Approaching encryption at ISDN speed using partial parallel modulus multiplication

An integrated co-processor architecture for a smartcard

“…As a second example, we look at modulo reduction methods, and in particular would want to discuss Montgomery's method [4] vs. the relaxed residuum method, as described in [5]. We have chosen the latter method for modulo reduction, where modulo reduction is achieved by two multiplications with constants, and one addition.…”

“…A very rough sketch of the modulo multiplication algorithm using the relaxed residuum method [5] which computes E=A·B mod N+e·N (with ev{0,1,2}) is given as follows: (1) C=A·B (2) D=C high ·M1 (3) E=D high ·NegN+C low M1 as well as NegN are constants depending on the modulus N. Computation of these constants is either trivial such as for NegN, or done by the co-processor. In the following C-like pseudo-code, function 'step(i)' just performs 'i' clock cycles.…”

Application of symmetric redundant residues for fast and reliable arithmetic

“…As evident from Table I and discussion of the examples it contains, redundant residue sets have been applied in an ad hoc fashion as tools for speeding up or simplifying circuit realizations 3,8,11,12,16,19 . Only very recently have these been explicitly recognized as redundant residues and, thus, received a unified treatment 14 .…”

“…We are investigating the possibility of a more direct approach, but for now we advocate the use of a method due to Posch and Posch 19,20 which performs reduction into an (h + 2)-bit TRU pseudoresidue (Table I) using a pair of short multiplications by constants. Let this TRU pseudoresidue be (Y h+1 .…”

The Chinese Remainder Theorem and its application in a high-speed RSA crypto chip

“…Although the intermediate results are not always fully reduced, continuing the exponentiation with an incomplete reduced intermediate result does not cause an error bigger than once the modulus. An exact proof with detailed error estimation can be found in [Dhe98] or [PP89]. If a final modular reduction is necessary after the modular exponentiation has finished, it can be performed by adding AE AE to the result.…”