The Application of Elliptic Curves Cryptography in Embedded Systems
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1 The Application of Elliptic Curves Cryptography in Embedded Systems Wang Qingxian School of Computer Science and Engineering University of Electronic Science and Technology China
2 Introduction to Cryptography Components of a Cryptosystem: a. Plain text b. Cipher c. Code d. Key Popular Schemes used: a. Public-key cryptography - Message is encrypted using a public key. Source: - It is decrypted using a private key. - Private key is related to public key. b. Three pass protocol - Message is encrypted by sender. - Message is super encrypted by receiver. - Sender decrypts message using private key. - Key exchange is not required. a/rot13.png 1-2
3 Strategies for Public key Infrastructure RSA (Ron Rivest, Adi Shamir and Len Adleman) - Product of two large prime numbers is used to create a public key and private key. - With suitably large prime numbers, the problem of factorization increases. - Key sizes increase as the need of security level increase. DSA (Digital Signal Algorithm) - Based on the problem of discrete log over finite field. - For a problem a^b = c, a and c are known, b is required. - Can be solved easily using logarithms. - For larger number the complexity increases and desired level of security is achieved. DSA and RSA are computationally intensive in terms of memory requirement and time. Embedded Systems 1-3
4 Elliptical Curves Basic Properties: Equation of an elliptic curve: y 2 = x 3 + ax + b The equation is defined for no repeated factors. Elliptic curve groups are additive groups. The addition of any two points on curve is defined geometrically. Law of addition: P+Q = R. The point R on the curve is reflected on x-axis to point R. Source: 2_1 1-4
5 Elliptical Curves for Cryptography Law for doubling a point on elliptic curve P+P = 2P = R. An essential property for cryptography is that a group has a finite number of points. An elliptic curve of underlying field Fp or F 2 m is used. The modified equation for each underlying field is: y 2 mod p = x 3 + ax + b mod p y 2 + xy = x 3 + ax 2 + b Elliptic Curve cryptography is based upon the complexity of discrete log problem. 1-5
6 Data security and Embedded systems Need of data security in Embedded Systems - Increase in number of wireless applications - Realization of these application on embedded systems platform. Current Security Algorithms and Embedded System - Time consuming signature generation and authentication. - Large key sizes - RSA and DSA provide a high level of security, but are expensive in terms of memory. - By reducing key size for RSA and DSA, security level is compromised. Elliptical Curve DSA (ECDSA) - Smaller key sizes without compromising level of security. - Quick signature generation and authentication. 1-6
7 ECDSA Implementation The elliptic curve discrete log problem: Given points P and Q in the group, find a number n such that Pn = Q. The private key used is Q. Signature generation: r = x_coord(k = kg) mod p (For any random number k, with G ) s = k^(-1) ( m + nr ) Signature verification: - For a user knowing private key Q and verifying signature for message m : K'= (s^(-1) m) G + (s^(-1) r) Q. r'= x_coord(k') Accept if r == r Source: Embedded.com 1-7
8 Why we obtain smaller key sizes using ECDSA? - RSA and DSA complexity increases as the numbers involved increases. - The use of finite field and modification in the equation of the curve. Source: Embedded.com 1-8
9 Advantages of using ECDSA on Embedded systems - Signature can be calculated before hand. - Smaller key size. - Less intensive modular operations. - Quick generation of signatures Implementation of ECDSA on TI MSP430x33x - Acceptable performance at lower cost. - Modified underlying field for faster arithmetic operations. - Signature generation time is 3.4 sec ECDSA on Palm PDAs - Signature generation time 0.9 sec - Signature verification time 2.4 sec bit ECDSA key provides same level of security as 1024 RSA key. 1-9
10 Conclusion Performance advantages of ECDSA - Computationally less intensive than RSA and DSA. - High security levels in constrained environments. - Reduction in key size without compromising the data integrity. - By having smaller key sizes and efficient signature generation ECDSA is extremely suitable for embedded applications. 1-10
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