How is Math used in Cryptology/ Cryptography

              Summary:

My topic is about how math is used in cryptography. Math is used in many cryptosystems to protect data like making public keys. Prime factorizing is one of the main factors of cryptography.There are so many branches of math that we use in cryptography (matrices, primes, ellipses, modular arithmetic and many more).

Abstract math is very Important in cryptography as well.

From Analytical number theory, tools like factorization and computing logarithms in a finite field. 

Combinatorial problems, like knapsack and subset-sum has been used in cryptosystem. You can find a very nice connection between subset-sum and Lattice based cryptography

Coding theory and many combinatorial designs (BIBDs, Orthogonal arrays) have been used in the constructing universal hash function families and thereby randomness extractor and pseudorandom number generators. They are mostly used in the unconditional setting.

Algebraic geometry have been used in elliptic curve cryptography.

Group theory and in general Algebraic number theory has been used (for example, hidden subgroup problem) to construct cryptographic primitives secure against quantum attack.

Discrete Fourier Analysis has been used to prove and construct hard-core predicates, something of great use in the theoretical cryptography

summary math fair

summary

In this project we have to calculate using algebra and geometric equations of making a successful basket and will be easier to show which positions is the best to shoot at. We can use geometry and algebra in every sport especially basketball. This project should in a basketball court and you must have a basketball with you and metric ruler to measure the distance. You have to place the person with basketball close to the hoop as shown in the photo. we used sine to calculate the ratio of the wiggle room which is the space between the two shots with angles 30, 45, 60, and 90 to find the  x total and the y total. This can help us find a better way to bank a basket.

A postulate is a statement that is assumed to be true (also called axioms) 

A theorem is a statement that is proved to be true by axioms and other proved facts (smaller theorems or theorems that support some other theorems are often called lemmas) 

A corollary is a direct consequence of a proven fact and are usually account by a short supporting statement

Difference Between A Postulate, Corollary and a Theorem

THEOREM : A mathematical statement that is proved using extremely careful and thoughtful mathematical reasoning. 

POSTULATE:  A statement that is assumed to be true without proof.

COROLLARY: A result in which proof is heavily on a given theorem.

What are the differences between theorems, corollaries, propositions, ?

A proposition is a statement which is offered up for investigation as to its truth or falsehood. The term theorem is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. A corollary is a proof which is a direct result, or a direct application, of another proof.It can be considered as being a proof for free on the back of a proof which has been paid for with blood, sweat and tears.The word is ultimately derived from the Latin corolla, meaning small garland, or the money paid for it. Hence has the sense something extra.

chehada assaf

a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory.

postulate: A postulate is a statement that is accepted without proof.Axiom is another name for postulate. For example, if you know that Pam is 5-feet tall and all her siblings are taller than her, you would believe her if she said that all of her siblings is at least 5.1'. Pam just stated a postulate, and you just accepted it without grabbing a tape measure to verify the height of her siblings.

corollary: A special case of a more general theorem which is worth noting separately. For example, the Pythagorean theorem is a corollary of the law of cosines.

proposition: A proposition is a declaration that can be either true or false, but not both. For example, “Today is Friday” is a proposition. This statement can be true or false, but not both.

Term 1- Assignment

Dear all;
Here is your 2nd assignment for term 1 .

Write down the difference between a theorem , a postulate, a corollary, and proposition ( if there is any)

Thanks in Advance

Keep the Good Work up :)