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RSA (Definition)

RSA is an example of public key cryptography. It is a widely used system, relying for its security on the difficulty of factoring a large number.

Alice forms her public and private keys as follows:

  • Chooses large primes $ p$ and $ q$, then form $ n=pq$.
  • Chooses e coprime with $ \phi (n) = (p-1)(q-1)$.
  • Publishes $ (n,e)$ as her public key.
  • Computes private key $ d$ such that $ de \equiv 1 \quad\textrm{mod } \phi(n)$.

To encrypt a message $ M$ (where $ M < n$) the user Bob forms $ C = M^e \quad\textrm{mod } n$.

To decrypt the message, Alice forms $ d(C) = C^d \quad\textrm{mod } n$.

This recovers message $ M$ because:

$\displaystyle d(C)$ $\displaystyle = C^d \quad\textrm{mod } n$    
  $\displaystyle = (M^e + rn)^d \quad\textrm{mod } n \qquad \textrm{for some }r$    
  $\displaystyle = (M^{(p-1)})^{t(q-1)}M \quad\textrm{mod } n \qquad \textrm{for some }t$    
  $\displaystyle \equiv (1+sp)^{t(q-1)}M \quad\textrm{mod } p \qquad \textrm{for some }s$    
  $\displaystyle \equiv M \quad\textrm{mod } p$    

So $ d(C) \equiv M \quad\textrm{mod } p$ and similarly, $ d(C) \equiv M \quad\textrm{mod } q$ so by the Chinese remainder theorem, $ d(C) \equiv M \quad\textrm{mod } n$, and since we know $ M<n$ we know that $ M = d(C)$.



"RSA" is owned by aoh45. [ full author list (2) ]
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Other names:  RSA cryptosystem
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Cross-references: Chinese remainder theorem, coprime, primes, chooses, number, public key cryptography
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This is version 2 of RSA, born on 2005-05-08, modified 2008-06-29.
Object id is 7025, canonical name is RSA.
Accessed 2317 times total.

Classification:
AMS MSC: 94A60 (Information and communication, circuits :: Communication, information :: Cryptography)
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