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RSA

RSA (Rivest–Shamir–Adleman) is a public-key cryptosystem that is widely used for secure data transmission.

The strength of RSA relies on the fact that you need to factor n to obtain d and there is no known algorithm that can do that efficiently for large numbers.

Introduction

RSA is an asymmetric cipher. The public key contains n and e, pubKey(n, e), and the private key contains n and d, privKey(n, d).

  • p and q are two large random primes that validate the following equation :

$$ n = p * q $$

  • Euler’s totient function : $$ \varphi(n)=(p - 1)(q - 1) $$

  • e (public key exponent) must verify :

$$ 1 < e < \varphi(n) $$ $$ gcd(e, \varphi(n)) = 1 $$ $$ d * e \equiv 1 [\varphi(n))] $$ $$ d \equiv invmod(e) [\varphi(n))] $$

  • m (plaintext) must verify (otherwise it will be trim by the modulus) :

$$ m < n $$

Info

Greatest common divisor (GCD) of two integers is the largest positive integer that divides each of the integers.

Encryption (public key)

  • c : ciphertext (encrypted message, the result of calculation)
  • m : cleartext (plain message to send)
  • e : exponent (from public key)
  • n : product of two large prime numbers (from public key)

$$ c = m^e [n] $$

Decryption (private key)

  • c : ciphertext (encrypted message, the result of calculation)
  • m : cleartext (plain message to send)
  • n : product of two large prime numbers (from public key)
  • d : exponent (from private key)

$$ m \equiv c^d [n] $$

References

  • https://bitsdeep.com/posts/attacking-rsa-for-fun-and-ctf-points-part-1/
  • https://en.wikipedia.org/wiki/RSA_(cryptosystem)
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