RSA cryptosystem

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foreword

According to your own learning needs, make a study note.
Reference materials involve a lot, too lazy to write.
There is a class on cryptography in blockchain in various cryptography books and MOOCs.

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1. The mathematical basis involved in RSA encryption

1. Euler function φ(n): less than n, greater than 0, and the number of numbers that are prime to n.

   If n=p ₁ ^l1 p ₂ ^l2 ……p _s ^ls , then
   

2. Euler's Theorem: If (a,n) = 1, then a ^φ(n) mod n = 1.

   Deduction of Euler's theorem:
   (1) a ^kφ(n) mod n = 1, if 0<a<n, (a,n) = 1;
   (2) a ^kφ(n)+1 mod n = a, if 0<a<n, (a, n) = 1;
   (3) a ^ed mod n = a, if 0<a<n, (a, n) = 1, ed mod φ(n) = 1;
   (4 )(a ^e mod n) ^d mod n= a, if 0<a<n, (a,n) = 1, ed mod φ(n) = 1.

2. RSA encryption mechanism

1. The process of RSA encryption in the process of transmitting messages

2. RSA key generation and information encryption process

(1) Randomly generate two large prime numbers p and q (1024 bits each), and calculate n = pq; (
2) Calculate

(3) Randomly generate an integer e that is mutually prime with φ(n), 1<e<φ (n);
(4) Calculate d, satisfying ed mod φ(n) = 1;
(5) Obtain public key <n, e>, private key <n, d>;
(6) For the transfer information M, record m is the plaintext and c is the ciphertext, then

Encryption: c=m ^e mod n public key: <n,e>;
decryption: m=c ^d mod n private key: <n,d>.

3. The attacker cracks the logic of the private key

• Clues for an attacker to crack the private key

public key<n,e>
ed mod φ(n) = 1

At this time, as long as φ(n) is known, the extended Euclidean algorithm can be used to solve d: If n=p ₁ ^l1 p ₂ ^l2 ……p _s ^ls , then

• The key to cracking the private key is to factorize n.

  Therefore, when n has only two large prime factors, it is extremely difficult to factorize n.

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3. The security of the RSA public key encryption system

• Selection of parameters

  It is generally believed that the value of n=pq in the RSA algorithm is 2048 bits, which is equivalent to 600 decimal integers.

1. Attack method

(1) Common mode attack (extended Euclidean algorithm)

(2) Direct decomposition modulus n attack (Euclidean algorithm)

(3) Timing attack

A timing attack is similar to a thief guessing a password by observing how long someone else turns the dial of a safe. This attack can be used not only to attack RSA, but also to attack any algorithm whose running time is not fixed.

Although timing attacks pose a serious threat, there are some simple and workable solutions:

• constant exponentiation time
• random delay
• hidden

(4) Low index attack (Chinese remainder theorem)

(5) Chosen Ciphertext Attack (CCA)

When conducting a CCA attack, the attacker selects some ciphertexts and obtains the corresponding plaintexts, which are obtained by decrypting with the private key of the target object.

Summarize

Here is only a brief description of the RSA operation ideas.