Abstract: A new hardware architecture is disclosed that performs the modular exponentiation operation, i.e., the computation of c=me mod n where c, m, e, n are large integers. The modular exponentiation operation is the most common operation in public-key cryptography. The new method, named the Spectral Modular Exponentiation method, uses the Discrete Fourier Transform over a finite ring, and relies on new techniques to perform the modular multiplication and reduction operations. The method yields an efficient and highly parallel architecture for hardware implementations of public-key cryptosystems which use the modular exponentiation operation as the basic step, such as the RSA and Diffie-Hellman algorithms. The method is extended to perform the multiplication operation in extension fields which is necessary to perform exponentiation or various other operations over these extension fields.

Abstract: Multi-precision multiplication methods over GF(2m) include representing a first polynomial and a second polynomial as an array of n words. A recursive algorithm may be used to iteratively decompose the multiplication into a weighted sum of smaller subproducts. When the size of the smaller subproducts is less than or equal to a predetermined size, a nonrecursive algorithm may be used to complete the multiplication. The nonrecursive algorithm may be optimized to efficiently perform the bottom-end multiplication. For example, pairs of redundant subproducts can be identified and excluded from the nonrecursive algorithm. Moreover, subproducts having weights in a special form may be efficiently calculated by a process that involves storing and reusing intermediate calculations.

Abstract: Multi-precision multiplication methods include storing a first operand and a second operand as a first array and a second array of n words. A first weighted sum is determined from multiple subproducts of corresponding words of the first operand and the second operand. The methods may further include iteratively determining a next weighted sum from a previous weighted sum and a recursively calculated intermediate product. The disclosed methods can be used in a variety of different applications (e.g., cryptography) and can be implemented in a number of software or hardware environments.

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 Montgomery multiplication process a multiplier operand in k-bit radix-digits, wherein k corresponds to a radix r=2k. A multiplicand operand and a modulus are processed word by word, and then additional bits of the multiplier operand are selected for processing. In a radix r=8 example, the multiplier operand is processed in 3 bit radix-8 digits. A processing kernel is configured to preprocess the modulus and/or the multiplier operand so that at least some values can be obtained from lookup tables.

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: Scalable Montgomery multiplication methods and apparatus are provided that are reconfigurable to perform Montgomery multiplication on operands having arbitrary data precision. The methods perform Montgomery multiplication by combining bit-wise and word-wise operations and exhibit pipelined and parallel operation. Apparatus include a control unit that directs bits of an operand to processing elements that receive words of a second operand and a modulus, and produce intermediate values of a Montgomery product. After an intermediate value of a word of a Montgomery product is obtained in a first processing element based on a selected bit of the first operand, the intermediate value is directed to a second processing element and is updated based on another selected bit of the first operand.

Abstract: Multi-precision multiplication methods over GF(2m) include representing a first polynomial and a second polynomial as an array of n words. A recursive algorithm may be used to iteratively decompose the multiplication into a weighted sum of smaller subproducts. When the size of the smaller subproducts is less than or equal to a predetermined size, a nonrecursive algorithm may be used to complete the multiplication. The nonrecursive algorithm may be optimized to efficiently perform the bottom-end multiplication. For example, pairs of redundant subproducts can be identified and excluded from the nonrecursive algorithm. Moreover, subproducts having weights in a special form may be efficiently calculated by a process that involves storing and reusing intermediate calculations.

Abstract: Multi-precision multiplication methods include storing a first operand and a second operand as a first array and a second array of n words. A first weighted sum is determined from multiple subproducts of corresponding words of the first operand and the second operand. The methods may further include iteratively determining a next weighted sum from a previous weighted sum and a recursively calculated intermediate product. The disclosed methods can be used in a variety of different applications (e.g., cryptography) and can be implemented in a number of software or hardware environments.

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 ⌉ .