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Online Cryptography Course Dan Boneh. Intro. Number T heory. Modular e’th roots. Modular e’th roots. We know how to solve modular linear equations: a⋅x + b = 0 in Z N Solution : x = −b⋅a -1 in Z N What about higher degree polynomials?

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modular e th roots1
Modular e’th roots

We know how to solve modular linear equations:

a⋅x + b = 0 in ZNSolution: x = −b⋅a-1 in ZN

What about higher degree polynomials?

Example: let p be a prime and c∈Zp . Can we solve:

x2 – c = 0 , y3 – c = 0 , z37 – c = 0 in Zp

modular e th roots2
Modular e’th roots

Let p be a prime and c∈Zp .

Def: x∈Zps.t.xe = c in Zp is called an e’th root of c .

Examples:

71/3 = 6 in

31/2= 5 in

11/3 = 1 in

21/2 does not exist in

the easy case
The easy case

When does c1/e in Zpexist? Can we compute it efficiently?

The easy case: suppose gcd( e , p-1 ) = 1 Then for all c in (Zp)*: c1/eexists in Zp and is easy to find.

Proof: let d = e-1 in Zp-1 . Then

d⋅e = 1 in Zp-1 ⇒

the case e 2 square roots
The case e=2: square roots

x

1

5

2

4

3

−x

10

9

8

7

6

If p is an odd prime then gcd( 2, p-1) ≠ 1

Fact: in , x ⟶ x2 is a 2-to-1 function

Example: in :

Def: x in is a quadratic residue (Q.R.) if it has a square root in

p odd prime ⇒ the # of Q.R. in is (p-1)/2 + 1

9

4

5

1

3

x2

euler s theorem
Euler’s theorem

Thm: x in (Zp)* is a Q.R. ⟺ x(p-1)/2= 1 in Zp(p odd prime)

Example:

Note: x≠0 ⇒ x(p-1)/2 = (xp-1)1/2 = 11/2 ∈ { 1, -1 } in Zp

Def: x(p-1)/2 is called the Legendre Symbol of x over p (1798)

in : 15, 25, 35, 45, 55, 65, 75, 85, 95, 105

= 1 -1 1 1 1, -1, -1, -1, 1, -1

computing square roots mod p
Computing square roots mod p

Suppose p = 3 (mod 4)

Lemma: if c∈(Zp)* is Q.R. then √c = c(p+1)/4 in Zp

Proof:

When p = 1 (mod 4), can also be done efficiently, but a bit harder

run time ≈ O(log3 p)

solving quadratic equations mod p
Solving quadratic equations mod p

Solve: a⋅x2+ b⋅x+ c = 0 in Zp

Solution: x = (-b ± √b2 – 4⋅a⋅c ) / 2a in Zp

  • Find (2a)-1 in Zpusing extended Euclid.
  • Find square root of b2 – 4⋅a⋅c in Zp(if one exists) using a square root algorithm
computing e th roots mod n
Computing e’th roots mod N ??

Let N be a composite number and e>1

When does c1/e in ZNexist? Can we compute it efficiently?

Answering these questions requires the factorization of N

(as far as we know)