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diff --git a/final/rsa-method.tex b/final/rsa-method.tex new file mode 100644 index 0000000..bf75cd0 --- /dev/null +++ b/final/rsa-method.tex @@ -0,0 +1,16 @@ +The encryption process begins with the selection of two large primes, $p$ and $q$, their product $n=pq$, and a fourth number $e$ relatively prime to $\phi(n)$. $n$ is public, whereas $p$ and $q$ are secret. + +\def\mod#1{\thinspace(mod\thinspace #1)} +\noindent Encryption is accomplished through the following three steps: +\pre{1.} Convert message to a number (like {\tt a} becomes $1$ and {\tt ab} becomes $130$, assuming a 128-character language) +\pre{2.} Break the converted message into blocks of size less than $n$. +\pre{3.} For each block B, an encrypted block C is created such that $$C \equiv B^e\thinspace(mod\thinspace n)$$. +\noindent To decrypt that message: +\pre{1.} Calculate an integer $d$ such that $de \equiv 1 \mod{\phi(n)}$ using the Euclidean algorithm. +\pre{2.} Convert back using $B \equiv C^d \mod{n}$. + +The decryption process described above makes use of Euler’s theorem. +Some decryption algorithms make use of other mathematical theorems of relation, including the Chinese Remainder Theorem. + +The RSA Algorithm, while nearly unbreakable, isn’t as untouchable as originally thought, shown by the example number $n=pq$ that Rivest, Shamir, and Adleman published as a challenge in ‘77 was broken in ‘94. +This proves that as computing power grows, the best cryptographers can do is increase the size of the secrets to make prime factorization as difficult as possible, or its analogue in more arcane algorithms. diff --git a/final/rsa.tex b/final/rsa.tex index 0b2962e..837a00e 100644 --- a/final/rsa.tex +++ b/final/rsa.tex @@ -1,8 +1,11 @@ In determining correctness, a major concern is determining that the message hasn't been tampered with by an intelligent intermediate. Public key cryptography tries to answer this problem by providing proof of authorship and, as an extension of ``normal'' encryption, preventing interception. -RSA is one such algorithm. +RSA (Rivest-Shamir-Adleman, named after its MIT faculty creators) is one such algorithm. It works by providing a set of public keys to all parties, and corresponding secret private keys. One of the simpler algorithms, it applies the NP-hard nature of factorizing a semiprime, Euler’s theorem, and the Euclidean Algorithm to encrypt communication. Because it is simple to devise, it has been included as a sample, in the form of a Python script which encrypts and decrypts messages, given a small RSA key (compared to those used in real applications). There are several optimizations (such as applying the Chinese Remainder Theorem) which can be used, but none have been applied to maintain the code's simplicity. + +\sinclude Methodology:rsa-method + |