Modulo operation

Abstract: New VLSI circuit architectures for addition and multiplication modulo (2/sup n/-1) and (2/sup n/+1) are proposed that allow the implementation of highly efficient combinational and pipelined circuits for modular arithmetic. It is shown that the parallel-prefix adder architecture is well suited to realize fast end-around-carry adders used for modulo addition. Existing modulo multiplier architectures are improved for higher speed and regularity. These allow the use of common multiplier speed-up techniques like Wallace-tree addition and Booth recoding, resulting in the fastest known modulo multipliers. Finally, a high-performance modulo multiplier-adder for the IDEA block cipher is presented. The resulting circuits are compared qualitatively and quantitatively, i.e., in a standard-cell technology, with existing solutions and ordinary integer adders and multipliers.

Abstract: The present invention is an elliptic curve cryptosystem that uses elliptic curves defined over finite fields comprised of special classes of numbers. Special fast classes of numbers are used to optimize the modulo arithmetic required in the enciphering and deciphering process. The class of numbers used in the present invention is generally described by the form 2 q -C where C is an odd number and is relatively small, for example, no longer than the length of a computer word (16-32 bits). When a number is of this form, modulo arithmetic can be accomplished using shifts and adds only, eliminating the need for costly divisions. One subset of this fast class of numbers is known as "Mersenne" primes, and are of the form 2 q -1. Another class of numbers that can be used with the present invention are known as "Fermat" numbers of the form 2 q +1. The present invention system whose level of security is tunable. q acts as an encryption bit depth parameter, such that larger values of q provide increased security. Inversion operations normally require an elliptic curve algebra can be avoided by selecting an inversionless parameterization of the elliptic curve. Fast Fourier transform for an FFT multiply mod operations optimized for efficient Mersenne arithmetic, allow the calculations of very large q to proceed more quickly than with other schemes.

Abstract: Two conversion techniques based on the Chinese remainder theorem are developed for use in residue number systems. The new implementations are fast and simple mainly because adders modulo a large and arbitrary integer M are effectively replaced by binary adders and possibly a lookup table of small address space. Although different in form, both techniques share the same principle that an appropriate representation of the summands must be employed in order to evaluate a sum modulo M efficiently. The first technique reduces the sum modulo M in the conversion formula to a sum modulo 2 through the use of fractional representation, which also exposes the sign bit of numbers. Thus, this technique is particularly useful for sign detection and for any operation requiring a comparison with a binary fraction of M. The other technique is preferable for the full conversion from residues to unsigned or 2's complement integers. By expressing the summands in terms of quotients and remainders with respect to a properly chosen divisor, the second technique systematically replaces the sum modulo M by two binary sums, one accumulating the quotients modulo a power of 2 and the other accumulating the remainders the ordinary way. A final recombination step is required but is easily implemented with a small lookup table and binary adders.