What is the standard definition of number ?

Not sure. For example, if there is a rewrite rule "x/0 = ∞" for x not zero, the symbol could start popping up in output expressions. If you feed that output expression into your function F(x), it depends on whether F will accept it as an argument, and if so, if can successfully associate an output to it. Not sure at all.

It won't since your language is a computer based language, you can only involve numbers or characters.
F(a)=F(b)=F(c)……………=1, F(abc)=3
F(abc) =F(a)+F(b)+F(c)
So inserting F(1/0) will be undefined not infinity.

rewrite rule: ab-->cd

Whether you want to replace --> with =, they are relations and quantification over infinite terms does not makes sense. That's my point.

It is just an example of an algebraic structure in which adding a concept of infinity keeps the entire system consistent. Otherwise, the domain has no identity element, and then is no longer a group.

I understand that cryptography requires an abelaian group for keeping all values of operations under the group but l can assure you the point at infinity is not related to infinity we are discussing here. There is no operation in solving weierstrass equations which involves the infinity symbol.

Therefore, it is a bit late in the game to argue whether extending a domain with the infinity symbol makes sense.

Finitist mathematics is not meant to discount the standard mathematics, we are just exploring a new world which is limited. Theory is more important than application

Whether you want to replace --> with =, they are relations and quantification over infinite terms does not makes sense. That's my point. — Wittgenstein

I certainly agree with that. In my impression, it is indeed not possible to use infinity in every slot where you can fit finite numbers.

Concerning "-->", yes, every arrow is fundamentally bidirectional, but when you rewrite expressions, you will generally only use one direction. So, a+b --> c obviously implies that c --> a+b. Still, only one of both will be useful in your rewrite strategy, in order to obtain a closed-form expression, or when trying to prove a theorem.

Finitist mathematics is not meant to discount the standard mathematics, we are just exploring a new world which is limited. — Wittgenstein

Galois fields? Pick a prime power p^n and carry out all arithmetic modulo p^n. Approximately all algebraic structure that exists over infinite/countable will turn out undamaged! ;-)

What is the standard definition of number ? — Wittgenstein

I can't give you a precise verbal description of the meaning of the word "number". What I can do is I can show you that people define the word "number" in such a way that it encompasses quantities that are not strictly finite, quantities such as Pi.

I think pi is finite, as it is less than 4, even 3.15 is greater than pi.You can give a mathematical definition of a number.

Galois fields? Pick a prime power p^n and carry out all arithmetic modulo p^n. Approximately all algebraic structure that exists over infinite/countable will turn out undamaged!

The cryptography technique using weierstrass equations that you mentioned uses finite field , like galois field.
Can l redirect your question back to you, as to suggest a cryptography system from an infinite field.
I think it is possible to use an infinite field but more difficult than a finite field.

I think it is possible to use an infinite field but more difficult than a finite field. — Wittgenstein

There's a stack exchange discussion on this question, "In cryptography, why do we reduce elliptic curves over finite fields?".

ECC over infinite fields are certainly not in use.

Imagine that P = s * G is the public key computed by multiplying a secret s of arbitrary size. A first step in the protocol will be to transmit P to the recipient while keeping s secret. If P can be arbitrary size, then you would need to transmit to the recipient a message of arbitrarily long size. I think that this problem alone is already a show stopper.

If you solve the problem by computing P = s * G mod 2^m, then you can guarantee that P will never be larger than m bits. That would already be one reason -- there could certainly be more -- to restrict calculations to finite fields.

How about having a matrice A.X=Y where A,X and Y have infinite number of rows and columns,X is the solution set which contains the secret, a row or a column in the matrix can specify the operations to perform on A to obtain A inverse. That can obviously be either transmitted via matrice element position or be a common understanding between the reciever and transmitter. This sounds really naive and stupid but l hope you can suggest improvements to using matrices in cryptography.

How about having a matrice A.X=Y where A,X and Y have infinite number of rows and columns,X is the solution set which contains the secret, a row or a column in the matrix can specify the operations to perform on A to obtain A inverse. — Wittgenstein

If X is the secret and Y is the public key, then A would be some kind of generator of sorts. That would require that each non-zero element of the domain of Y can be written as A*K for some value of K.

In ECC, the variable A (aptly named G) is a/the generator of the Galois field. Its multiples can produce every element in it.

If that would not be the case, then it makes particular values of Y impossible. If it makes enough values of Y impossible, then it makes it easier to work your way back from Y to X, and solve the elliptic curve discrete logarithm problem. It would certainly damage the intractability of the elliptic curve discrete logarithm problem and hence reduce the strength of the encryption algorithm.

Adding along a Weierstrasz elliptic curve thoroughly distorts the addition operation in arithmetic. That is the first source of encryption/distortion strength in ECC. A second source of strength/distortion, is that it also operates in a finite field that automatically wraps around the result of arithmetic calculations. For example, in a mod 7 Galois field, 2+4=>6 (smaller numbers) while 3+5=>1 (larger numbers), you get the strange result that adding smaller numbers gives a larger result. As such, it destroys any expectation of monotonicity in calculations.

You would need matrix A to unleash a strong source of distortion. Also elliptic?

This sounds really naive and stupid but l hope you can suggest improvements to using matrices in cryptography. — Wittgenstein

Daniel Bernstein is arguably the top guru in the field of practical design of algorithms. This is his home page. His most famous implementation is undoubtedly nacl.

The theoretical top number one is Adi Shamir. He came up with many of the theorems. He is also the "S" in the famous (but now probably outdated) RSA cryptosystem.

You are asking me to weigh in on stuff that is rather at their level. I am just a user of their stuff, really. I don't come up with the theorems by myself, and actually, not even the core implementations. I just build in the stuff into other things, hopefully, without creating too many additional issues ... ;-)