Number Theory and Algebra: A Brief Introduction Rana Barua Indian Statistical Institute Kolkata May 15, 2017 university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Elementary Number Theory: Modular Arithmetic Definition Let n be a positive integer and a and b two integers. We say that a is congruent to b modulo n and write a ≡ b mod n if n | ( b − a ) . Clearly, if a mod n = r 1 and b mod n = r 2 , then a ≡ b mod n iff r 1 = r 2 . Also, if a 1 ≡ b 1 mod n and a 2 ≡ b 2 mod n then a 1 ± a 2 ≡ b 1 ± b 2 mod n ; a 1 a 2 ≡ b 1 b 2 mod n . Let I Z n = { 0 , 1 , . . . , n − 1 } . Clearly, for any integer a there is a unique r ∈ I Z n s.t. a ≡ r mod n . We equip I Z n with two binary operations + and × which behave university-logo-isi exactly like the usual addition and multiplication, except that the results are reduced modulo n Rana Barua Number Theory and Algebra: A Brief Introduction
Groups and Fields Definition A non-empty set G with a binary operation + /. is called a group if the following properties hold. (Closure) For all a , b ∈ G , a + b ∈ G [ a . b ∈ G ] (Associativity) For all a , b , c ∈ G , a + ( b + c )) = ( a + b ) + c [ a . ( b . c ) = ( a . b ) . c ] (Existence of identity) There exist an element 0 ∈ G s.t. a + 0 = 0 + a = a for all a ∈ G [ There exist an element e ∈ G s.t. a . e = e . a = a ] (Existence of Inverse) For each a ∈ G there exists − a ∈ G s.t. a + ( − a ) = ( − a ) + a = 0. [ For each a ∈ G there exists a − 1 ∈ G s.t. a . a − 1 = a − 1 . a = e ] The group is said to be commutative if university-logo-isi a + b = b + a [ a . b = b . a ] for all a , b ∈ G . Rana Barua Number Theory and Algebra: A Brief Introduction
Groups and Fields(cont.) Definition A non-empty set G with a 2 binary operations + and . is called a field if the following properties hold. ( G , +) is a commutative group. ( G − { 0 } , . ) is a commutative group. For all a , b , c ∈ G ; a . ( b + c ) = a . b + a . c . university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Elementary Number Theory: The Field I Z n A useful result. Suppose gcd ( a , b ) = d . Then there exist integers λ, µ s.t. a λ + b µ = d . Corollary Suppose gcd ( a , n ) = 1 ./ Then there exist an integer b s.t. ab ≡ 1 mod n . Theorem Let p be a prime number. Then for any a ∈ I Z p − { 0 } there is a b ∈ I Z p − { 0 } s.t. ab ≡ 1 mod p . (In other words, I Z p is a field w.r.t. the above addition and university-logo-isi multiplication) Rana Barua Number Theory and Algebra: A Brief Introduction
Elementary Number Theory Euler phi-function : Let n be a positive integer. Define φ ( n ) = |{ j < n : gcd ( j , n ) = 1 }| . Properties: φ ( p α ) = p α ( 1 − 1 p ) . If gcd ( m , n ) = 1 then φ ( mn ) = φ ( m ) φ ( n ) . Consequently, if n = p e 1 1 p e 2 2 . . . p e k k then φ ( n ) = n ( 1 − 1 ) . . . ( 1 − 1 ) . p 1 p k university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Elementary Number Theory: Theorems of Euler and Fermat Theorem (Euler) For any integer a s.t. gcd ( a , n ) = 1 , we have a φ ( n ) ≡ 1 mod n . Proof : Let r ∈ I Z ∗ n s.t. a ≡ r mod n . Since I Z ∗ n is a group of order φ ( n ) , we have r φ ( n ) ≡ 1 mod n . So a φ ( n ) ≡ r φ ( n ) ≡ 1 mod n . Theorem (Fermat) Let p be a prime. Then for any integer a s.t. gcd ( a , p ) = 1 we have a p − 1 ≡ 1 mod p . university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Public Key Cryptosystems :Textbook RSA Key-Generation: Let N = pq be the product of two large primes. Choose e , d s.t. ed ≡ 1 mod φ ( N ) Public key: ( N , e ) Secret Key ( N , p , q , d ) Encryption : To encrypt a message M ∈ I Z ∗ N , compute y = M e mod N . Decryption : Given ciphertext y ∈ I Z ∗ N , compute M = y d mod N . university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Public Key Cryptosystems :RSA Correctness : Suppose y ≡ M e mod N . Since ed ≡ 1 mod φ ( N ) we have ed = t φ ( N ) + 1. Assume M ∈ I Z ∗ N . Then y d ≡ M ed ≡ ( M φ ( N ) ) t . M ≡ 1 . M mod N . Remark : If factorization of N is known or if φ ( N ) is known then RSA is completely broken university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
Public Key Cryptosystems :RSA Signature • RSA can be used as a signature scheme also. university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
More Number Theory: Quadratic Residues Definition Suppose p is an odd prime and a an integer. Then a is said to be a quadratic residue modulo p if a �≡ 0 mod p and a ≡ y 2 mod p for some y ∈ I Z p . Otherwise, a is said to be a quadratic non-residue modulo p Remark: Note that there are ( p − 1 ) / 2 QR modulo p in I Z p . Theorem (Euler’s Criterion) a is a quadratic residue modulo p iff p − 1 a ≡ 1 mod p . 2 university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
More Number Theory: Legendre Symbol Definition Suppose p is an odd prime and a an integer. Define the Legendre symbol as follows 0 if a ≡ 0 mod p � a � = + 1 if a is QR modulo p . p − 1 if a is QNR modulo p Definition i = 1 p e i Suppose, for n odd, n = Π k is a prime factorization and a i an integer. Define the Jacobi symbol as follows � a � e i � a � = Π k . i = 1 n p i university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
More Number Theory Theorem Suppose p is an odd prime and a an integer. Then � a � = a p − 1 / 2 mod p . p Remark: This result is used in the Solovay-Strassen Primality testing algorithm. university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
More Number Theory: The Chinese Remainder Theorem Theorem Suppose p , q are odd primes and a , b two integers. Let n = pq . Then the following system of congruence equations has a unique solution modulo n. X ≡ a mod n X ≡ b mod n . university-logo-isi Rana Barua Number Theory and Algebra: A Brief Introduction
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