma }(x)}\underline {v}$,implying p ∤ r₀.

• (iii) p ∤ r₁.Consider two subcases.

  • Subcase: $\underline {v}\bmod p$,$\underline {u}\bmod p$and ${h_{\sigma }(x)}\underline {u}\bmod p$are linearly independent over $\mathbb {Z}_{p}$.Then by Lemma 3, there exists $t_0\in \mathbb {Z}$satisfying

    $$\left\{ \begin{array}{lcl} \underline{u}(t_{0}) & \equiv & {a} \mod p,\\ {h_{\sigma}(x)}\underline{u}(t_{0}) & \equiv & -r_{0}{a}/r_{1} \mod p, \\ \underline{v}(t_{0}) & \not\equiv & 0 \mod p. \end{array}\right. $$

    Then${h_{\sigma }(x)}\underline {v}(t_0)\equiv r_0\underline {u}(t_0) + r_1{{h_{\sigma }(x)}}\underline {u}(t_0)\equiv 0\bmod p$. Notethat ${h_{\sigma }(x)}\underline {u}(t_0)\equiv -r_0{a}/r_1\not \equiv 0\bmod p$.Therefore, (3) is satisfiable.

  • Subcase: $\underline {v}\bmod p$,$\underline {u}\bmod p$and${h_{\sigma }(x)}\underline {u}\bmod p$are linearly dependentover $\mathbb {Z}_{p}$. By Statement(iii) of Lemma 4, $\underline {u}\not \sim {h_{\sigma }(x)}\underline {u}$.Then $\underline {v}\equiv d_0\underline {u} + d_1{{h_{\sigma }(x)}}\underline {u}\bmod p$for some$d_0,d_1\in \mathbb {Z}$. Furthermore, byLemma 3, there exists $t_0\in \mathbb {Z}$satisfying $\underline {u}(t_0)\equiv {a}\bmod p$and ${h_{\sigma }(x)}\underline {u}(t_0)\equiv -r_0{a}/r_1\bmod p$, andwe have ${h_{\sigma }(x)}\underline {v}(t_0)\equiv r_0\underline {u}(t_0) + r_1{{h_{\sigma }(x)}}\underline {u}(t_0)\equiv 0\bmod p$.Note that we have the following matrix equality

    $$\left( \begin{array}{c} \underline{v}\\ {h_{\sigma}(x)}\underline{v} \end{array}\right) \equiv \left( \begin{array}{cc} d_{0}&d_{1}\\r_{0}&r_{1} \end{array}\right) \left( \begin{array}{c} \underline{u}\\ {h_{\sigma}(x)}\underline{u} \end{array}\right) \mod p. $$

    Besides, by Statement (iii) of Lemma 4, we have$\underline {v}\not \sim {h_{\sigma }(x)}\underline {v}$. Hence,d₀r₁≢d₁r₀ mod p. Then$\underline {v}(t_0) = d_0\underline {u}(t_0) + d_1{{h_{\sigma }(x)}}\underline {u}(t_0) \equiv d_0a - d_1r_0{a}/r_1\not \equiv 0 \bmod p$.Therefore, (3) is satisfiable.

• (iv) p∣r₁, i.e.,${h_{\sigma }(x)}\underline {v} \equiv r_0\underline {u} \bmod p$, contradictory tothe given condition $\underline {u}\not \sim {h_{\sigma }(x)}\underline {v}$.

Therefore, all possible cases are exhausted, and if$\underline {u}\not \sim {h_{\sigma }(x)}\underline {v}$, thenone of Cases (i)–(iii) holds and (3) is satisfiable.

Similar to the above claim, the following claim also holds:

Claim. If $\underline {v}\not \sim {h_{\sigma }(x)}\underline {u}$,then for any $a\in {\mathbb {Z}_{p^{e}}^{*}}$,(4) is satisfiable.

If Condition 1 does not hold, then $\underline {u}\not \sim {h_{\sigma }(x)}\underline {v}$or $\underline {v}\not \sim {h_{\sigma }(x)}\underline {u}$.By the above two claims, at least one of (3) and (4) is satisfiable.□

Appendix B: Proof of Lemma 14

To prove Lemma 14, we prepare Lemma 18. By Lemma 18, given a₀,a₁,…,a_m satisfying $\left (a_{i-1},a_i\right )\in \widetilde {{R}}_{\epsilon }\left (\underline {u},\underline {v}\right )$, 1 ≤ i ≤ m, we can derive b₀,b₁,…,b_m satisfying b₀ = a₀ and $\left (b_{i-1},b_i\right )\in \widetilde {{R}}_{\epsilon + 1}\left (\underline {u},\underline {v}\right )$, 1 ≤ i ≤ m.

Lemma 18

Let$\underline {u}$, $\underline {v}$andℓbe given as in Lemma 14. Letℓ < 𝜖 < eanda ∈ A. If there exist$t_i \in \mathbb {Z}$, 0 ≤ i ≤ m, satisfying

$$\left\{ \begin{array}{ll} \underline{u}(t_{0})\equiv a\bmod p^{\epsilon},&\\ {h_{\sigma}(x)}\underline{u}(t_{i})\not\equiv0\bmod p, & 0\leq i\leq m,\\ \underline{u}(t_{i})\equiv\underline{v}(t_{i-1})\bmod p^{\epsilon}, &1\leq i\leq m, \end{array}\right. $$

then there exist$t_i^{\prime } \in t_i + \left (p^{n}-1\right ) p^{\epsilon -1} \mathbb {Z}$,0 ≤ i ≤ m, such that

$$\left\{ \begin{array}{ll} \underline{u}(t_{0}^{\prime})\equiv a\bmod p^{\epsilon+ 1},&\\ {h_{\sigma}(x)}\underline{u}(t_{i}^{\prime})\not\equiv0\bmod p, & 0\leq i\leq m,\\ \underline{u}(t_{i}^{\prime})\equiv\underline{v}(t_{i-1}^{\prime})\bmod p^{\epsilon+ 1}, &1\leq i\leq m. \end{array}\right. $$

Proof

For 0 ≤ i ≤ m,let $t_{i}^{\prime } = t_{i} + \left (p^{n}-1\right ){p^{\epsilon -1}d_{i}}$, where

$$\left\{ \begin{array}{ccl} d_{0} &\equiv & \left( a-\underline{u}(t_{0})\right)/ \left( p^{\epsilon}{h_{\sigma}(x)}\underline{u}(t_{0})\right) \mod p,\\ d_{i} &\equiv &\left( \underline{v}(t_{i-1}^{\prime})-\underline{u}({t_{i}})\right) /\left( p^{\epsilon}{h_{\sigma}(x)}\underline{u}(t_{i})\right) \bmod p, 1\leq i\leq m. \end{array}\right. $$

Because $\underline {u}(t_{0})\equiv a\bmod p^{\epsilon }$ and ${h_{\sigma }(x)}\underline {u}(t_0)\not \equiv 0\bmod p$, d₀is well-defined.By Lemma 5, $\underline {v}(t_{i-1}^{\prime })\equiv \underline {v}(t_{i-1}) \equiv \underline {u}(t_{i}) \bmod p^{\epsilon }$, 1 ≤ i ≤ m. As given above, ${h_{\sigma }(x)}\underline {u}(t_i)\not \equiv 0\bmod p$, 1 ≤ i ≤ m. So, d_i is well-definedfor any 1 ≤ i ≤ m. Then by Lemma 5,

$$ \left\{ \begin{array}{cclcl} \underline{u}(t_{0}^{\prime}) & \equiv & \underline{u}(t_{0}) + p^{\epsilon} d_{0}{h_{\sigma}(x)}\underline{u}(t_{0}) &\equiv &a \mod p^{\epsilon+ 1},\\ \underline{u}(t_{i}^{\prime}) & \equiv & \underline{u}(t_{i}) + p^{\epsilon} d_{i}{h_{\sigma}(x)}\underline{u}(t_{i}) &\equiv &\underline{v}(t_{i-1}^{\prime}) \mod p^{\epsilon+ 1}, 1\leq i\leq m. \end{array}\right. $$

Besides, by Statement (i) of Lemma 4, the sequence ${h_{\sigma }(x)}\underline {u}\bmod p$ is of period pⁿ − 1 and hence ${h_{\sigma }(x)}\underline {u}(t_i^{\prime })\equiv {h_{\sigma }(x)}\underline {u}(t_i)\not \equiv 0\bmod p$, 0 ≤ i ≤ m.□

Proof

(Proof of Lemma 14) Choose any a₀ ∈A and b = a₀ + dp^𝜖,where $d\in \mathbb {Z}$.

Suppose that Statement 1 holds for l = 𝜖. Then there exist k > 0 and $\left \{{a_1,\dots ,a_k}\right \}\subseteq \left \{{a_0\gamma ^{i}:i\in \mathbb {Z}}\right \}+p^{\ell } \mathbb {Z}_{p^{e}}$ such that a_k = a₀ + dp^𝜖− 1,γ^k ≡ 1 mod p^ℓ, and $\left (a_{i-1},a_i\right )\in \widetilde {{R}}_{\epsilon }\left (\underline {u},\underline {v}\right )$, 1 ≤ i ≤ k. By the definition of $\widetilde {{R}}_{\epsilon }\left (\underline {u},\underline {v}\right )$, for any 1 ≤ i ≤ k, there exists t_i− 1,0 such that $\underline {u}(t_{i-1,0}) \equiv a_{i-1}\bmod p^{\epsilon }$, $\underline {v}(t_{i-1,0}) \equiv a_i \bmod p^{\epsilon }$ and ${h_{\sigma }(x)}\underline {u}(t_{i-1,0})\not \equiv 0\mod p$. So, we have

$$\begin{array}{@{}rcl@{}} {\kern3.8pc}\left\{ \begin{array}{rllr} {h_{\sigma}(x)}\underline{u}(t_{i,0})&\not\equiv& 0\mod p, \qquad\qquad\qquad ~~~~~~0\leq i< k,&\qquad\qquad{\kern7.4pt}\text{(14a)}\\ \underline{u}(t_{i,0}) &\equiv& \underline{v}(t_{i-1,0}) \mod p^{\epsilon},\qquad\qquad 1\leq i< k,&\text{(14b)}\\ \underline{u}(t_{0,0}) &\equiv& a_{0} \mod p^{\epsilon},&\text{(14c)}\\ \underline{v}(t_{k-1,0}) &\equiv& a_{0}+dp^{\epsilon-1}\mod p^{\epsilon}.&\text{(14d)} \end{array}\right. \end{array} $$

The proof consists of three steps. In the first step, we find ti, j′,0 ≤ i < k,0 ≤ j < p − 1, to form a chain oftransitive pairs in $\widetilde {{R}}_{\epsilon + 1}\left (\underline {u},\underline {v}\right )$, i.e.,

$$\left\{ \begin{array}{ll} \left( \underline{u}(t_{i-1,j}^{\prime}),\underline{u}(t_{i,j}^{\prime})\right)\in\widetilde{{R}}_{\epsilon+ 1}\left( \underline{u},\underline{v}\right),&1\leq i<k,0\leq j<p;\\ \left( \underline{u}(t_{k-1,j-1}^{\prime}),\underline{u}(t_{0,j}^{\prime})\right)\in\widetilde{{R}}_{\epsilon+ 1}\left( \underline{u},\underline{v}\right),&1\leq j<p. \end{array} \right. $$

The second step shows $\underline {v}(t_{k-1,p-1}^{\prime })-\underline {u}(t_{0,0}^{\prime })\equiv dp^{\epsilon }\bmod p^{\epsilon + 1}$. Using the above chain of transitive pairs, in the third step we prove that Statement 1 holds for l = 𝜖 + 1 with arbitrarily chosen a₀ and $b\in a_0+p^{\epsilon }\mathbb {Z}_{p^{e}}$ above.

Step 1: :

Let

$$\left\{ \begin{array}{ccll} r_{i} & = &\gamma^{i} d / \left( {h_{\sigma}(x)}\underline{u}(t_{i,0})\right)\bmod p,&0\leq i<k,\\ t_{i,j} &=& t_{i,0} + jr_{i} p^{\epsilon-2}\left( p^{n}-1\right),& 0\leq i < k, 1\leq j<p. \end{array} \right. $$

For 1 ≤ i < k and 1 ≤ j < p, we have

$$\begin{array}{@{}rcl@{}} \underline{v}({t_{i-1,j}}) &\equiv & \underline{v}\left( {t_{i-1,0} + (p^{n}-1){jr_{i-1}p^{\epsilon-2}}}\right)\\ &\equiv & \underline{v}({t_{i-1,0}}) + jr_{i-1}p^{\epsilon-1} {h_{\sigma}(x)}\underline{v}({t_{i-1,0}}) \qquad \text{~by Lemma 5} \\ &\equiv & \underline{u}({t_{i,0}}) + jr_{i-1}\gamma p^{\epsilon-1} {h_{\sigma}(x)}\underline{u}({t_{i-1,0}})\qquad \text{ \;\,by (14b) and } \underline{v}\equiv \gamma\underline{u}\bmod p\\ &\equiv &\underline{u}({t_{i,0}}) + j r_{i} p^{\epsilon-1}{h_{\sigma}(x)}\underline{u}({t_{i,0}})\\ &\equiv &\underline{u}\left( {t_{i,0} + (p^{n}-1){jr_{i}p^{\epsilon-2}}}\right)\qquad \qquad\qquad \text{by Lemma 5}\\ &\equiv& \underline{u}\left( t_{i,j}\right) \mod p^{\epsilon}, \end{array} $$

(15)

and for 1 ≤ j < p, we have

$$\begin{array}{@{}rcl@{}} \underline{v}(t_{k-1,j-1})&\equiv &\underline{v}\left( t_{k-1,0} + (j-1) r_{k-1} p^{\epsilon-2}\left( p^{n}-1\right)\right)\\ &\equiv&\underline{v}(t_{k-1,0}) + (j-1)r_{k-1}p^{\epsilon-1}{h_{\sigma}(x)}\underline{v}(t_{k-1,0})\qquad\quad\text{by Lemma 5}\\ &\equiv& \underline{u}(t_{0,0}) + dp^{\epsilon-1} + (j-1)r_{k-1}\gamma p^{\epsilon-1}{h_{\sigma}(x)}\underline{u}(t_{k-1,0})\\ &&\hspace{0.36\textwidth}{\text{by (14c), (14d) and } \underline{v}\equiv \gamma\underline{u}\bmod p}\\ &\equiv& \underline{u}(t_{0,0}) + j r_{0}p^{\epsilon-1}{h_{\sigma}(x)}\underline{u}(t_{0,0})\qquad\qquad\qquad\quad\text{by } \gamma^{k}\equiv1\bmod p\\ &\equiv & \underline{u}\left( t_{0,0} + jr_{0}p^{\epsilon-2}\left( p^{n}-1\right)\right)\qquad\qquad\qquad\qquad\qquad\text{by Lemma 5} \\ &\equiv & \underline{u}(t_{0,j}) \mod p^{\epsilon}. \end{array} $$

(16)

By Statement (i) of Lemma 4, the sequence ${h_{\sigma }(x)}\underline {u}\bmod p$ is of period pⁿ − 1, and henceby (14a), for 1 ≤ i < k, 1 ≤ j < p, we have

$$ {h_{\sigma}(x)}\underline{u}(t_{i,j})\equiv{h_{\sigma}(x)}\underline{u}(t_{i,0})\not\equiv 0\mod p. $$

(17)

By (14a), (15), (16), (17) and Lemma 18, there exist $t^{\prime }_{i,j}\in t_{i,j} + (p^n-1) p^{\epsilon -1}\mathbb {Z}$, 0 ≤ i < k,0 ≤ j < p, such that

$$\begin{array}{@{}rcl@{}} {\kern2.4pc}\left\{ \begin{array}{rllr} {h_{\sigma}(x)}\underline{u}(t_{i,j}^{\prime})&\not\equiv& 0\mod p,\qquad\qquad\qquad\;\; 0\leq i< k,0\leq j< p,&\quad\quad{\kern5.9pt}\text{(18a)}\\ \underline{u}(t_{i,j}^{\prime}) & \equiv& \underline{v}(t_{i-1,j}^{\prime}) \mod p^{\epsilon+ 1},\qquad 1\leq i< k,0\leq j< p,&\text{(18b)}\\ \underline{u}(t_{0,j}^{\prime})& \equiv& \underline{v}(t_{k-1,j-1}^{\prime}) \mod p^{\epsilon+ 1}, \qquad\qquad\;\;\;\; 1\leq j< p,&\text{(18c)}\\ \underline{u}(t_{0,0}^{\prime}) & \equiv& a_{0} \mod p^{\epsilon+ 1}.&\text{(18d)} \end{array}\right. \end{array} $$

Step 2: :

Denote $\underline {w}=(\underline {v}-\gamma \underline {u})/p^{\ell }$. ByLemmas 13 and 5, for 0 ≤ i < k,we have

$$\begin{array}{@{}rcl@{}} &&\underline{v}{(t_{i,j}^{\prime})} - \underline{v}(t_{i,0}^{\prime}) \\ &\equiv&\gamma(\underline{u}({t_{i,j}^{\prime}})-\underline{u}({t_{i,0}^{\prime}})) + p^{\ell}(\underline{w}(t_{i,j}^{\prime})-\underline{w}(t_{i,0}^{\prime})) \\ &\equiv&\gamma(\underline{u}({t_{i,j}^{\prime}})-\underline{u}({t_{i,0}^{\prime}})) + p^{\ell+\epsilon-1}jr_{i} {h_{\sigma}(x)}\underline{w}({t_{i,0}^{\prime}})\mod p^{\min\left\{{\epsilon+\ell,e}\right\}}. \end{array} $$

(19)

Iteratively using (18b) and (19), for 0 ≤ j < p we have

$$\begin{array}{@{}rcl@{}} &&\underline{v}({t_{k-1,j}^{\prime}})-\underline{v}({t_{k-1,0}^{\prime}})\equiv \gamma^{k}(\underline{u}({t_{0,j}^{\prime}})-\underline{u}({t_{0,0}^{\prime}}))\\ &&+ j p^{\ell+\epsilon-1} {\sum}_{i = 0}^{k-1} \gamma^{k-1-i}r_{i} {h_{\sigma}(x)}\underline{w}({t_{i,0}^{\prime}})\mod p^{\epsilon+ 1}. \end{array} $$

(20)

Recalling γ^k ≡ 1 mod p^ℓ,for 1 ≤ j ≤ p − 1 we have

$$\begin{array}{@{}rcl@{}} &&(\underline{v}({t_{k-1,j}^{\prime}})-\underline{u}({t_{0,j}^{\prime}}))-(\underline{v}({t_{k-1,0}^{\prime}})-\underline{u}({t_{0,0}^{\prime}}))\\ &\equiv&(\underline{v}({t_{k-1,j}^{\prime}})-\underline{v}({t_{k-1,0}^{\prime}}))-(\underline{u}({t_{0,j}^{\prime}}) - \underline{u}({t_{0,0}^{\prime}})) \\ &\equiv&(\gamma^{k}-1)(\underline{u}({t_{0,j}^{\prime}})-\underline{u}({t_{0,0}^{\prime}})) + j p^{\ell+\epsilon-1} \sum\limits_{i = 0}^{k-1} \gamma^{k-1-i}r_{i} {h_{\sigma}(x)}\underline{w}({t_{i,0}^{\prime}}) \text{ by (20)} \\ &\equiv&(\gamma^{k}-1)p^{\epsilon-1}jr_{0}{h_{\sigma}(x)}\underline{u}({t_{0,0}^{\prime}})\\ && + j p^{\ell+\epsilon-1} \sum\limits_{i = 0}^{k-1} \gamma^{k-1-i}r_{i} {h_{\sigma}(x)}\underline{w}({t_{i,0}^{\prime}}) \bmod p^{\epsilon+ 1}. \text{ by Lemma 5} \end{array} $$

(21)

Denote ${\Delta }^{\prime }=\underline {v}({t_{k-1,p-1}^{\prime }})-\underline {u}({t_{0,0}^{\prime }})$.By Lemma 5, (14c) and (14d),

$$ \underline{v}(t_{k-1,0}^{\prime})-\underline{u}(t_{0,0}^{\prime}) \equiv\underline{v}(t_{k-1,0})-\underline{u}(t_{0,0})\equiv dp^{\epsilon-1}\bmod p^{\epsilon}. $$

(22)

Then we have

$$\begin{array}{@{}rcl@{}} {\Delta}^{\prime} &\equiv & \sum\limits_{j = 0}^{p-1}(\underline{v}({t_{k-1,j}^{\prime}}) - \underline{u}({t_{0,j}^{\prime}}))\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{by (18c)}\\ &\equiv & dp^{\epsilon} + (\gamma^{k}-1)p^{\epsilon-1}r_{0}{h_{\sigma}(x)}\underline{u}({t_{0,0}^{\prime}})\sum\nolimits_{j = 1}^{p-1}j \\ && + p^{\ell+\epsilon-1}\sum\limits_{i = 0}^{k-1}\gamma^{k-1-i}r_{i}{h_{\sigma}(x)}\underline{w}({t_{i,0}^{\prime}})\sum\nolimits_{j = 1}^{p-1}j \mod p^{\epsilon+ 1}.\text{ by (21) and (22)} \end{array} $$

Since p∣(γ^k − 1)and${\sum }_{j = 1}^{p-1}j \equiv p(p-1)/2\equiv 0\bmod p$, wehave Δ^′≡dp^𝜖 mod p^𝜖+ 1.Thus, by (18d), we have

$$ \underline{v}(t_{k-1,p-1}^{\prime})\equiv \underline{u}(t_{0,0}^{\prime})+{\Delta}^{\prime} \equiv a_{0} + dp^{\epsilon}\equiv b\mod p^{\epsilon+ 1}. $$

(23)

Step 3: :

Let a0′ = a₀,apk′ = b, and$a^{\prime }_{i+kj} = \underline {u}(t_{i,j}^{\prime })$, where 0 ≤ i < k,0 ≤ j < p and (i, j) ≠ (0, 0). Then we have $a^{\prime }_1\dots ,a^{\prime }_{pk}$ satisfying

$$\left\{ \begin{array}{ll} a^{\prime}_{pk} = b,&\\ \gamma^{pk}\equiv(\gamma^{k})^{p}\equiv1\mod p^{\ell},&\\ \left( a^{\prime}_{j-1},a^{\prime}_{j}\right)\in\widetilde{{R}}_{\epsilon+ 1}\left( \underline{u},\underline{v}\right), 1\leq j\leq pk.& \text{by (18) and (23)} \end{array}\right. $$

Since $\underline {v}\equiv \gamma \underline {u}\bmod p^{\ell }$,𝜖 ≥ℓ + 1 and $\left (a^{\prime }_{j-1},a^{\prime }_j\right )\in \widetilde {{R}}_{\epsilon + 1}\left (\underline {u},\underline {v}\right )$, where 1 ≤ j ≤ pk, we have $a^{\prime }_j\in \left \{{a_0\gamma ^t:t\in \mathbb {Z}}\right \}+p^{\ell } \mathbb {Z}_{p^{e}}$ for 1 ≤ j ≤ pk. Therefore, Statement 1 holds for l = 𝜖 + 1.□