elliptic curve cryptography

Elliptic Curve Cryptography Ajithkumar Vyasarao

CYSINFO CYBER SECURITY MEETUP – 17TH SEPTEMBER 2016

#Who am I

• Ajithkumar Vyasarao • B.Sc., LL.M, MCA

• PG Diploma in Cyber Laws from NALSAR Hyderabad

• PG Diploma in IPRL from National Law School of India University, Bangalore

• Life Member Cryptology Research Society of India

• Pursuing research in the area of Cryptography


Agenda

• Introduction

• Elliptic Curves Over the Real Numbers

• Elliptic Curves Over a Finite Field

• Elliptic Curve Cryptography Demo

• Elliptic Curve Cryptography – Key Exchange


Introduction

• Major issue with use of public key is the size of numbers used.

• ECC belongs to the category of Public-key Cryptography, performs the computations using elliptic curve arithmetic instead of integer or polynomial arithmetic.

• ECC provides equally good security compared to RSA, but uses smaller key size.

• Notable Advantages of ECC • Uses smaller keys, ciphertexts and signatures. • ECC supports, very fast key generation. • ECC scores over RSA because of its moderately fast encryption and decryption. • ECC computations are uses less memory and CPU cycles compared to RSA, hence

suited for securing Mobile Handheld devices.


Comparable Key Sizes for Equivalent Security

Symmetric scheme

(key size in bits)

ECC-based scheme

(size of n in bits)

RSA/DSA

(modulus size in bits)

56 112 512

80 160 1024

112 224 2048

128 256 3072

192 384 7680

256 512 15360

William Stallings Table 10.3 - “ Comparable Key Sizes in Terms of Computational Effort for Cryptanalysis"


Introduction

• An elliptic curve is defined by an equation in two variables with coefficients.

• Elliptic curves are not ellipses. Elliptic curves are described by cubic equations similar to those used for calculating the circumference of an ellipse

• Elliptic curve cryptography makes use of elliptic curves, in which the variables and coefficients are all restricted to elements of a finite field.


ECC over Real Numbers

• Elliptic curve over real numbers are nothing but set of points (x,y) which satisfy an elliptic curve equation y2 = x3 + ax + b, where x, y, a and b are real numbers.

• Supplying different set of values for a and b results in a different elliptic curve.

• For example a = -4 and b = 0.67 gives the elliptic curve with equation y2 = x3 - 4x + 0.67

• If the cubic polynomial x3+ax+b has no repeated roots, we say the elliptic curve is non-singular.

• A necessary and sufficient condition for the cubic polynomial x3+ax+b to have distinct roots is 4a^3 + 27 b^2 not equal to 0.

• we’ll always assume the elliptic curves are non-singular.


P + Q = R is the additive property defined geometrically CYSINFO CYBER SECURITY MEETUP – 17TH SEPTEMBER 2016

Elliptic Curves Over Finite Fields

• Instead of choosing the field of real numbers, we can create elliptic curves over other fields!

• Let a and b be elements of Zp for p prime, p>3. An elliptic curve E over Zp is the set of points (x,y) with x and y in Zp that satisfy the equation

together with a single element , called the point at infinity.

• As in the real case, to get a non-singular elliptic curve, we’ll require 4a3 + 27 b2 (mod p) 0 (mod p).

• Elliptic curves over Zp will consist of a finite set of points


Elliptic Curves Over Finite Fields

• Just as in the real case, we can define addition of points on an elliptic curve E over Zp, for prime p>3.

• This is done in the essentially the same way as the real case, with appropriate modifications.


Point Addition over Finite Field Fp

• Point addition and doubling are now slightly different visually.

• Lines drawn on this graph will wrap around the horizontal and vertical directions, just like in a game of Asteroids, maintaining the same slope.

• So adding points (2, 22) and (6, 25) looks like this:

• The third intersecting point is (47, 39) and its reflection point is (47, 28).


Ref: http://www.coindesk.com/math-behind-bitcoin/

Elliptic Curve Cryptography Discrete Logarithm Problem [ ECCDLP ] • Addition is simple

P + P = 2P

Multiplication is faster , it takes only 8 steps to compute 100P, using point doubling and add

1. P * 2 = 2P

2. P + 2P = 3P

3. 3P * 2 = 6P

4. 6P *2 = 12P

5. 12P * 2 =24 P

6. P + 24 P = 25 P

7. 25P * 2 = 50 P

8. 50P *2 = 100 P


Elliptic Curve Cryptography Discrete Logarithm Problem [ ECCDLP ] • Division is slow,

• In ECC Q is defined as product of n*P is another point on the curve Q = nP

given initial point P and final point Q, it is hard to compute ‘n’ which serves as a secret key.

Brute force method, start with P, every step multiply P with number 1, 2 and so on,

For each step compare result of P*x where x=1,2,3,… with Q

This problem is known as discrete log problem, difficult to beak


Elliptic Curve point addition (R)- Demo


Elliptic Curve scalar multiplication (R)- Demo


Elliptic Curve point addition (Fp)- Demo


Elliptic Curve scalar multiplication (Fp)- Demo


ECC Application

• ECC is being used in many places such as • PDAs

• VOIP

• Smart cards

• Mobile devices


Diffie-Hellman Key Exchange – ECC scenario

• Alice and Bob two parties need to exchange secret key

1. Both Alice and Bob agree upon starting point P point on elliptic curve publicly defined y2 = x3 - 4x + 0.67

2. Alice selects his private ‘α’ and computes αP shares this with bob

3. Bob selects his private ‘β’ and computes βP shares with Alice

4. Alice receives βP and computes βPα by multiplying with his private

5. Bob receives αP and computes αPβ by multiplying with his private

6. It is obvious βPα = αPβ , hence both Alice and Bob have same key which serves as private key for further encryption and decryption


Security Aspect

• Attacks on groups of elliptic curves are weaker than available factoring algorithms attacks

• Best known attacks on elliptic curves based on cryptographic criterions are the Baby-Step Giant-Step and Pollard-Rho method

• Complexity of these methods are approximately √ p .

• An elliptic curve using a prime p with 160 bit ,roughly 2160 points, provides a security of 280 steps on an average that is required by an attacker.

• An elliptic curve using a prime p with 256 bit, roughly 2256 points, provides a security of 2128 steps on an average.


Reference

1. Network security and cryptography - William Stallings

2. Kwangjo Kim, Cryptography, Open Access Journal

3. Elliptic Curve Cryptography - Lecture notes by Wayne Patterson

4. Elliptic Curve Cryptography: a gentle introduction, http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/

5. Certicom, https://www.certicom.com/ecc


https://www.certicom.com/ecc

https://www.certicom.com/ecc


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