AbstractPublic-key crypto-algorithms are widely employed for
authentication, signatures, secret-key generation and access
control. The new range of public-key sizes for RSA and
DSA has gone up to 1024 bits and beyond. The elliptic-curve
key range is from 162 bits to 256 bits. Many varied software
and hardware algorithms are being developed for implementation
for smart-card crypto-coprocessors and for public-key
infrastructure. We begin with an algorithm from Aryabhatiya
for solving the indeterminate equation a · x + c =
b · y of degree one (also known as the Diophantine
equation) and its extension to solve the system of two residues
X mod mi = Xi (for i = 1,2). This contribution known
as the Aryabhatiya algorithm (AA) is very profound in the sense that
the problem of two congruences was solved with just one modular
inverse operation and a modular reduction to a smaller modulus
than the compound modulus. We extend AA to any set of t residues, and this is stated as
the Aryabhata remainder theorem (ART). An iterative algorithm is
also given to solve for t moduli mi (i = 1, 2,... ,
t). The ART, which has much in common with the extended
Euclidean algorithm (EEA), Chinese remainder theorem (CRT) and
Garner's algorithm (GA), is shown to have a complexity
comparable to or better than that of the CRT and GA.
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