Design Of Finite Field Multiplier Over GF(2~m)

Finite field arithmetic has gained much attention in cryptography and error correcting codes, especially in public key cryptography based on complex arithmetic such as Elliptic Curve Cryptosystems. Multiplication, as a basic and complex operation in finite field arithmetic, has been widely studied. With the increasing digit-processing speed and real-time requirements, it is very important to design and fabricate finite field multiplier in hardware to meet performance. On the other hand, as one important way of side-channel attacks, fault-based cryptanalysis has been proven to be a great threat to cryptosystems security, which makes error detection/correction be another aspect must be considered in design of finite field multiplier. In this dissertation, we investigate efficient and secure design of multiplier over finite field GF(2m).The main works are as follows.1. Based on even-type Gaussian normal basis(GNB), an efficient bit parallel systolic multiplier with error detection is proposed, which has the properties of regularity and modularity. Compared with similar works, the proposed multiplier has advantage in reducing space-time complexity. Moreover, using two different methods, error detection capability of multiplier is achieved. Analysis shows that any single-cell fault can be detected, which can efficiently defeat fault-based attacks.2. Based on Montgomery method, a new bit parallel systolic even-type GNB multiplier is developed. In previous works, Montgomery method has been extensively studied in modular integer multiplication and polynomial basis multiplication over GF(2m), but never for normal basis multiplication. The proposed multiplier is the first work in this aspect. What's more important, this multiplier significantly reduced space-time complexity compared with previous works and has more practical applications in the future.3. A bit parallel systolic shifted polynomial basis multiplier with error detection is proposed, which designs for finite fields defined by irreducible trinomials. The error detection scheme is based on Hamming codes and any two-bit fault can be detected, which has a better capability in defeating fault-based analysis. Compared with similar multiplier in previous works, the proposed multiplier has advantage in reducing time complexity with comparable space complexity.4. With respect to Dickson polynomial basis, two subquadratic space complexity multipliers are proposed for finite fields constructed by two classes of irreducible Dickson polynomials. By using the Dickson polynomial basis, finite field multiplication is converted to special Toeplitz matrix-vector multiplication, which can be computed by block segmentation and recursive computation approach. By acting proper transformation on the Toeplitz matrix to utilize some good properties of block segmentation function, then with the help of new block recombination method, the proposed multipliers achieve less space complexity compared with similar works.