Monthly Archives: July 2007

Galois Field and the Advanced Encryption Standard

I got a taste of some interesting mathematics stuff while i was preparing for my internals last weekend.Network Security has Advanced Encryption part of my syllabus and to the greatest of comedies finite field arithmetic was not taught as it was not part of the syllabus.

Now  Field in an modern algebra is an algebraic structure in which addition,multiplication is supported with the rules of a ring i.e addition and multiplication should be commutative.Every element except zero has multiplicative inverse in the set.Division is also possible inside the field to some extent.Examples of field are complex numbers,rational numbers etc.

But above mentioned are basically infinte fields  but in cryptography we are mainly concerned with finite fields.It is refered to as Galois Field now for every prime number p there exists a finite field with p^n elements where p is the charecteristic of the field.Now any arithmentic operation is performed modulo p.So the simplest of finite field is 2^1 which contains the elements 0,1.

This gives the Cayle’s table for 2^1 finite field with the addition and multiplication operations.

 + | 0 1        · | 0 1
 --+----        --+----
 0 | 0 1        0 | 0 0
 1 | 1 0        1 | 0 1

Now why cant we use p as a non prime number this is because if p is non prime there cant be a  multiplicative inverse as the gcd(anynumber,p) may not be equal to one that is they may not be relatively prime to each other.In modulor arithmetic inverse of a number mod p exists if they are relatively prime hence its only a ring if p is not prime.

In AES we use the 2^8 GF which can be represented in polynomial or binary form as like 1100000 representing the polynomials whose power should be less than 8 and coefficients 0 or 1.So if we ad to polynomials in galois field if we get 2 as coefficient we take mod 2 and the sum is 0.

It was interesting studying such refereshing maths probably it would be great to know things or resources that can be interesting for a computer professional.


Presidential Dilemma-Who is the so called Best President?

Presidential elections have never been more contraversial than this time around and never before has political leadership has made a mockery of things some of the comments made by political leaders has been devoid of stature and dignity.BJP which argued against a second term for K.R Narayanan pitched for a second term for A.P.J Abdul Kalam Sushma Swaraj commented saying “How come you compare  A.P.J with K.R.Narayanan” I ask “Why not? ” in the process of petty politics even those dead are not spared at times.Ok Let s do a analysis on A.P.J and K.R.Narayanan.The strong point of A.P.J is that he visisted lots of college spoke a lot on 2020 world super power(god how many of them are gonna mke india a super power :)) and he had a strong student support this was mainly due to the anguish of middle class youth towards Indian youth.But being a president with almost null power you cant do much but as a cobler you need to mend shoes.Consider a cobler who says i sing well yet i dont stich shoes well thats what precisely A.P.J did he could not put effective debates on issues such as Economic Inequalities,education system or untouchability and broad social issues.Thanks to A.P.J that Savarkar(prime conspirator in Gandhi murder) photo hangs in the parliament against Gandhi.K.R Narayana refused to give Bharath Ratna to Savarkar and during the post godhra Gujarat riots wrote a letter to Vajpayee asking him to send the army to save innocent lives.A.P.J signed the dismissal order of Bihar government which has been struck down as unconstitutional by the supreme court.

Movies its an addiction difficult to cure

I always watch quite a lot movies the bizarre thing is most movies i liked most of my friends get bored off many a times i used to curiously suggets some movies and get screwed for my taste.Recently Sivaji starring RajiniKanth(aka superstar) and directed by Shankar was relaesed and it was the most over rated over hyped movie i have seen and didnt like it much.I got a chance to see some good hollywood movies which were really good.

Schindler’s List:God this was a long overdue and i got a chance to have it this month.A real life story of Oscar Schindler who saved many jews during the holocaust.

V for Vendatta:This movie impressed me quite a bit but really hard to follow.Its againts totalitarian governments.

Farenheit 9/11:This is a scintillating documentary be Michael Moore on the controversies of september attacks.Have a go at it cool movie.

The Butterfly Effect:This is a movie based on chaos theory take an example if you have a chance to change certain events of your life what could be your present try this movie a cool one.

P.S:Not to mention waiting for the release of Dasavatharam by kamal hassan being an ardent kamal fan(read it veriyan).