Number Theory and Algebra: A Brief Introduction Rana Barua Indian - - PowerPoint PPT Presentation

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Number Theory and Algebra: A Brief Introduction

Rana Barua

Indian Statistical Institute Kolkata

May 15, 2017

Rana Barua Number Theory and Algebra: A Brief Introduction

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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 = r1 and b mod n = r2, then a ≡ b mod n iff r1 = r2. Also, if a1 ≡ b1 mod n and a2 ≡ b2 mod n then a1 ± a2 ≡ b1 ± b2 mod n; a1a2 ≡ b1b2 mod n. Let I Zn = {0, 1, . . . , n − 1}. Clearly, for any integer a there is a unique r ∈ I Zn s.t. a ≡ r mod n. We equip I Zn with two binary operations + and × which behave exactly like the usual addition and multiplication, except that the results are reduced modulo n

Rana Barua Number Theory and Algebra: A Brief Introduction

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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.

Rana Barua Number Theory and Algebra: A Brief Introduction

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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.

Rana Barua Number Theory and Algebra: A Brief Introduction

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Elementary Number Theory: The Field I Zn

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. ab ≡ 1 mod n. Theorem Let p be a prime number. Then for any a ∈ I Zp − {0} there is a b ∈ I Zp − {0} s.t. ab ≡ 1 mod p. (In other words, I Zp is a field w.r.t. the above addition and multiplication)

Rana Barua Number Theory and Algebra: A Brief Introduction

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Elementary Number Theory

Euler phi-function: Let n be a positive integer. Define φ(n) = |{j < n : gcd(j, n) = 1}|. Properties: φ(pα) = pα(1 − 1

p).

If gcd(m, n) = 1 then φ(mn) = φ(m)φ(n). Consequently, if n = pe1

1 pe2 2 . . . pek k then

φ(n) = n(1 − 1 p1 ) . . . (1 − 1 pk ).

Rana Barua Number Theory and Algebra: A Brief Introduction

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Elementary Number Theory: Theorems of Euler and Fermat

Theorem (Euler) For any integer a s.t. gcd(a, n) = 1, we have aφ(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 aφ(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 ap−1 ≡ 1 mod p.

Rana Barua Number Theory and Algebra: A Brief Introduction

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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 = Me mod N. Decryption: Given ciphertext y ∈ I Z ∗

N, compute

M = yd mod N.

Rana Barua Number Theory and Algebra: A Brief Introduction

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Public Key Cryptosystems :RSA

Correctness: Suppose y ≡ Me mod N. Since ed ≡ 1 mod φ(N) we have ed = tφ(N) + 1. Assume M ∈ I Z ∗

N.

Then yd ≡ Med ≡ (Mφ(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

Rana Barua Number Theory and Algebra: A Brief Introduction

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Public Key Cryptosystems :RSA Signature

• RSA can be used as a signature scheme also.

Rana Barua Number Theory and Algebra: A Brief Introduction

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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 ≡ y2 mod p for some y ∈ I

• Zp. 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 Zp. Theorem (Euler’s Criterion) a is a quadratic residue modulo p iff a

p−1 2

≡ 1 mod p.

Rana Barua Number Theory and Algebra: A Brief Introduction

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More Number Theory: Legendre Symbol

Definition Suppose p is an odd prime and a an integer. Define the Legendre symbol as follows a p

• =

   if a ≡ 0 mod p +1 if a is QR modulop −1 if ais QNR modulop . Definition Suppose, for n odd, n = Πk

i=1pei i

is a prime factorization and a an integer. Define the Jacobi symbol as follows a n

• = Πk

i=1

a pi ei .

Rana Barua Number Theory and Algebra: A Brief Introduction

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More Number Theory

Theorem Suppose p is an odd prime and a an integer. Then a p

• = ap−1/2 mod p.

Remark: This result is used in the Solovay-Strassen Primality testing algorithm.

Rana Barua Number Theory and Algebra: A Brief Introduction

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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.

Rana Barua Number Theory and Algebra: A Brief Introduction