chapter 8

Cryptography and Security Services: Mechanisms and Applications

Manuel [email protected]

M. Mogollon – 1

Chapter 8Elliptic Curve Cryptography

2M. Mogollon – 2

Elliptic Curve Elliptic Curve Cryptography

Session 6 – Contents

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

3M. Mogollon – 3


Cryptography Basics

4M. Mogollon – 4


Security Services Security Mechanisms

Encryption

Hash Functions

Digital Signatures

Security Tokens

Digital SignaturesNon-Repudiation

Access

Authentication

Integrity

Confidentiality

5M. Mogollon – 5


Types of Crypto Systems

• 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.

6M. Mogollon – 6


Symmetric Key Crypto System

• 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).

PlaintextPlaintext Encryption Algorithm

Encryption Algorithm

Ciphertext

Encipher Decipher

Secret Key

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

Asdfe8i4*(74mjsd(9&*nng654mKhnamshy75*72mnasjadif3%j*j^3cdf(#4215kndh_!8g,kla/”2acd:{qien*38mnap4*h&fk>0820&ma012M


7M. Mogollon – 7


Asymmetric Key Crypto System(Public Key Algorithm)

• 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

PlaintextPlaintext Encryption Algorithm

Encryption Algorithm

Ciphertext

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

Asdfe8i4*(74mjsd(9&*nng654mKhnamshy75*72mnasjadif3%j*j^3cdf(#4215kndh_!8g,kla/”2acd:{qien*38mnap4*h&fk>0820&ma012M


8M. Mogollon – 8


Combining Symmetric and Asymmetric Ciphers

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

Symmetric Encryption

Ciphertext Block

IV + +

Secret Key

IV+ +

Secret KeyUse a symmetric algorithm to encipher

and decipher a secure transaction.

Encipher Decipher

Client Web Server




Master Key Generation

Pre-Master Key

Master Key Generation

Pre-Master Key

Integrity (HMAC)

Integrity (HMAC)

Cleartext Block

Cleartext Block

Ciphertext Block

Cleartext Block

Cleartext Block

Ciphertext Block

Ciphertext Block

9M. Mogollon – 9


Types of Public-key Cryptography

• Exponentiation Ciphers RSA.

• Discrete logarithm systems ElGamal public-key encryption, Digital Signature Algorithm (DSA),

Diffie-Hellman key exchange.

• Elliptic curve cryptography

10M. Mogollon – 10


Public Key Encryption

Encipher

Decipher

Alice’s Private Key

Alice’s Public Key

Encipher

Decipher

Bob’s Public Key

Bob’s Private Key

Encipher

Decipher

Bob’s Private Key

Bob’s Public Key

Sender (Alice) Receiver (Bob)

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.

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

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

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.



Elliptic Curve Concepts



What is Elliptic Curve Cryptography?

• 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.

ECC Applications

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



ECC Applications

• 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 Curves

• 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

• 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 F2

m. The finite field is also denoted as GF(q).



Finite Fields

• 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 F2

m consists of the 2m possible bit strings of length m.For example, F2

3 = {000, 001, 010, 011, 100, 101, 110, 111}



Group Fields in EC

• 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.



Elliptic Curves and Points

• 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:

baxxy 32

baxxxyy 232



Elliptic Curve ArithmeticPoint 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.

R

Q

P

- 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)

E: y2 = x3 - 9x + 6

- R

PR

P (0.0, 2.45)-R (3.38, -3.76)R (3.38, 3.76)2P = R = (3.38, 3.76)

E: y2 = x3 - 9x + 6



Elliptic Curve Arithmetic

• Doubling a Point in Fp

Provided thatthen,

where

and

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

0Py

),(),(),( RRPPPP yxRyxPyxP

pxx PR mod22

pyxxy PRPR mod)(

py

ax

P

P mod)2(

)3( 2



Elliptic Curves Arithmetic

• 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 ).

QP ),(),(),( RRQQPP yxRyxQyxP

pxxx QPR mod2

pyxxy PRPR mod)(

pxxyy

PQ

PQ mod)()(



Elliptic Curve Arithmetic

Point Addition in Fp

• Adding P to -P.

P

P (-1.85, 4.05)-P (-1.85, -4.05)P + (-P) = O, the point at infinity

E: y2 = x3 - 9x + 6

-P



EC Points Points in the Elliptic Curvey^2 = x^3 + x + 1 (mod 23)

02468

1012141618202224

0 2 4 6 8 10 12 14 16 18 20

• 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).



Point and Curve Order

Point Order Point Order Point Order Point Order

(0,1) 28 (9,16) 28 (7,11) 14 (13,16) 7(0,22) 28 (18,3) 28 (7,12) 14 (17,3) 7(1,7) 28 (18,20) 28 (12,4) 14 (17,20) 7

(1,16) 28 (19,5) 28 (12,19) 14 (11,3) 4(3,10) 28 (19,18) 28 (5,4) 7 (11,20) 4

(3,13) 28 (6,4) 14 (5,19) 7 (4,0) 1 (infinity)

(9,7) 28 (6,19) 14 (13,7) 7

• 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



Point Order



Selecting an EC for Cryptography

• 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).

ppFEpp q 21)(#21



Selecting a Generator Point

• 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

http://www.cryptomathic.com/labs/ellipticcurvedemo.html




Discrete Logarithmic Problem

• 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 = xk mod 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?



EC Discrete Logarithmic Problem

• 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.

)( pFE )( pFEP)( pFEQ



Elliptic Curve Public-Key Cryptography

• 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



Brute Force Attack

• 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.



Brute Force Attack

• 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



Breaking the Code

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.



Public Key Systems Key Size Comparisons

Security (Bits)

Symmetric Encryption Algorithm

Hash Algorithm

Minimum Size of Public keys (Bits)

Diffie-Hellman and RSA Modulus Size

ECC

80 SKIPJACK SHA-1 1024 1024 160

112 3DES 2048 2048 224

128 AES-128 SHA-256 3072 3072 256

192 AES-192 SHA-384 7680 7680 384

256 AES-256 SHA-512 15360 15360 512

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.

3/23/1 ))2log((log NNn



Elliptic Curve Cryptography



Domain Parameters

• 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 Fp and 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,

q Defines 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.

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

a, b The coefficients defining the elliptic curve E, elements of Fq.G A 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.n The order of the base point G.h Called the cofactor, h = #E(Fq)/n, where n is the order of the base point G.

h is normally a small number.

qF )( pFE )(2mFE



ECC Cryptography

• 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

• 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



ECC Cryptography

• 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.



Key Generation

• 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.

pbaxxy QQQ mod32 mFinbaxxyxy QQQQQ 232



• 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.

ECC ElGamal EncryptionAlice Bob

pGPrivPub AA modT and PubA do not need to be secret.

• 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.

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)}



• 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)

ECC ElGamal EncryptionAlice Bob

T and PubA do not need to be secret

• 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.

CM, PubB

23mod)16,13(23mod)19,5(4

A

A

PubPub

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



Diffie-Hellman Key Exchange System

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.

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

prime integer

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

secret.Alice Bob

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

prime integer

pGrivPPub AA mod pGrivPPub BB mod

AB PrivPubZZ BA PrivPubZZ

Sender and receiver agree on the same domain parameters.

ubBPubAP



Diffie-Hellman Key Exchange System

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

ubBPubAP

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

pGrivPPub AA mod pGrivPPub BB mod

AB PrivPubz BA PrivPubz

23mod)16,13(23mod)19,5(4 APub 23mod)23,17(23mod)19,5(2 BPub

23mod)19,5(23mod4)3,17( z 23mod)19,5(23mod2)16,13( z

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



• 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).

ECCDSA Signature GenerationAlice Bob

pGPrivPub AA modT 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.

]2,2[ nk

),(* 11 yxGk

nsc mod1

ncmHu mod.)(1

ncru mod.2

nxv mod0

Ao PubuGuyx **),( 210

nk mod1

nxr mod1

(r, s)

nrPrivmHks A mod}.)({1



• 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).

ECCDSA Signature GenerationAlice Bob

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 .

nsc mod1

ncmHu mod.)(1 ncru mod.2

Ao PubuGuyx **),( 210

23mod)16,13(23mod)19,5(4 APub

)7,13()19,5(.3.),( 11 Gkyx

7mod57mod27mod3 1

7mod67mod13 r

nk mod1

7mod27mod1757mod)6.410(5 s

7mod47mod37mod2 1 c

7mod57mod4.101 u

7mod37mod4.62 u

7mod67mod13mod0 pxv

)7,13()20,17()20,17(),()16,13(.3)19,5(.5),(

0

0

o

o

yxyx

nrPrivmHks A mod}.)({1



Cipher Suite

• 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.



To Probe Further• 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

http://citeseer.nj.nec.com/lop00overview.html

http://www.secg.org/secg_docs.htm


http://www.certicom.com/index.php?action=ecc,ecc_tutorial

chapter 8

Documents

security policy

security architecture

security authority

provision of security

network security chapter

type of crypto system

secret key encryption

enciphering key secret