Public-key cryptography is also called public-key cryptography. Public-key cryptography is the most important invention and progress of modern cryptography. In 1976, Whitfield Diffie and Martin Hellman published the epoch-making article "New directions in cryptography" Laid the foundation of the public key cryptosystem. Public-key cryptosystems are generally divided into three categories according to the difficult problems they are based on: large prime number decomposition problems, discrete logarithm problems, and elliptic curves.
1: The
specific description of the factorization of large numbers :
Ⅰ) Given two prime numbers p, q, it is easy to calculate the product p·q=n;
Ⅱ) Given a large integer n, find the prime factor p, q of n such that n=p Q is very difficult. The
factorization of large numbers is an unsolved problem in the international mathematics community for hundreds of years, and it is also the basis for the establishment of the public-key RSA algorithm cryptosystem in modern cryptography. "Characteristics of the composite number mode of factorization of large numbers" found the distribution and characteristics of prime number factors and the connection mechanism between them from the fixed points of the RSA algorithm. Based on this, the problem of factorization of large numbers was transformed into two containing The problem of finding common factors between the numbers of prime factors turns the most difficult problem of factorization of large numbers into a series of elementary mathematical problems of algorithms. This is undoubtedly an important achievement and progress in the study of factorization of large numbers.
2: The
known discrete logarithm of the finite cyclic group G={g∧k∣k=0,1,2,...} and its generator g and order n=∣G∣.
Ⅰ) Given integer a, calculate The element g∧a=h is very easy;
Ⅱ) Given the element h, calculate the integer x, 0≤x≤n, so that g∧x=h is very difficult, and its difficulty is the same as the difficulty of factoring the product of prime numbers in RSA Magnitude.
3: Elliptic curve
point group
E(F_p)={(x,y)∈F_p×F_p∣y²=x³+ax+b,a,b∈F_p}∪{O}, the elliptic curve point group on the finite field F_p of the known elliptic curve ,
The order of point P=(x,y) is a large prime number.
Ⅰ) Given an integer a, calculate the integer x so that xP=(x_a,y_a)=Q is easy;
Ⅱ) Given a point Q, calculate the integer x such that xP=Q is very difficult.
Example 3 P=10823 is a prime number, the elliptic curve point group
E(F_p)={(x,y)∈F_p×F_p∣y²=x³+3x+ in the finite field F_p=Z/pZ 7}∪{O}, ∣E(F_p)∣=100482=2·3·16747. The generator of E(F_p) is P_0=(1,8811). Point P=6P_0=(62046,14962) order It is a prime number 16747.
Ⅰ) Given a=1007, it is easy to calculate aP=(80726,17229)=Q;
Ⅱ) Given a point Q=(80726,17229), it is difficult to find the integer x such that xP=Q. In
summary, Understanding mathematical principles may be more brainstorming, but as users, we don’t actually need to fully grasp the principles. We just need to remember the most important thing, that is, the security of the private key is the most important in the public key key system. If there is no corresponding security mechanism to protect the private key in the operating environment, an encryption chip must be used to store the private key, including the operation of the private key must be executed in the encryption chip, otherwise the private key will be leaked and the entire security system will be breached.
Reference materials:
"Concise Mathematical Foundation of Information Security", Chen Gongliang, Higher Education Press, January 1, 2011.