Some mathematical foundations of cryptography

Mainly some content about number theory and modern algebra, which is too abstract. Here are some theorems from textbooks and reference materials from the Internet, so that you can review them later~
Textbook "Basics of Coding Theory" by Chen Lusheng and "Information Security Mathematics" Basics" Chen Gongliang

Number Theory

4. Quadratic congruence and square residue

The general form of quadratic congruence:

quadratic remainder

Definition

Discussion modulo the prime number ppQuadratic congruence of$p$$x^2\equiv a\left(modp\right),（a,p）=1$ (1):

Corollary of the squared residue:

Define the Legendre symbol to determine whether the integer a is a quadratic residue modulo an odd number p:

Euler's discriminant rule:
Some properties of the Legendre symbol:

These properties are generally used to calculateLegendre Legendre symbolThe following are also some properties and theorems for calculating Legendre symbol;


quadratic reciprocity law.

Note that p and q are odd prime numbers that are relatively prime. If not, they can be split using the above properties and
defined in Legendre symbol. The modulo p is extended to the general case modulo m, andthe Jacobian symbolto determine.

Some properties of Jacobian symbols:

important lemma theorems for calculations:

5. Original roots and indicators

Discussion about $a^n\equiv 1\left(modm\right)$ problem
definition $or{d}_m(a):$

The order is the smallest positive integer that satisfies 4.1, only when the order isφ (m) \varphi(m)Only when$\phi$$($$m$$)$ can we say that a is modulo mmThe primitive root of$m,$

that is, n must be$or{d}_{m}Only\; multiples\; of\; (a)$ can make the formula true.

Because of Euler's theorem, $a^{\phi \left(m\right)}\equiv 1\left(modm\right)$ so in calculating$or{d}_m\left(a\right)$ Whenφ (m) \varphi(m)Find among the factors of$\phi$$($$m$$)\; .$

Regarding (ii):

Can be used to simplify calculations . For example:

first calculate $or{d}_m\left(a\right)$ , find the value with large exponent 23456 mod$or{d}_m\left(a\right)$ The same value thenconverts the large exponent to the small exponent

Corollary:



Perform $Perform\; standard\; factorization\; on\phi (m)\; ,\; and\; then\; determine$

abstract algebra

group

half group

The definition of a semigroup: Satisfies the associative law of operations. The operations here need to be understood abstractly.

A semigroup that satisfies the commutative law is called a commutative semigroup:

Unitary: There exists $e\in S$ for any$a\in S$ capital$a\ast e=e\ast a=a$

group

The difference between a group and a semigroup: a unitary group + identity element e + each element has an inverse element

Theorem: The inverse element exists and is unique

Subgroup:

Theorem: The identity element of a subgroup of a group is also the identity element of the group, and the inverse element in the subgroup is also the inverse element in the group.

Necessary conditions for judging subgroups:

order of group elements

exists such that $a^n=n\; for\; which\; e$ is established is calledthe order of element a.If it does not exist, then the order of group element a is said to be infinite.
Properties of group element order:

isomorphism of groups

For two groups of isomorphic mapping, if e is the identity element of a group, then$f\left(e\right)$ is the identity element of another group, and there exists$f(a^{-1})=f(a{)}^{-1}$
For two formally different groups, if they are isomorphic, then we can abstractly regard them as groups that are essentially the same. The only difference is the symbols used.

Cyclic group *

Cyclic group $G$ has a generator a, and the order of this generator is n, that is,$a^n=e$ , this group$The\; order\; of\; G$ is n, and the number of finite order cyclic groups is also n:
$n=generator\; order=order\; of\; group=$
Theorem 2.8 of$the\; number\; of\; elements\; of\; a\; finite-order\; cyclic\; group$ : The subgroup of a cyclic group is also a cyclic group
Theorem 2.9:

Understanding: A factor m of n, n-order cyclic groupGGA cyclic subgroup of order m of$G\; exists\; and\; is\; unique.$

Accompaniment and business group

That is, take the group GGThe elements in$G$ serve as representative elements, and the subgroup HHThe new set formed by performing group operations on all elements of$H\; is\; called\; a\; coset.$
Theorem 2.10

Corollary 2.2Suppose$<G,+>$ is a$p$ finite ringlet group, if$p$ is a prime number, then$<G,+>$ is a cyclic group.
A finite commutative group of prime order is a cyclic group.

ring

ring definition

The ring defines two operations + and * on the set, which satisfies the commutative group for addition and the semigroup for multiplication. And multiplication satisfies the left distributive law and the right distributive law for addition.
If a multiplicative identity element exists for a ring, it is said to be a unitary ring.
Definition : A ring containing only a finite number of elements is called a finite ring

whole ring

If a ring does not have zero factors, then the elimination law of this ring with respect to multiplication holds, and vice versa.
Definition 2.17 A unitary commutative ring that does not contain zero factors is called an integral ring, and the entire ring satisfies the multiplication law.
Sub-ring :

ideal

Taking an element from a subring of a ring, taking any element from the ring, and performing multiplication operations still belongs to the subring $I$ , then this subring is an ideal of the ring.
For a ring{ R , + , ∗ } \{R,+,*\}{ R,+，∗} ,$0$ and R are two ideals of R, called$R$ ’sordinary ideal, R’s other ideals are calledtrue ideals

Isomorphism of Rings

If two rings that are formally different are isomorphic, then we can abstractly regard them as essentially the same rings. For two isomorphic rings, the only difference is that the symbols of their corresponding elements are different. They are essentially the same. Isomorphic rings can be regarded as the same ring.

area

Definition : A field containing only a finite number of elements is called a finite field. A finite field is also called a Galois field. A finite field containing q elements is denoted $F_q$Or $GF\left(q\right)$
Theorem 2.17: A domain must be an integral
Theorem 2.18: A finite integral ring must be a domain.

child area

domain characteristics

Satisfy $i.e\_=$The smallest positive integernn$that\; is\; 0$$n$ is domain$Characteristics\; of\; F$ , e is the multiplicative identity element, and 0 is the additive identity element. If there is no characteristic, it is 0. The characteristics of the rational number field, the real number field, and the complex number field are all 0.
Theorem 2.20:

The characteristics of the finite field must beprime numbers
of Theorem 2.21


:

domain isomorphism

The definition of isomorphism of a domain is exactly the same as that of a ring. Two isomorphic domains are just different in their corresponding symbols. The essence of isomorphic domains is the same. In the future, we will often call isomorphic domains same domain.
$Theorem2.23:\; SupposeFandF\text{'}are\; two\; isomorphic\; fields,\; thenFand{F}^{\prime }has\; the\; same\; characteristics.$Isomorphic domain characteristics have the
same automorphic mapping: Isomorphic mapping from F to itself.
Theorem 2.24

Prime domain:

Polynomials over the field

About the definition of polynomials on the field:

About the definition of multiplication and addition of polynomials on the field:

ensure that the highest order term of the two expressions exists

. The polynomial coefficient defined on $F_{2}The\; polynomial\; coefficients\; defined\; on[x]\; can\; only\; be0,1$

Highest common factor and lowest common multiple

irreducible polynomial

The unique factorization theorem The formal derivative

of the heavy factor
$The\; formal\; derivative\; off(x)\; is\; expressed\; asf{}^{The\; formal\; derivative\; of\; the\; \prime }(x)$

polynomial satisfies the
necessary and sufficient condition that there is no multiple factor:
$f\left(x\right)$ and$f^{\prime }\left(x\right)$is relatively prime, that is,$gcd(f(x),f^{\prime }(x))=1$
Remainder theorem

Sufficient and necessary conditions for roots on domain F:


Split domain

F is a domain. For any polynomial$There\; is\; a\; split\; domain\; for\; f\left(x\right)$ , and$Any\; two\; split\; domains\; of\; f\left(x\right)$ are isomorphic.

Polynomial Ideals and Quotient Rings

finite field theory