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generator and a high-accuracy subsampling (pooling)
block, both in AQFP, the implemented CNN achieves a
four-orders-of-magnitude higher energy efficiency than
the CMOS-based implementation while maintaining a
96% accuracy for the MNIST dataset.

SC is another example of nonconventional arithmetic applied to design CNNs [130], [131], [133]-[137]. In
[130], the polling layers are implemented with an improved version of the SC max unit based on the FSMs
presented in Section II-C [132]. The SC-based ReLU,
SReLU, is also implemented through an FSM that mimics the three key operations of the artificial neuron:
integration, output generation, and decrement. To improve the signal-to-noise ratio, weight normalization
and upscaling are performed, while the overhead of
binary-to-stochastic conversion is reduced by sharing
number generators. In [131], the linear rectifier used for
the activation function is realized through an FSM, and
pooling is achieved by simply aggregating streams to
compute either the average or the max function. Compared with the existing CNN deterministic architectures, the stochastic-based architecture can achieve,
at the cost of a slight degradation in computing accuracy, significantly higher energy efficiency at low cost.
The SC-DCNN is another design and optimization
framework of SC-based CNNs, which adopts a bottomup approach [133]. Connecting different basic function
blocks with joint optimization and applying efficient
weight storage methods, the area and power (energy)
consumption are reduced while maintaining high network accuracy. Integral SC representations, supported
on integer stochastic streams obtained by the summation of two or more binary stochastic streams, have also
been proposed for designing efficient stochastic-based
DNN architectures [134]. Hybrid stochastic-binary neural networks, particularly for near-sensor computing,
have been proposed [135]. Combining signal acquisition and SC in the first ANN layer, binary arithmetic is
applied to the remaining layers to avoid compounding
accuracy losses. It is also shown that retraining these
remaining ANN layers can compensate for precision
loss introduced by SC. Improved SC-based multipliers
and their vector extension have also been proposed
for application to DNNs [136]. Moreover, the SC-based
convolution can be improved in CNNs by producing
the result of the first multiplication and performing the
subsequent multiplications with the differences of the
successive weights [137].
An SC-based DL framework was implemented with
AQFP technology [79], a superconducting logic family
that achieves high energy efficiency, as described in Section III-A. Through an ultraefficient stochastic number

B. Cryptography
Difficult mathematical problems are used to ensure the
security of public-key cryptographic systems. These
systems are designed so that the derivation of the private key from the public key is as difficult as computing the solution to problems such as the factorization
of large integers (e.g., RSA [138]) or the discrete logarithm (e.g., Elliptic Curve (EC) cryptography [170]). The
number of steps required to solve these problems is a
function of the length of the input; thus, the dimension
of keys is enlarged to make the systems secure (e.g., private keys larger than 512 and 1024 bits).
Modular reduction (modulo N ) is a fundamental arithmetic operation in public cryptography for mapping results back to the set of representatives " 0, f, N - 1 , .
The Montgomery algorithm for modular multiplication
replaces the costly reduction for a general value N with
a reduction for L, typically a power of 2 [139]. However,
since in RNS, the reduction modulo power of 2 is inefficient, an RNS variation of the Montgomery modular multiplication algorithm relies on the BE method presented
in II-B by choosing L = M 1 and GCD ^ M 1, M 2 h = 1 [140],
[141]. Similarly, the reduction modulo L corresponds to
the reduction modulo of the dynamic range M 1, which
automatically relies on the modulo operation in each
one of the RNS channels of S 1 . Moreover, GN HM1 and
G M 1HM2 exist, which are required for the precomputed
constants used in the modified Montgomery modular
multiplication algorithm [142].
A survey on the usage of RNS to support these classic public-key algorithms can be found in [142]. Multimoduli RNS architectures have also been proposed to
obfuscate secure information. The random selection of
moduli is proposed for each key bit operation, which
makes it difficult to obtain the secret key, preventing
power analysis while still providing all the benefits of
RNS [143].
However, if quantum computers become available, this
type of public-key cryptography, based on the factorization of large integers or the discrete logarithm, will no longer be secure [144]. Therefore, there is a need to develop
arithmetic techniques for alternate systems, usually designated as postquantum cryptography [145]. Quite recently,
another survey was published on the application of RNS
and SC to this type of emergent cryptography [147]. We
present two representative examples of the application
of RNS and SC to the Goldreich-Goldwasser-Halevi (GGH)

pu 2
log 2 ^conv h = log 2 ^2 pu 1 + 2 pu 2h = log 2 ;2 pu 1 c 1 + 2 pu 1 mE
2
pu 2
2
= pu 1 + log c 1 + pu 1 m
2
. pu 1 + BitShift ^1, pu 2 - pu 1 h

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