Modular Hardware Architecture for Somewhat Homomorphic Function Evaluation

This post is based on the paper https://eprint.iacr.org/2015/337

This paper proposes an instruction-set hardware architecture for all building blocks required in polynomial ring based fully homomorphic schemes and uses the hardware architecture to instantiate the somewhat homomorphic encryption scheme YASHE on a large Virtex 7 FPGA. The building blocks present in the architecture are sufficiently generic to allow implementation of other homomorphic schemes based on RLWE or NTRU.

System Setup

  

The computations are performed in a modular polynomial rings of the form R = Z[x]/(f(x)) where f(x) is a monic irreducible polynomial Φ_d(x) of degree n. To utilize single instruction multiple data (SIMD) operations f(x) is a cyclotomic polynomial. The plaintext space is R₂.

The used parameter supports homomorphic evaluations of SIMON-64/128 (having a multiplicative depth of 44); in particular d = 65535 (and thus the degree of f(x) is n= 2¹⁵), log₂(q) = 1228 and the discrete Gaussian distribution with parameter is 8. The chosen parameter set meets at least 128-bit security and SIMD in 2048 slots.

High-level optimizations

Two main operations in the YASHE scheme are the YASHE.Add for homomorphic addition and the YASHE.Mult for homomorphic multiplication. While YASHE.Add is relatively simple, YASHE.Mult is very costly since one first needs to compute c₁ · c₂ over the integers, then scale the result by t/q and round, before mapping back into the ring R_q. For the polynomial multiplication of such a large degree n = 2¹⁵,  an NTT is used over other polynomial multiplication algorithms thanks to its almost linear time complexity. Moreover to tackle the problem of long integer arithmetic CRT is used to split the coefficients into 30-bit small coefficients. Application of the CRT helps to achieve parallelization and to utilize the small DSP multipliers available in FPGAs.

High-level Architecture

The instruction-set architecture shown in the figure has three main components: a computer, a HE-coprocessor and an external memory. The HE-coprocessor is implemented on an FPGA and supports NTT of polynomials, coefficient wise additions, subtractions and multiplications, computation of the residues using the CRT, computation of the coefficients modulo a large modulus from the CRT-residues, and scaling of the coefficients. The computer works in master-mode and instructs the HE-coprocessor to perform the required operations. Since the FPGA has a limited amount of internal memory in the form of block RAMs, only few residue polynomials can be kept during a computation. A large external memory is used to store the set of cipher-text polynomials.

The HE-coprocessor has a set of parallel processors, where each parallel processor is composed of several cores. The parallel processors are independent of each other and supports a high degree of parallelism and scalability. Since the architecture is very large, special care is taken to reduce the routing complexity. For example, a routing-friendly parallel NTT algorithm is used in the HE-coprocessor.    

Experimental Results

The designed HE-coprocessor with 8 parallel processors, each processor having 16 parallel cores, 16 small-CRT units and 2 large-CRT units was compiled for the Xilinx Virtex-7 XC7V1140T-2 FPGA, the largest device of the Virtex-7 FPGA family. The architecture consumes 23 % of the available registers, 50 % of LUTs, 53 % of DSP slices, and 38 % of block RAMs. This implementation evaluates SIMON-64/128 in approximately 157.7s (at 143MHz) and it processes 2048 cipher-texts at once giving a relative time of only 77ms per block. This is 26.6 times faster than the leading software implementation on a 4-core Intel Core-i7 processor running at 3.4GHz. The timing values do not consider the overhead of data transfer between the external memory and the HE-coprocessor.

 Future Work

1. The architecture does not implement the required interface between the external memory and the FPGA-based HE-coprocessor. The cost of data transfer is very important considering the vast amount of data exchange during various operations; and hence the interface should be properly designed. It should be noted that most of the memory access can be performed in parallel with the computation using a ping-pong scheme of two sets of block RAMs in the FPGA. The FPGA and the master-computer operate on these two sets alternatively between two consecutive instructions: when the FPGA operates on the first set, the master-computer operates on the second, and vice versa. Since only 38% of the available block RAMs are utilized in the proposed architecture, such a ping-pong styled memory access seems practical in order to reduce the cost of memory access.

2. The computation time can be reduced by performing low-level optimizations in the architecture. For example, it might be possible to increase the operating frequency beyond 200 MHz by optimizing the long critical paths that are present in the architecture. The cycle requirement of the NTT algorithm can also be reduced. Moreover, since the implemented architecture consumes only ~50% of the available resources, it might be possible to put more number of parallel processors in the FPGA. This will require area reduction of each processor and a better placement-routing optimization.

3. The results are obtained from a single FPGA. Since the HE-coprocessor is composed of independent parallel processors, several FPGAs can be used to scale down the computation time. 

Sujoy Sinha Roy