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Chapter 8

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Chapter 8

Elliptic Curve Cryptography

- Cryptography Basics
- Elliptic Curve (EC) Concepts
- Finite Fields
- Selecting an Elliptic Curve
- Cryptography Using EC
- Digital Signature

Digital Signatures

Security Tokens

Access

Authentication

Confidentiality

Non-Repudiation

Digital Signatures

Encryption

Hash Functions

Integrity

- Symmetric Cryptography – Secret Key
- A single key serves as both the encryption and the decryption key.
- Initial arrangements need to be made for individuals to share the secret key.
- Stream Ciphers and Block Ciphers (DES, AES)

- Asymmetric Cryptography – Public-Key
- One key is used to encipher and another to decipher.
- Privacy is achieved without having to keep the enciphering key secret because a different key is used for deciphering.
- Pohlig Hellman, Schnorr, RSA, ElGamal, and Elliptic Curve Cryptography (ECC) are popular asymmetric crypto systems.

As the market requirements for secure products has exponentially increased, our strategy will be to ….

As the market requirements for secure products has exponentially increased, our strategy will be to ….

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Secret Key

- Security is based on the secret key, not on the encryption algorithm.
- The sharing of secret keys is necessary.
- Strengths: Fast, good for encrypting large amounts of data.
- Weakness: Key delivery.
- There are two types of symmetric crypto systems: Stream Cipher (RC4) and Block Ciphers (DES, AES, RC5, CAST, IDEA).

Ciphertext

Plaintext

Plaintext

Encryption Algorithm

Encryption Algorithm

Encipher

Decipher

As the market requirements for secure products has exponentially increased, our strategy will be to ….

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Encipher

Decipher

- Public key encryption involves two mathematically related keys.
- Either key can be used to encipher.
- One of the keys can be made public and the other kept private.
- Strengths: No key delivery issues, can be used for non-repudiation.
- Weakness: Slow, inefficient for large amounts of data, computationally expensive.
- Algorithms: RSA, ElGamal, Schnorr, Pohlig-Hellman, Elliptic Curve Cryptography.
- Used mainly for key exchange or digital signatures.

One Key to Encipher

Another Key to Decipher

Ciphertext

Plaintext

Plaintext

Encryption Algorithm

Encryption Algorithm

Client

Web Server

Exchange (wrap / transport ) or agree (Diffie-Hellman) on a pre-master key.

Pre-Master Key

Pre-Master Key

Master Key Generation

Integrity (HMAC)

Integrity (HMAC)

Master Key Generation

Encipher

Decipher

Cleartext Block

Cleartext Block

Cleartext Block

Cleartext Block

+

+

+

+

IV

IV

Use a symmetric algorithm to encipher and decipher a secure transaction.

Symmetric Encryption

Symmetric Encryption

Symmetric Encryption

Symmetric Encryption

Secret Key

Secret Key

Ciphertext Block

Ciphertext Block

Ciphertext Block

Ciphertext Block

- Exponentiation Ciphers
- RSA.

- Discrete logarithm systems
- ElGamal public-key encryption, Digital Signature Algorithm (DSA), Diffie-Hellman key exchange.

- Elliptic curve cryptography

Receiver (Bob)

Sender (Alice)

Alice’s Private Key

Alice’s Public Key

Non-Repudiation of Origin (Authenticity) Anyone who has Alice’s public key will be able to decipher the message. Alice cannot deny that she sent the message.

Encipher

Decipher

Alice’s Public Key

Alice’s Private Key

Bob will not be able to decipher the message because he doesn’t have Alice’s private key.

Encipher

Decipher

Bob’s Public Key

Bob’s Private Key

Confidentiality ─ Bob will be the only one able to decipher the message because only he has his private key.

Decipher

Encipher

Bob’s Private Key

Bob’s Public Key

Enciphering is not possible because Alice doesn’t have Bob’s private key.

Encipher

Decipher

- elliptic curve cryptography / (abbr. ECC)(1) an encryption system that uses the properties of elliptic curve and provides the same functionality of other public key cryptosystems; (2) A public key crypto system that provides, bit-by-bit key size, the highest strength of any cryptosystem known today.

- ECC with 160-bit key size offers the same level of security as RSA with 1024-bit key size.
- Smaller key size provides
- Storage efficiencies
- Bandwidth savings
- Computational efficiencies

- ECC implementation is beneficial in applications where bandwidth, processing capacity, power availability, or storage are constrained.
- ECC includes key distribution, encryption, and digital signatures.

- Which leads to
- Higher speeds
- Lower power consumptions
- Code size reductions

- Applications requiring intensive public-key operations.
- Web servers.

- Applications with limited power, computational power, speed transfer, memory storage, or bandwidth.
- Wireless communications
- PDAs

- Applications rigid constrains on processing power, parameter storage, and code space.
- Smart card and tokens.

- Elliptic Curve Cryptography uses plane curves, which are sets of points satisfying the equation F (x, y) = 0.
- Examples of plane curves are:
- Lines (2x + y = a)
- Conic sections (3x2 + 5y2 = a)
- Cubic curves (y2 + xy = x3 + ax2 + b), which include elliptic curves.

- Finite fields are fields that are finite.
- A field is a set F in which the usual mathematical operations (addition, subtraction, multiplication, and division by nonzero quantities) are possible; these operations follow the usual commutative, associative, and distributive laws.
- Rational numbers (fractions), real numbers, and complex numbers are elements of infinite fields.
- A discrete logarithm (DL) and elliptic curve (EC) cryptography schemes are always based on computations in a finite field in which there are only a finite number of quantities.
- For cryptography applications, the finite fields that are usually used are the field of characteristic (congruences).
- The finite field used in DL and EC are the field of prime characteristic Fp and the field of characteristic two F2m. The finite field is also denoted as GF(q).

- Characteristic Prime Finite Fields
- The finite field Fp is the prime finite field containing p elements. If p is an odd prime number, then there is a unique field Fp that consists of the set of integers{0, 1, 2 ,..., p – 1}.

- Characteristic Two Finite Fields
- A characteristic two finite field (also known as a binary finite field) is a finite field whose number of elements is 2m. If m is a positive integer greater than 1, the binary finite field F2m consists of the 2mpossible bit strings of length m.
For example, F23 = {000, 001, 010, 011, 100, 101, 110, 111}

- A characteristic two finite field (also known as a binary finite field) is a finite field whose number of elements is 2m. If m is a positive integer greater than 1, the binary finite field F2m consists of the 2mpossible bit strings of length m.

- There are two essential properties of group fields when they are used in elliptic curve cryptography:
- A group should have a finite number of points. An elliptic curve has infinite number of points, but an elliptic curve over Fq has a finite number of elements.
- The operation that is used should be easy to compute but very difficult and time consuming to reverse.

- The scalar integer multiplication of an elliptic curve point, P, which is defined as the repeated addition of the point with itself, Q = kP, is an operation that is easy to compute but very difficult and time consuming to reverse.

- There are several ways of defining equations for elliptic curves, but the most common are the Weierstrass equations.
- ECC may be implemented over Fq, where q is an odd prime p, or 2m.
- If ECC is implemented over Fp, the following equation is used:
- If ECC is implemented over F2m, the following equation is used:

Point Addition in Fp

- The group law is defined by P + Q–R = 0; therefore, P + Q = R, where the negative of the point R(x, y) is the point R (x, –y).
- Given two points on the curve P and Q, the line through them meets the curve at a third point –R. The reflection of R gives the point R, which is equal to P + Q.
- The tangent line through P gives the point –R.

E: y2 = x3 - 9x + 6

E: y2 = x3 - 9x + 6

P (0.0, 2.45)

-R (3.38, -3.76)

R (3.38, 3.76)

2P = R = (3.38, 3.76)

- R

P (0.0, 2.45)

Q (-3.24, -1.17)

-R (4.49, 7.47)

R (4.49, -7.49)

P + Q = R = (4.49, -7.49)

R

P

P

Q

- R

R

- Doubling a Point in Fp
Provided that

then,

where

and

λ is the slope of the line through P(xP , yP).

- Point Addition in Fp
Similar to the addition of two points in plane geometry. Forthen,

where

and

λ is the slope of the line through P(xP , yP) and Q(xQ , yQ ).

E: y2 = x3 - 9x + 6

P (-1.85, 4.05)

-P (-1.85, -4.05)

P + (-P) = O, the point at infinity

P

-P

Point Addition in Fp

- Adding P to -P.

The points are symmetric because in elliptic curves, for every point P, there must exist another point –P.

The point P(0, 1) generates a maximal subgroup because it generates the maximum number of points, 28 (27 plus the point at infinity).

The curve order is 28 and is denoted as #E(Fp).

- For any point in y2 = x3 + x + 1 (mod 23), the value of k such that kP = O is not always the same. The order of points varies; it can be 28, 14, 7 or 4.
- The maximum point order is the curve order.

See next slide

- There are several procedures to select an elliptic curve for cryptographic purposes. The following are some of the criteria:
- Select a large prime number, p, to be used as the module.
- Select the coefficients a and b randomly and define E Fp:y2 = x3 + ax + b.
- Calculate the curve order #E(Fq).
- Check that #E(Fq) is divisible by a large prime number.
- Check that the largest prime divisor of #E(Fq) does not divide qv-1 for v = 1, 2, 3, ……<large limit>.

- Another way to select the elliptic curve is by selecting the curve order first:
- Select a large prime number, p, to be used as the module.
- Select the curve order, #E(Fp), such that
- Check that #E(Fp) is divisible by a large prime number, r.
- Check that r does not divide pv-1 for v = 1, 2, 3, ……10.
- Use the Atkin-Morain algorithm to find parameters a and b in Fp such that the elliptic curve E has an order of #E(Fp).

Select a random point G on E(Fp) and a large prime number n that divides #E(Fp).

Check that the nG = O, n being the point order.

The size of the odd prime modulus in bits is 15

Curve generated using Cryptomathic on line generator at http://www.cryptomathic.com/labs/ellipticcurvedemo.html#Key-Generation

- In the multiplicative group Zp* discrete logarithm (Diffie-Hellman, ElGamal, DSS), the following is the discrete logarithm problem:
- Given elements y and x of the group, and a prime p, find a number k such that y = xkmod p.
- For example, if y = 2, x = 8, and p = 341, then find k such that 2 ≡ 8k mod 341.
- In the Diffie-Hellman discrete logarithm, y is the public key, g is a large random number, p is the modulo, and k is the private key that the cryptanalyst is trying to find out.

Which one is the correct Private Key?

- Given an elliptic curve , a point of an order n, and a point , determine the integerk, 0≤ k ≥ n-1, such that Q = kP, provided that such integer k exists.
- Q is the public key and k is the private key.
- The scalar integer multiplication of an elliptic curve point, P is defined as the process of adding P to itself k times. Q = kP is analogous to exponentiation in a discrete logarithm cryptosystem, i.e., it is an operation that is easy to compute but very difficult and time consuming to reverse.

The scalar integer multiplication of an elliptic curve point, P is defined as the process of adding P to itself k times. Q = k P.

When the point (0,1) is added to itself 13 times the result is the point (9, 16).

Q = k P = 13 * (0,1) = (9,16)

Select Q = Public Key = (9,16)k = Private Key = 13

There is not a known algorithm to attack ECC

Brute force attack

Starting with point (0,1), add (0,1) to itself until (9,16) is found.

Stop when Q = d P = (9, 16)

The order of the base point is 28

It would take a system doing a million addition/sec, 14 microseconds to try 50% of all possible points.

The size of the odd prime modulus in bits is 5.

There is not a known algorithm to attack ECC

Brute force attack

Starting with point P, add P to itself until Q is found.

Stop when kP = Q

The order of the base point is 1.73*1046

It would take a system doing a million addition/sec (3.15*1018 additions/year) 1032 years to try 50% of all possible points.

The size of the odd prime modulus in bits is 161. Equivalent to RSA 1024

April 27, 2004Certicom Corp. (TSX: CIC), the authority for strong, efficient cryptography, today announced that Chris Monico, an assistant professor at Texas Tech University, and his team of mathematicians have successfully solved the Certicom Elliptic Curve Cryptography (ECC) 109-bit Challenge. The effort required 2600 computers and took 17 months. For comparison purposes, the gross CPU time used would be roughly equivalent to that of an Athlon XP 3200+ working nonstop for about 1200 years.

Blake, Seroussi, and Smart (1999, p9) compared the two algorithms known to break ECC and discrete algorithms. Simplifying the formulas and making several approximations, they arrived at the following formula comparing key-length for similar levels of security:

where β ≈ 4.91. The parameters n and N are the “key sizes” of ECC and DL cryptosystems.

- Parties using elliptic curve cryptography need to share certain parameter, the “Elliptic Curve Domain Parameters”.
- The EC domain parameters may be public; the security of the system does not rely on these parameters being secret.
- The domain consists of six parameters which are calculated differently for Fpand F2m . It precisely specify an elliptic curve and base point.
- The six domain parameters are the following:
T = (q; FR; a, b; G; n; h), in which,

qDefines the underlying finite field Fq. The field size is defined by the module, so, q = p or q = 2m ; p>3 should be a prime number.

FRField representation of the method used for representing field elements in , either or .

a, bThe coefficients defining the elliptic curve E, elements of Fq.

GA distinguished point, G=(xG ,yG), on an elliptic curve called the base point or generating point defined by two field elements xG and yG in Fq.

nThe order of the base point G.

hCalled the cofactor, h = #E(Fq)/n, where n is the order of the base point G. h is normally a small number.

- Encryption
- EC Integrated Encryption Scheme (ECIES)
- Variant of ElGamal public-key encryption
- Proposed by Bellare and Rogaway
- Variant of ElGamal public-key encryption schme
- ANSI X9.63, ISO/IEC 15946-3, and IEEE P1363a draft

- Provably Secure Encryption Curve (PSEC)
- Fujisaki and Okamoto
- Evaluated by NESSIE and CRYPTREC

- EC Integrated Encryption Scheme (ECIES)
- Key Exchange
- Station-to-Station Protocol
- Diffie, van Oorschot, and Wiener
- Discrete logarithm-base key agreement
- ANSI X9.63

- ECMQV
- Meneses, Qu, and Vanstone
- ANSI X9.63, IEEE 1363-2000, and ISO/IEC 15946-3

- Station-to-Station Protocol

- Digital Signature
- Elliptic Curve Digital Signature Algorithm (ECDSA)
- Analog to the Digital Signature Algorithm (DSA)
- Secure Hash Algorithm (SHS-1)
- ANSI X9.62, FIPS 186-2, IEEE1363-2000 and ISO/IEC 15946-2

- EC Korean Certificate-based Digital Signature Algorithm (EC-KCDSA)
- Lim and Lee
- ISO/IEC 15946-2.

- Elliptic Curve Digital Signature Algorithm (ECDSA)

- The public and private keys of an entity A are associated with a particular set of elliptic curve domain parameters (q; FR; a; b; G; n; h). To generate a key pair, entity Alice does the following:
- Selects a random or pseudo-random integer d in the interval [1, n - 1].
- Computes Q = d * G.
- Has Q as public key, PubA, and d as private key, PrivA.
- Checks that xG and yG are elements of the elliptic curve equation by calculating or .

- Example:
- For E(F23): y2 = x3 + x + 1, #E(F23) =28. Then, n=7, since n should be a prime factor of 28.
- The cofactor h is equal to 28 / 7 = 4.
- A point with an order of 7 should be selected.
- The point G could be (5, 19), one of several points with n = 7. The domain parameter T = (p; a; b; G; n; h) is T = [23; 1; 1; (5,19); 7, 4 ].
- Select d = 4, so Q = 4 (5, 19). (13, 16).
- Alice’s public key is PubA = Q = (13, 16) and her private key is PrivA = 4.

Alice

Bob

- Let T = (p; a; b; G; n; h) and be Alice’s public key.
- Alice deciphers the message by
- Multiplying her private key PrivA by (PrivB . G).
- Subtracting the above result from M + PrivB . PubA.

- Bob selects a random number as his private key and generates his public key using the same elliptic curve and G point.
- Bob enciphers the message, M, by doing
- CM = [{PrivB* G}, {M + PrivB*PubA}]
- Bob sends his PubB and cipher message to Alice.

T and PubA do not need to be secret.

CM, PubB

CM = [{PrivB* G}, {M + PrivB*PubA}]

M = {M + PrivB * PubA} – { PrivA * PrivB * G}

Since PubA= PrivA * G, then,

M = {M + PrivB * (PrivA . G)} – { PrivA * (PrivB * G)}

Alice

Bob

- Let T = [23; 1; 1; (5,19); 7; 4 ] and select 4 as the PrivA, as the public key.
- Alice deciphers the message by
- Multiplying her private key 4 by (18,11) = (5, 4).
- Subtracting the above result from (17, 20)M = (17,20) – (5, 4)
- M = (17,20) + (5, -4) = (8, 20)

- Bob selects 4 as his private key.
- The message is the point (8,20).
- Bob enciphers the message by
- CM = [{5*(5, 19)}, {(8, 20) + 5* (13, 16)}]
- Bob sends his PubB and cipher messageCM = [(17, 20), (18,11)] to Alice.

T and PubA do not need to be secret

CM, PubB

Note: The cofactor h =4 in T is not related to the PrivA, which was selected at random and happens to be 4, also.

Sender and receiver agree on the same domain parameters.

T = (p; a; b; G; n; h), does not need to be secret.

Bob

Alice

T = (p; a; b; G; n; h)PrivB = Random large prime integer

T = (p; a; b; G; n; h)PrivA = Random large prime integer

Alice and Bob convert the shared secret value z to an octet string Z and use Z as the shared secret key for symmetric encryption algorithms to secure their communications.

Bob

Alice

T = [23; 1; 1; (5,19); 7; 4]

T = [23; 1; 1; (5,19); 7; 4]

Note: The cofactor h =4 in T is not related to the PrivA, which was selected at random and happens to be 4, also.

Alice

Bob

- T = (p; a; b; G; n; h) and
is Alice’s public key.

- Selects a random integer
- Computes
- Computes
- Computes
- The signature for the message m is the pair of integers (r, s).

T and PubA do not need to be secret.

- Verifies Alice’s signature(r, s) on the message m as follows:
- Computes H(m) and
- Computes
- Computes
- Accepts the signature if v = r.

(r, s)

Alice

Bob

- Let T = [23; 1; 1; (5,19); 7; 4] and
- Select k = 3
- Compute
- Compute
- Compute
- The signature for the message m is the pair of integers (r, s), (6, 2).

- Bob verifies Alice’s signature(6, 2) on the message m as follows:
- Compute H(m) and
- Compute
- Compute
- Compute
- Accept the signature becausev = 6 mod 7 = r .

- There are many algorithms that can be used for encryption, key exchange, message digest, and authentication; the level of security for each of these algorithms varies. Establishing a connection between two entities requires that they tell each other what crypto algorithms they understand. Normally one of the entities involved in the communication proposes a list of algorithms, and the other entity selects the algorithms supported by both. The selected algorithms may not have matching levels of security, reducing the overall security of the communication.
- A cipher suite is a collection of cryptographic algorithms that matches the level of security of all the algorithms listed in the cipher suite. To enable secure communications between two entities, they exchange information about which cipher suites they have in common, and they then use the cipher suite that offers the highest level of security.

- Hankerson, D., Meneses, A., Vanstone S. (2004). Guide to Elliptic Curve Cryptography. New York: Springer-Verlag.
- Blake, I., Seroussi G., Smart, N. (1999). Elliptic Curves in Cryptography. Cambridge, United Kingdom: Cambridge University Press.
- Rosing, M. (1999). Implementing Curve Cryptography. Greenwich, CT: Manning Publications.
- Lopez, J., Dahab, R., An overview of Elliptic Curve Cryptography, Institute of computting , State University of Campinas, sao Paulo Brazil, may 2, 2000. (Retrieved September 26, 2003 from http://citeseer.nj.nec.com/lop00overview.html)
- Brown, M., Cheung, D., Hankerson, D., Lopez, J., Kirkup, M., Menezes, A., PGP in Constrained Wireless Devices, Proceedings of the 9th USENIX Security Symposium, August 2000.
- Certicom Research, Standard for Efficient Cryptograph (SEC 1): Elliptic Curve Cryptograph, September 20, 2000. (Retrieved September 26, 2003 from http://www.secg.org/secg_docs.htm)
- Certicom Research, Current Public-Key Crypto Systems, April 1997. (Retrieved on September 20, 2000 from )
- Cryptomathic, Ellipt Curve Online Key Generation athttp://www.cryptomathic.com/labs/ellipticcurvedemo.html#Key-Generation
- Certicom Elliptic Curve Tutorial at http://www.certicom.com/index.php?action=ecc,ecc_tutorial
- IEEE P1363, Standard Specifications for Public key Cryptography, draft 2000