Abstract: Scalable and unified multipliers for multiplication of cryptographic parameters represented as elements of either of the prime field (GF(p)) and the binary extension field (GF(2m)) include processing elements arranged to execute in pipeline stages. The processing elements are configurable to perform operations corresponding to either the prime field or the binary extension field. In an example, the processing elements include a dual-field adder having a field-select input that permits selection of a field arithmetic. In a representative example, multipliers are implemented as integrated circuits having processing units that each receive a single bit of one operand and partial words of the remaining operand.

Abstract: Methods and apparatus for modular arithmetic operations with respect to a modulus p include representing operands as a series of s w-bit numbers, wherein s = ? k w ? . Operations are executed word by word and a carry, borrow, or other bit or word is obtained from operations on most significant words of the operands. Depending on the value of this bit or word, an operation-specific correction factor is applied. Cryptographic systems include computer executable instructions for such methods. Bit-level operations are generally avoided and the methods and apparatus are applicable to systems based on, for example, public-key cryptographic algorithms defined over the finite field GF(p).

Abstract: Cryptographic methods and apparatus are provided for determination of multiplicative inverses. A Montgomery radix is selected based on a wordsize, permitting word-wise Montgomery multiplication. Using word-wise Montgomery multiplication, methods and apparatus determine various multiplicative inverses with reduced computation time.

Abstract: Methods for generating elliptic curves of known order over finite fields include selecting a discriminant and a class polynomial from respective sets of discriminants and class polynomials. Based on the selected values, an order of an elliptic curve is determined and the elliptic curve is specified based on a root of the class polynomial. The order of the elliptic curve is adjusted based on a twist operation. The methods are implemented in, for example, computer executable instructions stored on a computer readable medium. Elliptic curve generators based on the methods are provided as well as cryptographic systems including such generators.

Abstract: Methods and apparatus for modular arithmetic operations with respect to a modulus p include representing operands as a series of s w-bit numbers, wherein 1 s = ⌈ k w ⌉ .