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Modular exponentiation
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Spectral Modular Exponentiation
"... We describe a new method to perform the modular exponentiation operation, i.e., the computation of c = m e mod n, wherec, m, e and n are large integers. The new method uses the discrete Fourier transform over a finite ring, and relies on new techniques to perform multiplication and reduction operati ..."
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Cited by 4 (3 self)
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We describe a new method to perform the modular exponentiation operation, i.e., the computation of c = m e mod n, wherec, m, e and n are large integers. The new method uses the discrete Fourier transform over a finite ring, and relies on new techniques to perform multiplication and reduction
An algorithm for modular exponentiation
 Information Processing Letters
, 1998
"... A practical technique for improving the performance of modular exponentiations (ME) is described. The complexity of the ME algorithm is O modular multiplications (MMs), where n is the length of the exponent, requiring an O ( n 2) precomputed lookup table size with very small constant of proportiona ..."
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Cited by 31 (9 self)
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A practical technique for improving the performance of modular exponentiations (ME) is described. The complexity of the ME algorithm is O modular multiplications (MMs), where n is the length of the exponent, requiring an O ( n 2) precomputed lookup table size with very small constant
An Algorithm for Modular Exponentiation
"... A practical technique for improving the performance of modular exponentiations (ME) is described. The complexity of the ME algorithm is O n ⎛⎝logn⎠ modular multiplications (MMs), where n is the length of the exponent, requiring an O(n 2) precomputed lookup table size with very small const ..."
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A practical technique for improving the performance of modular exponentiations (ME) is described. The complexity of the ME algorithm is O n ⎛⎝logn⎠ modular multiplications (MMs), where n is the length of the exponent, requiring an O(n 2) precomputed lookup table size with very small
FPGA Implementation of Modular Exponentiation
"... Abstract. An e cient implementations of the main building block in the RSA cryptographic scheme is achieved by mapping a bitlevel systolic array for modular exponentiation onto Xilinx FPGAs. One XC6000 chip, or 4 Kgates accommodates 132bit long integers. 16 Kgates is required for modular exponenti ..."
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Abstract. An e cient implementations of the main building block in the RSA cryptographic scheme is achieved by mapping a bitlevel systolic array for modular exponentiation onto Xilinx FPGAs. One XC6000 chip, or 4 Kgates accommodates 132bit long integers. 16 Kgates is required for modular
Fast batch verification for modular exponentiation and digital signatures
, 1998
"... Abstract Many tasks in cryptography (e.g., digital signature verification) call for verification of a basicoperation like modular exponentiation in some group: given ( g, x, y) check that gx = y. Thisis typically done by recomputing gx and checking we get y. We would like to do it differently,and f ..."
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Cited by 143 (2 self)
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Abstract Many tasks in cryptography (e.g., digital signature verification) call for verification of a basicoperation like modular exponentiation in some group: given ( g, x, y) check that gx = y. Thisis typically done by recomputing gx and checking we get y. We would like to do it differently
Fast Modular Exponentiation
"... The wellknown binary method computes C = M E #mod N# using an average number of 1:5#n , 1# multiplications, where n is the number of bits in the binary expansion of E. When the exponent is recoded using the canonical bit recoding technique then the average number of multiplications can be reduc ..."
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Cited by 3 (0 self)
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The wellknown binary method computes C = M E #mod N# using an average number of 1:5#n , 1# multiplications, where n is the number of bits in the binary expansion of E. When the exponent is recoded using the canonical bit recoding technique then the average number of multiplications can be reduced to 1:33#n , 1#. We show that a further reduction is achieved if the bits of the exponent are scanned at d#1 bits at a time: for n =2 10 , for example, the average number of multiplications becomes 1:212#n , 1# with d =5.Furthermore, given any ##0, the computation can be done using an average of #1 + ###n , 1# multiplications for large n by taking d = d 1 # e.
Modular exponentiation using parallel multipliers
 Proceedings of the 2003 IEEE International Conference on Field Programmable Technology (FPT
, 2003
"... A field programmable gate array (FPGA) semisystolic implementation of a modular exponentiation unit, suitable for use in implementing the RSA public key cryptosystem is presented. The design is carefully matched with features of the FPGA architecture, utilizing embedded 18×18bit multipliers on the ..."
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Cited by 9 (0 self)
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A field programmable gate array (FPGA) semisystolic implementation of a modular exponentiation unit, suitable for use in implementing the RSA public key cryptosystem is presented. The design is carefully matched with features of the FPGA architecture, utilizing embedded 18×18bit multipliers
Montgomery Modular Exponentiation on Reconfigurable Hardware
, 1999
"... It is widely recognized that security issues will play a crucial role in the majority of future computer and communication systems. Central tools for achieving system security are cryptographic algorithms. For performance as well as for physical security reasons, it is often advantageous to realize ..."
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Cited by 43 (3 self)
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perform modular exponentiation with very long integers. This operation is at the heart of many practical publickey algorithms such as RSA and discrete logarithm schemes. We combine the Montgomery modular multiplication algorithm with a new systolic array design, which is capable of processing a variable
Efficient Software Implementations of Modular Exponentiation
"... Abstract. RSA computations have a significant effect on the workloads of SSL/TLS servers, and therefore their software implementations on general purpose processors are an important target for optimization. We concentrate here on 512bit modular exponentiation, used for 1024bit RSA. We propose opti ..."
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Cited by 1 (1 self)
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Abstract. RSA computations have a significant effect on the workloads of SSL/TLS servers, and therefore their software implementations on general purpose processors are an important target for optimization. We concentrate here on 512bit modular exponentiation, used for 1024bit RSA. We propose
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