Number Theory and Algebra: A Brief Introduction
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1 Number Theory and Algebra: A Brief Introduction Indian Statistical Institute Kolkata May 15, 2017
2 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 if n (b a). a b mod n 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 IZ n = {0, 1,..., n 1}. Clearly, for any integer a there is a unique r IZ n s.t. a r mod n. We equip IZ n with two binary operations + and which behave exactly like the usual addition and multiplication, except that the results are reduced modulo n
3 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 a + b = b + a [a.b = b.a] for all a, b G.
4 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.
5 Elementary Number Theory: The Field IZ n A useful result. Suppose gcd(a, b) = d. Then there exist integers λ, µ s.t. aλ + bµ = d. Corollary Suppose gcd(a, n) = 1./ Then there exist an integer b s.t. Theorem ab 1 mod n. Let p be a prime number. Then for any a IZ p {0} there is a b IZ p {0} s.t. ab 1 mod p. (In other words, IZ p is a field w.r.t. the above addition and multiplication)
6 Elementary Number Theory Euler phi-function: Let n be a positive integer. Define Properties: φ(p α ) = p α (1 1 p ). If gcd(m, n) = 1 then φ(n) = {j < n : gcd(j, n) = 1}. Consequently, if n = p e 1 1 pe pe k k φ(mn) = φ(m)φ(n). then φ(n) = n(1 1 p 1 )... (1 1 p k ).
7 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 IZ n s.t. a r mod n. Since IZ 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.
8 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 IZ N, compute y = M e mod N. Decryption: Given ciphertext y IZ N, compute M = y d mod N.
9 Public Key Cryptosystems :RSA Correctness: Suppose y M e mod N. Since ed 1 mod φ(n) we have ed = tφ(n) + 1. Assume M IZ 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
10 Public Key Cryptosystems :RSA Signature RSA can be used as a signature scheme also.
11 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 IZ 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 IZ p. Theorem (Euler s Criterion) a is a quadratic residue modulo p iff a p mod p.
12 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 = p +1 if a is QR modulop 1 if ais QNR modulop. Definition Suppose, for n odd, n = Π k i=1 pe i i is a prime factorization and a an integer. Define the Jacobi symbol as follows ( a ) ( ) a ei = Π k i=1. n p i
13 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.
14 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.
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