Security Model for Public Key Encryption Setup - Challenger runs KeyGen($\lambda$) to obtain (pk,sk) and passes a public key pk to Adversary Phase 1 - A asks decryption queries o...

The Diophantine Equation ax+by=c 디오판토스 방정식 : 2개 이상의 미지수를 갖는 방정식의 정수해를 찾는 문제 선형 디오판토스 방정식 : $ax+by=c$ We already proved in Theorem 2.3: If $d=gcd(a,b)$ , then $ax+by=d$ has a solution...

2.4.1 The Euclidean Algorithm $(a \le b < 0)$ Lemma (will be proved) $ a =qb + r \Rightarrow gcd(a,b) = gcd(b,r) $ Proof of $gcd(a,b) = r_{n}$ pf example 숫자가 클수록 모든 positive divisor을 구하...

Definition $ \text{Given } a,b \in \mathbb{Z} \text{ (at least one of them is not zero), } $ $ a,b \text{ are called relatively prime } \overset{\text{def}}{\Longleftrightarrow} gcd(a,b)=1 $ Theo...

Definition Let $a,b\in \mathbb{Z} $. The greatest common divisor of a and b is the positive integer d satisfying $ d \mid a \text{ and } d\mid b$ (common divisor) $ \text{if } c \mid a \t...

Definition ex 4|12, 4$\mid$(-12), 3 $\nmid$ 10 Properties of a|b pf 정리 $ \mid $ 기호 $ a\mid b $ 성질 마지막 선형결합을 나누는 성질은 중요하게 쓰인다

Theorem 2.1. Division Algorithm pf Corollary ex Let a=61 and b=-7 then 61 = (-8)(-7) + 5, hence q = -8 and r = 5 Division Algorithm 에서 나오는 even, odd number 모든 정수는 이 둘중 하나로 표현이 된다는 것 e...

David M.Burton - Seventh Edition Theorem 1.1. Well-Ordering Principle Well ordering axiom for $\mathbb{N}$ S : non empty subset of {0,1,2 …} $\Rightarrow$ S has the smallest element in S ($\Leftri...

앞부분 코드 def gcd(a,b): if b==0: return a return gcd(b,a%b) def extended_euclidean(a,b): if b==0: return a,1,0 gcd,x1,y1=extended_euclidean(b,a%b) x=y1 y=x1-(...

register input_data = "..." def decode_caesar_cipher(data, shift): decoded = "" for char in data: if char.isalpha(): # Shift character and wrap around alphabet ...