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Secure Multiparty Computation.md * DELETE FILE : _posts/Lecture Notes/Modern Cryptography/2023-09-19-symmetric-key-encryption.md * chore: remove files * [PUBLISHER] upload files #197 * PUSH NOTE : 수학 공부에 대한 고찰.md * PUSH NOTE : 09. Lp Functions.md * PUSH ATTACHMENT : mt-09.png * PUSH NOTE : 08. Comparison with the Riemann Integral.md * PUSH ATTACHMENT : mt-08.png * PUSH NOTE : 04. Measurable Functions.md * PUSH ATTACHMENT : mt-04.png * PUSH NOTE : 06. Convergence Theorems.md * PUSH ATTACHMENT : mt-06.png * PUSH NOTE : 07. Dominated Convergence Theorem.md * PUSH ATTACHMENT : mt-07.png * PUSH NOTE : 05. Lebesgue Integration.md * PUSH ATTACHMENT : mt-05.png * PUSH NOTE : 03. Measure Spaces.md * PUSH ATTACHMENT : mt-03.png * PUSH NOTE : 02. Construction of Measure.md * PUSH ATTACHMENT : mt-02.png * PUSH NOTE : 01. Algebra of Sets and Set Functions.md * PUSH ATTACHMENT : mt-01.png * PUSH NOTE : Rules of Inference with Coq.md * PUSH NOTE : 블로그 이주 이야기.md * PUSH NOTE : Secure IAM on AWS with Multi-Account Strategy.md * PUSH ATTACHMENT : separation-by-product.png * PUSH NOTE : You and Your Research, Richard Hamming.md * PUSH NOTE : 10. Digital Signatures.md * PUSH ATTACHMENT : mc-10-dsig-security.png * PUSH ATTACHMENT : mc-10-schnorr-identification.png * PUSH NOTE : 9. Public Key Encryption.md * PUSH ATTACHMENT : mc-09-ss-pke.png * PUSH NOTE : 8. Number Theory.md * PUSH NOTE : 7. Key Exchange.md * PUSH ATTACHMENT : mc-07-dhke.png * PUSH ATTACHMENT : mc-07-dhke-mitm.png * PUSH ATTACHMENT : mc-07-merkle-puzzles.png * PUSH NOTE : 6. Hash Functions.md * PUSH ATTACHMENT : mc-06-merkle-damgard.png * PUSH ATTACHMENT : mc-06-davies-meyer.png * PUSH ATTACHMENT : mc-06-hmac.png * PUSH NOTE : 5. CCA-Security and Authenticated Encryption.md * PUSH ATTACHMENT : mc-05-ci.png * PUSH ATTACHMENT : mc-05-etm-mte.png * PUSH NOTE : 1. OTP, Stream Ciphers and PRGs.md * PUSH ATTACHMENT : mc-01-prg-game.png * PUSH ATTACHMENT : mc-01-ss.png * PUSH NOTE : 4. Message Authentication Codes.md * PUSH ATTACHMENT : mc-04-mac.png * PUSH ATTACHMENT : mc-04-mac-security.png * PUSH ATTACHMENT : mc-04-cbc-mac.png * PUSH ATTACHMENT : mc-04-ecbc-mac.png * PUSH NOTE : 3. Symmetric Key Encryption.md * PUSH ATTACHMENT : is-03-ecb-encryption.png * PUSH ATTACHMENT : is-03-cbc-encryption.png * PUSH ATTACHMENT : is-03-ctr-encryption.png * PUSH NOTE : 2. PRFs, PRPs and Block Ciphers.md * PUSH ATTACHMENT : mc-02-block-cipher.png * PUSH ATTACHMENT : mc-02-feistel-network.png * PUSH ATTACHMENT : mc-02-des-round.png * PUSH ATTACHMENT : mc-02-DES.png * PUSH ATTACHMENT : mc-02-aes-128.png * PUSH ATTACHMENT : mc-02-2des-mitm.png * PUSH NOTE : 18. Bootstrapping & CKKS.md * PUSH NOTE : 17. BGV Scheme.md * PUSH NOTE : 16. The GMW Protocol.md * PUSH ATTACHMENT : mc-16-beaver-triple.png * PUSH NOTE : 15. Garbled Circuits.md * PUSH NOTE : 14. Secure Multiparty Computation.md * PUSH NOTE : 13. Sigma Protocols.md * PUSH ATTACHMENT : mc-13-sigma-protocol.png * PUSH ATTACHMENT : mc-13-okamoto.png * PUSH ATTACHMENT : mc-13-chaum-pedersen.png * PUSH ATTACHMENT : mc-13-gq-protocol.png * PUSH NOTE : 12. Zero-Knowledge Proofs (Introduction).md * PUSH ATTACHMENT : mc-12-id-protocol.png * PUSH NOTE : 11. Advanced Topics.md * PUSH NOTE : 0. Introduction.md * PUSH NOTE : 02. Symmetric Key Cryptography (1).md * PUSH NOTE : 09. Transport Layer Security.md * PUSH ATTACHMENT : is-09-tls-handshake.png * PUSH NOTE : 08. Public Key Infrastructure.md * PUSH ATTACHMENT : is-08-certificate-validation.png * PUSH NOTE : 07. Public Key Cryptography.md * PUSH NOTE : 06. RSA and ElGamal Encryption.md * PUSH NOTE : 05. Modular Arithmetic (2).md * PUSH NOTE : 03. Symmetric Key Cryptography (2).md * PUSH ATTACHMENT : is-03-feistel-function.png * PUSH ATTACHMENT : is-03-cfb-encryption.png * PUSH ATTACHMENT : is-03-ofb-encryption.png * PUSH NOTE : 04. Modular Arithmetic (1).md * PUSH NOTE : 01. Security Introduction.md * PUSH ATTACHMENT : is-01-cryptosystem.png * PUSH NOTE : Search Time in Hash Tables.md * PUSH NOTE : 랜덤 PS일지 (1).md * chore: rearrange articles * feat: fix paths * feat: fix all broken links * feat: title font to palatino
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@@ -90,7 +90,7 @@ For even better (maybe faster) results, we need the help of elementary number th
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> a^{p-1} \equiv 1 \pmod p.
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> $$
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*Proof*. (Using group theory) The statement can be rewritten as follows. For $a \neq 0$ in $\mathbb{Z}_p$, $a^{p-1} = 1$ in $\mathbb{Z}_p$. Since $\mathbb{Z}_p^\ast$ is a (multiplicative) group of order $p-1$, the order of $a$ should divide $p-1$. Therefore, $a^{p-1} = 1$ in $\mathbb{Z}_p$.
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*Proof*. (Using group theory) The statement can be rewritten as follows. For $a \neq 0$ in $\mathbb{Z} _ p$, $a^{p-1} = 1$ in $\mathbb{Z} _ p$. Since $\mathbb{Z} _ p^\ast$ is a (multiplicative) group of order $p-1$, the order of $a$ should divide $p-1$. Therefore, $a^{p-1} = 1$ in $\mathbb{Z} _ p$.
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Here is an elementary proof not using group theory.
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@@ -117,7 +117,7 @@ For direct calculation, we use the following formula.
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> **Lemma.** For $n \in \mathbb{N}$, the following holds.
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>
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> $$
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> \phi(n) = n \cdot \prod_{p \mid n} \left( 1 - \frac{1}{p} \right)
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> \phi(n) = n \cdot \prod _ {p \mid n} \left( 1 - \frac{1}{p} \right)
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> $$
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>
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> where $p$ is a prime number dividing $n$.
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@@ -131,31 +131,31 @@ So to calculate $\phi(n)$, we need to **factorize** $n$. From the formula above,
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### Reduced Set of Residues
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Let $n \in \mathbb{N}$. The **complete set of residues** was denoted $\mathbb{Z}_n$ and
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Let $n \in \mathbb{N}$. The **complete set of residues** was denoted $\mathbb{Z} _ n$ and
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$$
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\mathbb{Z}_n = \left\lbrace 0, 1, \dots, n-1 \right\rbrace.
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\mathbb{Z} _ n = \left\lbrace 0, 1, \dots, n-1 \right\rbrace.
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$$
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We also often use the **reduced set of residues**.
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> **Definition.** The **reduced set of residues** is the set of residues that are relatively prime to $n$. We denote this set as $\mathbb{Z}_n^\ast$.
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> **Definition.** The **reduced set of residues** is the set of residues that are relatively prime to $n$. We denote this set as $\mathbb{Z} _ n^\ast$.
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>
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> $$
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> \mathbb{Z}_n^\ast = \left\lbrace a \in \mathbb{Z}_n \setminus \left\lbrace 0 \right\rbrace : \gcd(a, n) = 1 \right\rbrace.
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> \mathbb{Z} _ n^\ast = \left\lbrace a \in \mathbb{Z} _ n \setminus \left\lbrace 0 \right\rbrace : \gcd(a, n) = 1 \right\rbrace.
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> $$
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Then by definition, we have the following result.
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> **Lemma.** $\left\lvert \mathbb{Z}_n^\ast \right\lvert = \phi(n)$.
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> **Lemma.** $\left\lvert \mathbb{Z} _ n^\ast \right\lvert = \phi(n)$.
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We can also show that $\mathbb{Z}_n^\ast$ is a multiplicative group.
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We can also show that $\mathbb{Z} _ n^\ast$ is a multiplicative group.
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> **Lemma.** $\mathbb{Z}_n^\ast$ is a multiplicative group.
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> **Lemma.** $\mathbb{Z} _ n^\ast$ is a multiplicative group.
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*Proof*. Let $a, b \in \mathbb{Z}_n^\ast$. We must check if $ab \in \mathbb{Z}_n^\ast$. Since $\gcd(a, n) = \gcd(b, n) = 1$, $\gcd(ab, n) = 1$. This is because if $d = \gcd(ab, n) > 1$, then a prime factor $p$ of $d$ must divide $a$ or $b$ and also $n$. Then $\gcd(a, n) \geq p$ or $\gcd(b, n) \geq p$, which is a contradiction. Thus $ab \in \mathbb{Z}_n^\ast$.
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*Proof*. Let $a, b \in \mathbb{Z} _ n^\ast$. We must check if $ab \in \mathbb{Z} _ n^\ast$. Since $\gcd(a, n) = \gcd(b, n) = 1$, $\gcd(ab, n) = 1$. This is because if $d = \gcd(ab, n) > 1$, then a prime factor $p$ of $d$ must divide $a$ or $b$ and also $n$. Then $\gcd(a, n) \geq p$ or $\gcd(b, n) \geq p$, which is a contradiction. Thus $ab \in \mathbb{Z} _ n^\ast$.
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Associativity holds trivially, as a subset of $\mathbb{Z}_n$. We also have an identity element $1$, and inverse of $a \in \mathbb{Z}_n^\ast$ exists since $\gcd(a, n) = 1$.
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Associativity holds trivially, as a subset of $\mathbb{Z} _ n$. We also have an identity element $1$, and inverse of $a \in \mathbb{Z} _ n^\ast$ exists since $\gcd(a, n) = 1$.
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Now we can prove Euler's generalization.
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@@ -167,13 +167,13 @@ Now we can prove Euler's generalization.
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> a^{\phi(n)} \equiv 1 \pmod n.
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> $$
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*Proof*. Since $\gcd(a, n) = 1$, $a \in \mathbb{Z}_n^\ast$. Then $a^{\left\lvert \mathbb{Z}_n^\ast \right\lvert} = 1$ in $\mathbb{Z}_n$. By the above lemma, we have the desired result.
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*Proof*. Since $\gcd(a, n) = 1$, $a \in \mathbb{Z} _ n^\ast$. Then $a^{\left\lvert \mathbb{Z} _ n^\ast \right\lvert} = 1$ in $\mathbb{Z} _ n$. By the above lemma, we have the desired result.
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*Proof*. (Elementary) Set $f : \mathbb{Z}_n^\ast \rightarrow \mathbb{Z}_n^\ast$ as $x \mapsto ax \bmod n$, then the rest of the reasoning follows similarly as in the proof of Fermat's little theorem.
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*Proof*. (Elementary) Set $f : \mathbb{Z} _ n^\ast \rightarrow \mathbb{Z} _ n^\ast$ as $x \mapsto ax \bmod n$, then the rest of the reasoning follows similarly as in the proof of Fermat's little theorem.
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Using the above result, we remark an important result that will be used in RSA.
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> **Lemma.** Let $n \in \mathbb{N}$. For $a, b \in \mathbb{Z}$ and $x \in \mathbb{Z}_n^\ast$, if $a \equiv b \pmod{\phi(n)}$, then $x^a \equiv x^b \pmod n$.
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> **Lemma.** Let $n \in \mathbb{N}$. For $a, b \in \mathbb{Z}$ and $x \in \mathbb{Z} _ n^\ast$, if $a \equiv b \pmod{\phi(n)}$, then $x^a \equiv x^b \pmod n$.
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*Proof*. $a = b + k\phi(n)$ for some $k \in \mathbb{Z}$. Then
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@@ -192,44 +192,44 @@ by Euler's generalization.
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> - $(\mathsf{G3})$ $G$ has an **identity** element $e$ such that $e * a = a * e = a$ for all $a \in G$.
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> - $(\mathsf{G4})$ There is an **inverse** for every element of $G$. For each $a \in G$, there exists $x \in G$ such that $a * x = x * a = e$. We write $x = a^{-1}$ in this case.
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$\mathbb{Z}_n$ is an additive group, and $\mathbb{Z}_n^\ast$ is a multiplicative group.
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$\mathbb{Z} _ n$ is an additive group, and $\mathbb{Z} _ n^\ast$ is a multiplicative group.
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## Chinese Remainder Theorem (CRT)
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> **Theorem.** Let $n_1, \dots, n_k$ be integers greater than $1$, and let $N = n_1n_2\cdots n_k$. If $n_i$ are pairwise relatively prime, then the system of equations $x \equiv a_i \pmod {n_i}$ has a unique solution modulo $N$.
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> **Theorem.** Let $n _ 1, \dots, n _ k$ be integers greater than $1$, and let $N = n _ 1n _ 2\cdots n _ k$. If $n _ i$ are pairwise relatively prime, then the system of equations $x \equiv a _ i \pmod {n _ i}$ has a unique solution modulo $N$.
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>
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> *(Abstract Algebra)* The map
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>
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> $$
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> x \bmod N \mapsto (x \bmod n_1, \dots, x \bmod n_k)
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> x \bmod N \mapsto (x \bmod n _ 1, \dots, x \bmod n _ k)
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> $$
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>
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> defines a ring isomorphism
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>
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> $$
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> \mathbb{Z}_N \simeq \mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \cdots \times \mathbb{Z}_{n_k}.
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> \mathbb{Z} _ N \simeq \mathbb{Z} _ {n _ 1} \times \mathbb{Z} _ {n _ 2} \times \cdots \times \mathbb{Z} _ {n _ k}.
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> $$
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*Proof*. (**Existence**) Let $N_i = N/n_i$. Then $\gcd(N_i, n_i) = 1$. By the extended Euclidean algorithm, there exist integers $M_i, m_i$ such that $M_iN_i + m_in_i= 1$. Now set
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*Proof*. (**Existence**) Let $N _ i = N/n _ i$. Then $\gcd(N _ i, n _ i) = 1$. By the extended Euclidean algorithm, there exist integers $M _ i, m _ i$ such that $M _ iN _ i + m _ in _ i= 1$. Now set
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$$
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x = \sum_{i=1}^k a_i M_i N_i.
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x = \sum _ {i=1}^k a _ i M _ i N _ i.
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$$
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Then $x \equiv a_iM_iN_i \equiv a_i(1 - m_in_i) \equiv a_i \pmod {n_i}$ for all $i = 1, \dots, k$.
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Then $x \equiv a _ iM _ iN _ i \equiv a _ i(1 - m _ in _ i) \equiv a _ i \pmod {n _ i}$ for all $i = 1, \dots, k$.
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(**Uniqueness**) Suppose that we have two distinct solutions $x, y$ modulo $N$. $x, y$ are solutions to $x \equiv a_i \pmod {n_i}$, so $n_i \mid (x - y)$ for all $i$. Therefore we have
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(**Uniqueness**) Suppose that we have two distinct solutions $x, y$ modulo $N$. $x, y$ are solutions to $x \equiv a _ i \pmod {n _ i}$, so $n _ i \mid (x - y)$ for all $i$. Therefore we have
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$$
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\mathrm{lcm}(n_1, \dots, n_k) \mid (x - y).
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\mathrm{lcm}(n _ 1, \dots, n _ k) \mid (x - y).
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$$
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But $n_i$ are pairwise relatively prime, so $\mathrm{lcm}(n_1, \dots, n_k) = N$ and $N \mid (x-y)$. Hence $x \equiv y \pmod N$.
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But $n _ i$ are pairwise relatively prime, so $\mathrm{lcm}(n _ 1, \dots, n _ k) = N$ and $N \mid (x-y)$. Hence $x \equiv y \pmod N$.
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*Proof*. (**Abstract Algebra**) The above uniqueness proof shows that the map
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$$
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x \bmod N \mapsto (x \bmod n_1, \dots, x \bmod n_k)
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x \bmod N \mapsto (x \bmod n _ 1, \dots, x \bmod n _ k)
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$$
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is injective. By pigeonhole principle, this map must also be surjective. This map is also a ring homomorphism, by the properties of modular arithmetic. We have a ring isomorphism.
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@@ -260,19 +260,19 @@ int chinese_remainder_theorem(vector<int>& remainder, vector<int>& modulus) {
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}
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```
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The `modular_inverse` function uses the extended Euclidean algorithm to find $M_i$ in the proof. For large moduli and many equations, $N_i = N / n_i$ results in a very large number, which is hard to handle (if your language has integer overflow) and takes longer to compute.
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The `modular_inverse` function uses the extended Euclidean algorithm to find $M _ i$ in the proof. For large moduli and many equations, $N _ i = N / n _ i$ results in a very large number, which is hard to handle (if your language has integer overflow) and takes longer to compute.
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A better way is to construct the solution **inductively**. Find a solution for the first two equations,
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$$
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\begin{array}{c}
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x \equiv a_1 \pmod{n_1} \\
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x \equiv a_2 \pmod{n_2}
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\end{array} \implies x \equiv a_{1, 2} \pmod{n_1n_2}
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x \equiv a _ 1 \pmod{n _ 1} \\
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x \equiv a _ 2 \pmod{n _ 2}
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\end{array} \implies x \equiv a _ {1, 2} \pmod{n _ 1n _ 2}
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$$
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and using the result, add the next equation $x \equiv a_3 \pmod{n_3}$ and find a solution.[^1]
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and using the result, add the next equation $x \equiv a _ 3 \pmod{n _ 3}$ and find a solution.[^1]
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Lastly, the ring isomorphism actually tells us a lot and is quite effective for computation. Since the two rings are *isomorphic*, operations in $\mathbb{Z}_N$ can be done independently in each $\mathbb{Z}_{n_i}$ and then merged back to $\mathbb{Z}_N$. $N$ was a large number, so computations can be much faster in $\mathbb{Z}_{n_i}$. Specifically, we will see how this fact is used for computations in RSA.
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Lastly, the ring isomorphism actually tells us a lot and is quite effective for computation. Since the two rings are *isomorphic*, operations in $\mathbb{Z} _ N$ can be done independently in each $\mathbb{Z} _ {n _ i}$ and then merged back to $\mathbb{Z} _ N$. $N$ was a large number, so computations can be much faster in $\mathbb{Z} _ {n _ i}$. Specifically, we will see how this fact is used for computations in RSA.
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[^1]: I have an implementation in my repository. [Link](https://github.com/calofmijuck/BOJ/blob/4b29e0c7f487aac3186661176d2795f85f0ab21b/Codes/23000/23062.cpp#L38).
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