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Montgomery reduction apparatus
| Details |
Inventors: Shimbo, Atsushi; Kawamura, Shinichi;
Assignee: Kabushiki Kaisha Toshiba (Kawasaki, JP)
Primary Examiner: Smithers; Matthew
Assistant Examiner:
Attorney, Agent or Firm: Finnegan, Henderson, Farabow, Garrett & Dunner, L.L.P.
In a Montgomery reduction apparatus for receiving positive integers C and p and calculating D=C.multidot.R.sup.-1 mod p by using R defined as R=2.sup.n using an integer n falling within the range n.gtoreq.L with a bit length being represented by L when p is expressed in binary notation, this invention includes an (.alpha., .beta.) extraction section for calculating an integer pair (.alpha., .beta.) satisfying C=.alpha.R+.beta. on the basis of C and R, a multiplication section for obtaining .epsilon..beta. by multiplying .epsilon. satisfying R.sup.-1 =.epsilon.(mod p) and .beta. calculated by the (.alpha., .beta.) extraction section, an addition section for obtaining .alpha.+.epsilon..beta. by adding .alpha. calculated by the (.alpha., .beta.) extraction section and .epsilon..beta. calculated by the multiplication section, and a calculation section for obtaining a remainder D=.alpha.+.beta..multidot..epsilon.(mod p) which is congruent to .alpha.+.epsilon..beta. obtained by the addition section with respect to p as a modulus and is not more than p. |
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DETAILED DESCRIPTION OF THE INVENTION A Montgomery reduction apparatus and method of the present invention aim at more efficiently implementing Montgomery reduction as an algorithm for efficient remainder calculation.
More specifically, efficient arithmetic operation is implemented by setting a modulus p to a value suitable for an arithmetic expression in Montgomery reduction.
Montgomery reduction is an algorithm for executing remainder calculation in a remainder computation system using an odd integer as a modulus.
Although the modulus p need not be an odd number, it is assumed in this embodiment for the sake of descriptive convenience that the modulus p is an odd prime number.
When Montgomery reduction is to be used, a constant R must be obtained from the bit size of the modulus p.
The constant R needs to meet conditions that R>p and R and p are relatively prime.
In Montgomery reduction, the remainder and quotient must be obtained by using R.
To execute these processes efficiently, the size of R is generally set to a multiple of the word length of the calculation unit to be mounted.
More specifically, when the word length of the calculation unit is represented by w, R=(2.
sup.
w).
sup.
L (L is the minimum integer that satisfies R>p) is often set.
For the sake of simplicity, assume in the following description that a bit size n of p is a multiple of w, and R=2.
sup.
n.
Note that these conditions are assumed for the sake of descriptive convenience, but the present invention itself is not limited by them.
In Montgomery reduction, D=CR.
sup.
- mod p is calculated with respect to a product C of positive integers A and B that are smaller than the modulus p.
In this case, R.
sup.
- represents a multiplicative inverse element of R which satisfies R.
multidot.
R.
sup.
-1 =1 mod p.
Each embodiment of the present invention will be described below.
(First Embodiment of Present Invention) FIG.
1 is a view for explaining the principle of a Montgomery reduction calculation method in the first embodiment of the present invention
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