stochastic bitstreams so as to achieve a trade-off between precision and bitstream length. The following sections show how SC can be applied to design invertible logic on the current CMOS technology, based on Boltzmann machines [78], and to homomorphic encryption. D. Hyperdimensional Representation and Arithmetic The circuits of the brain are composed of billions of neurons, with each neuron typically connected to up to 10, 000 other neurons. This situation leads to more than 100 trillion modifiable synapses [69]. High-dimensional modeling of neural circuits not only supports artificial neural networks but also inspires parallel distributed processing. It explores the properties of high-dimensional spaces, with applications in, for example, classification, pattern identification and discrimination. HDC, inspired by the brain-like computing paradigm, is supported by random high-dimensional vectors [70]. It looks at computing with very wide words, represented by high-dimensional binary vectors, which are typically called hyperdimensional vectors, with a dimensionality on the order of thousands of bits. It is an alternative to Support Vector Machines (SVMs) and CNN-based approaches for supervised classification. With Associative Memory (AM), a pattern X can be stored using another pattern A as the address, and X can be retrieved later from the memory address A. However, X can also be retrieved by addressing the memory with a pattern Al similar to A. The high number of bits in HDC are not used to improve the precision of the information to represent, as it would be difficult to give an exact meaning to a 10,000-bit vector in the same way a traditional computing model does. Instead, they ensure robustness and randomness by i) being tolerant to errors and component failure, which come from redundancy in the representation (many patterns mean the same thing and thus are considered equivalent) and ii) having highly structured information, as in the brain, to deal with the arbitrariness of the neural code. HDC starts with vectors drawn randomly from hyperspace and uses AM to store and access them. A space of n =10,000-dimensional vectors and independent and identically distributed (i.i.d.) components drawn from a normal distribution with mean n = 0 can be considered as a typical example. Points in the space can be viewed as corners of a 10,000-dimensional unit hypercube, for which the distance d ^ X, Y h between two vectors X, Y is expressed using the Hamming metric. It corresponds to the length of the shortest path between the corner points along the edges (the distance is often normalized 20 IEEE CIRCUITS AND SYSTEMS MAGAZINE to the number of dimensions so that the maximum distance, when the values of all bits differ, is 1). Assuming that the distance ^k h from any point of the space to a randomly drawn point follows a binomial distribution, with p ^0 h = 0.5 and p ^1 h = 0.5: f ^kh = c 10, 000 10, 000! m 0.5 k 0.5 10,000 - k = 0.5 10,000 k k! ^10, 000 - k h ! (32) with mean n = 10, 000 # 0.5 = 5, 000 and variance VAR = 10, 000 # 0.5 # 0.5 = 2, 500 (the standard deviation v = 50). This means that if vectors are randomly selected, they differ by approximately 5,000 bits, with a normalized distance of 0.5. These vectors are therefore considered " unrelated " [70]. Although it is intuitive that half the space has a normalized distance from a point of less than 0.5, this statement is not true if we observe that it is only less than a thousand-millionth closer than 0.47 and another thousand-millionth farther than 0.53. This distribution provides robustness to the hyperdimensional space, given that " nearly " all of the space is concentrated around the 5,000 mean distance (0.5). Hence, a 10,000-bit vector representing an entity may see a large number of bits, e.g., a third, changing their values, by errors or noise, and the resulting vector still identifies the correct one because it is far from any " unrelated " vector. The vector representation of patterns enables the use of the body of knowledge on vector and matrix algebra to implement hyperdimensional arithmetic. For example, the componentwise addition of a set of vectors results in a vector with the same dimensionality that may represent that set. The sum-vector values can be normalized to yield a mean vector. Moreover, a binary vector can be obtained by applying a threshold. If there are no duplicated elements in a set, the sum-vector is a possible representative of the set that makes up the sum [70]. Another important arithmetic operation in HDC is vector multiplication, which, as for SC with BR, can be implemented with the bitwise logic XNOR operator (see Fig. 8(c)). Reverting the order of the BR from ^1, -1 h to ^0, 1 h, the ordinary multiplication of binary vectors X and Y can be implemented by the bitwise eXclusive-OR (XOR). It is easy to show that XOR is commutative and that each vector is its own inverse ^ X ) X = 0 h, where 0 is the unit vector ^ X ) 0 = X h . Multiplication can be considered as a mapping of points in the space: multiplying vector X by M maps the vector into a new vector X M as far from X as the number of 1s in M (represented by < M < ) (33). If M is a random vector, with half of the bits taking, on average, a value 1, X M is " unrelated " to X, and it can be said that multiplication randomizes it. d^ X M , X h = < X M ) X < = < M ) X ) X < = < M < (33) FIRST QUARTER 2021

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