[Tom-Finke]

Implements #50

See https://eprint.iacr.org/2018/1041.pdf

[codecov]

Codecov Report

❌ Patch coverage is 90.75630% with 11 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/linear_algebra.jl	93.54%	6 Missing ⚠️
src/types.jl	60.00%	4 Missing ⚠️
src/openfhe.jl	87.50%	1 Missing ⚠️

Files with missing lines	Coverage Δ
src/SecureArithmetic.jl	100.00% <ø> (ø)
src/arithmetic.jl	100.00% <100.00%> (ø)
src/openfhe.jl	96.06% <87.50%> (+0.01%)	⬆️
src/types.jl	87.50% <60.00%> (-12.50%)	⬇️
src/linear_algebra.jl	93.54% <93.54%> (ø)

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[Tom-Finke]

Mat Mul unit test failing with InexactError: Int64(2.321928094887362), originating in

SecureArithmetic.jl/src/linear_algebra.jl

Line 42 in a8dab40

for k in 0:Int(log2(d÷l))-1

The algorithm assumes that d/l is a power of 2, i.e. that the smaller dimension fits a power of 2 times into the bigger dimension. They mention this only in Section 5 Implementation of Homomorphic Matrix Operations:

For simplicity, we assume that d is a power-of-two integer. In general, we can pad zeros at the end of each row to set d as a power of two

Options:

• Pad the matrix with zeros, so that the power of two assumption is satisfied. This is what openfhe_numpy does, see https://github.com/openfheorg/openfhe-numpy/blob/45857c644bc75321d54f6cdd49cc63f3f948b224/openfhe_numpy/utils/packing.py#L123-L127.
• Adjust the repeated doubling algorithm so that it can work with non-power-of-two $d/l$

Since we only have to pad the smaller dimension $l$, which is the number of columns, this padding is cheap even for for encrypted matrices in column order. We simply add additional columns by resizing the 1D array representing the matrix.
In the worst case, the the only power of two that divides $d$ is 1 and we would end up calculating a $d \times d * d \times d$ matrix product.
A way around this would be to also pad the left matrix such that $d$ is a power of two, but this is expensive on an encrypted matrix, since we would have to pad both rows and columns, which entails expensive masking and rotations.

[github-actions]

Benchmark Results: SecureArithmetic Matrix Multiplication

Warning: Benchmark run exited with status 1.

Summary

FHE Params	Matrix Size	Mode	Wall Time (ms)	Alloc	#Mult	#Rotate	Result Level
d15_n2048	4×4	secure_secure	180.7	623.6 KiB	57	36	4
d15_n2048	4×4	secure_plain	71.4	282.7 KiB	20	12	3
d15_n2048	4×4	plain_secure	121.2	446.4 KiB	41	24	4
d15_n2048	4×4 * 4×1	secure_secure	120.5	450.9 KiB	39	26	7
d15_n2048	4×4 * 4×1	secure_plain	55.7	227.4 KiB	15	10	4
d15_n2048	4×4 * 4×1	plain_secure	93.7	370.2 KiB	29	20	7
d15_n2048	8×8	secure_secure	415.0	1.5 MiB	129	84	4
d15_n2048	8×8	secure_plain	158.5	928.3 KiB	44	28	3
d15_n2048	8×8	plain_secure	277.4	1.1 MiB	93	56	4
d15_n2048	8×8 * 8×1	secure_secure	237.8	962.1 KiB	79	54	9
d15_n2048	8×8 * 8×1	secure_plain	105.5	635.8 KiB	29	20	5
d15_n2048	8×8 * 8×1	plain_secure	176.8	838.3 KiB	57	40	9
d15_n2048	16×16	secure_secure	857.1	3.5 MiB	273	180	4
d15_n2048	16×16	secure_plain	330.1	4.2 MiB	92	60	3
d15_n2048	16×16	plain_secure	575.5	3.4 MiB	197	120	4
d15_n2048	16×16 * 16×1	secure_secure	405.5	2.1 MiB	155	106	11
d15_n2048	16×16 * 16×1	secure_plain	197.5	2.6 MiB	55	38	6
d15_n2048	16×16 * 16×1	plain_secure	289.9	2.3 MiB	109	76	11
d15_n2048	32×32	secure_secure	1021.1	9.9 MiB	189	186	2
d15_n2048	32×32	secure_plain	426.9	27.3 MiB	64	62	1
d15_n2048	32×32	plain_secure	684.2	15.4 MiB	157	124	2
d15_n2048	32×32 * 32×1	secure_secure	526.2	5.7 MiB	97	103	2
d15_n2048	32×32 * 32×1	secure_plain	232.3	16.1 MiB	33	36	1
d15_n2048	32×32 * 32×1	plain_secure	406.6	10.7 MiB	65	72	2
d10_128c	4×4	secure_secure	2164.2	9.0 MiB	57	36	4
d10_128c	4×4	secure_plain	859.3	3.1 MiB	20	12	3
d10_128c	4×4	plain_secure	1458.2	6.1 MiB	41	24	4
d10_128c	4×4 * 4×1	secure_secure	1398.1	6.5 MiB	39	26	7
d10_128c	4×4 * 4×1	secure_plain	660.9	2.6 MiB	15	10	4
d10_128c	4×4 * 4×1	plain_secure	1105.5	5.0 MiB	29	20	7
d10_128c	8×8	secure_secure	4935.2	21.1 MiB	129	84	4
d10_128c	8×8	secure_plain	1957.0	7.5 MiB	44	28	3
d10_128c	8×8	plain_secure	3331.9	14.3 MiB	93	56	4
d10_128c	8×8 * 8×1	secure_secure	2548.5	13.6 MiB	79	54	9
d10_128c	8×8 * 8×1	secure_plain	1264.0	5.3 MiB	29	20	5
d10_128c	8×8 * 8×1	plain_secure	1911.0	10.2 MiB	57	40	9
d10_128c	16×16	secure_secure	10556.1	45.7 MiB	273	180	4
d10_128c	16×16	secure_plain	4223.4	18.3 MiB	92	60	3
d10_128c	16×16	plain_secure	7150.4	31.6 MiB	197	120	4
d10_128c	16×16 * 16×1	secure_plain	2488.7	11.5 MiB	55	38	6
d10_128c	32×32 * 32×1	secure_plain	4775.4	33.0 MiB	105	72	7

Runs Without a Result

4 of 45 runs did not produce a result.

FHE Params	Matrix Size	Mode	Error
d10_128c	16×16 * 16×1	secure_secure	SetLevel(): The current multiplicative depth [10] is insufficient; increase it.
d10_128c	16×16 * 16×1	plain_secure	SetLevel(): The current multiplicative depth [10] is insufficient; increase it.
d10_128c	32×32 * 32×1	secure_secure	SetLevel(): The current multiplicative depth [10] is insufficient; increase it.
d10_128c	32×32 * 32×1	plain_secure	SetLevel(): The current multiplicative depth [10] is insufficient; increase it.

Full results (all columns)

fhe_params,matrix_size,mode,status,error,wall_time_s,alloc_bytes,eval_add_count,eval_add_time_s,eval_sub_count,eval_sub_time_s,eval_mult_count,eval_mult_time_s,eval_rotate_count,eval_rotate_time_s,eval_negate_count,eval_negate_time_s,resize_count,resize_time_s,result_level
d15_n2048,4×4,secure_secure,ok,,0.180741597,638576,35,0.003480401,0,0.0,57,0.044468673,36,0.10870433,0,0.0,0,0.0,4
d15_n2048,4×4,secure_plain,ok,,0.071369414,289488,13,0.001840383,0,0.0,20,0.01201881,12,0.036764218,0,0.0,0,0.0,3
d15_n2048,4×4,plain_secure,ok,,0.121215752,457152,25,0.002329129,0,0.0,41,0.025064639,24,0.072361363,0,0.0,0,0.0,4
d15_n2048,4×4 * 4×1,secure_secure,ok,,0.120485046,461760,28,0.004230431,0,0.0,39,0.027586695,26,0.071043298,0,0.0,2,0.000103125,7
d15_n2048,4×4 * 4×1,secure_plain,ok,,0.055730187,232848,11,0.002187073,0,0.0,15,0.00749446,10,0.029910873,0,0.0,2,0.000414617,4
d15_n2048,4×4 * 4×1,plain_secure,ok,,0.093749079,379072,21,0.003220044,0,0.0,29,0.021406869,20,0.051725108,0,0.0,2,9.6877e-5,7
d15_n2048,8×8,secure_secure,ok,,0.415021956,1529648,79,0.005507443,0,0.0,129,0.099143287,84,0.253719082,0,0.0,0,0.0,4
d15_n2048,8×8,secure_plain,ok,,0.158529913,950608,29,0.002520587,0,0.0,44,0.026185425,28,0.083655119,0,0.0,0,0.0,3
d15_n2048,8×8,plain_secure,ok,,0.277373597,1188608,57,0.003711997,0,0.0,93,0.056755064,56,0.168521956,0,0.0,0,0.0,4
d15_n2048,8×8 * 8×1,secure_secure,ok,,0.237771523,985184,56,0.006245954,0,0.0,79,0.051749277,54,0.143453153,0,0.0,2,0.000125259,9
d15_n2048,8×8 * 8×1,secure_plain,ok,,0.105537904,651088,21,0.003069196,0,0.0,29,0.013312289,20,0.057770808,0,0.0,2,0.000387396,5
d15_n2048,8×8 * 8×1,plain_secure,ok,,0.176835784,858400,41,0.004483763,0,0.0,57,0.039859617,40,0.098281573,0,0.0,2,0.000104078,9
d15_n2048,16×16,secure_secure,ok,,0.857067291,3633944,167,0.008802296,0,0.0,273,0.2030537,180,0.527795893,0,0.0,0,0.0,4
d15_n2048,16×16,secure_plain,ok,,0.330119827,4423440,61,0.003550587,0,0.0,92,0.054334271,60,0.174854609,0,0.0,0,0.0,3
d15_n2048,16×16,plain_secure,ok,,0.575517871,3613320,121,0.006029529,0,0.0,197,0.118572789,120,0.349674372,0,0.0,0,0.0,4
d15_n2048,16×16 * 16×1,secure_secure,ok,,0.405514881,2162224,108,0.00788021,0,0.0,155,0.090155761,106,0.237506303,0,0.0,2,0.000170238,11
d15_n2048,16×16 * 16×1,secure_plain,ok,,0.197549804,2734488,39,0.004228329,0,0.0,55,0.02337621,38,0.108989141,0,0.0,2,0.000424953,6
d15_n2048,16×16 * 16×1,plain_secure,ok,,0.289943005,2461760,77,0.005802838,0,0.0,109,0.071071508,76,0.146963086,0,0.0,2,0.000153952,11
d15_n2048,32×32,secure_secure,ok,,1.021148639,10339000,157,0.007861391,0,0.0,189,0.207471223,186,0.568927325,0,0.0,0,0.0,2
d15_n2048,32×32,secure_plain,ok,,0.426857902,28658960,63,0.00287879,0,0.0,64,0.032568035,62,0.1901612,0,0.0,0,0.0,1
d15_n2048,32×32,plain_secure,ok,,0.684217428,16103976,125,0.00600447,0,0.0,157,0.092762325,124,0.378004954,0,0.0,0,0.0,2
d15_n2048,32×32 * 32×1,secure_secure,ok,,0.52615251,5972320,105,0.00498404,0,0.0,97,0.065097199,103,0.320474668,0,0.0,2,0.000156696,2
d15_n2048,32×32 * 32×1,secure_plain,ok,,0.232302645,16894528,37,0.001787262,0,0.0,33,0.004366755,36,0.108861112,0,0.0,2,0.000405223,1
d15_n2048,32×32 * 32×1,plain_secure,ok,,0.406610142,11244016,73,0.0031799,0,0.0,65,0.057389073,72,0.218591504,0,0.0,2,0.000161535,2
d10_128c,4×4,secure_secure,ok,,2.164211516,9485936,35,0.040131469,0,0.0,57,0.54283922,36,1.249257866,0,0.0,0,0.0,4
d10_128c,4×4,secure_plain,ok,,0.859261952,3238608,13,0.020617569,0,0.0,20,0.146636739,12,0.413570652,0,0.0,0,0.0,3
d10_128c,4×4,plain_secure,ok,,1.458176403,6355392,25,0.026419424,0,0.0,41,0.30819818,24,0.832061505,0,0.0,0,0.0,4
d10_128c,4×4 * 4×1,secure_secure,ok,,1.398132727,6851520,28,0.04810795,0,0.0,39,0.324841863,26,0.787540105,0,0.0,2,0.000392814,7
d10_128c,4×4 * 4×1,secure_plain,ok,,0.660857127,2690448,11,0.025722313,0,0.0,15,0.091594392,10,0.325795078,0,0.0,2,0.003896478,4
d10_128c,4×4 * 4×1,plain_secure,ok,,1.105549573,5294272,21,0.036355021,0,0.0,29,0.255985256,20,0.578892086,0,0.0,2,0.000396658,7
d10_128c,8×8,secure_secure,ok,,4.935176355,22173488,79,0.061906284,0,0.0,129,1.189385473,84,2.903323483,0,0.0,0,0.0,4
d10_128c,8×8,secure_plain,ok,,1.957000529,7831888,29,0.027760709,0,0.0,44,0.325688429,28,0.952685759,0,0.0,0,0.0,3
d10_128c,8×8,plain_secure,ok,,3.33186089,14951168,57,0.041496372,0,0.0,93,0.701078348,56,1.915700951,0,0.0,0,0.0,4
d10_128c,8×8 * 8×1,secure_secure,ok,,2.548519902,14256224,56,0.063689677,0,0.0,79,0.577314413,54,1.404289678,0,0.0,2,0.000354786,9
d10_128c,8×8 * 8×1,secure_plain,ok,,1.263985546,5566288,21,0.034161374,0,0.0,29,0.161842167,20,0.636961778,0,0.0,2,0.003809056,5
d10_128c,8×8 * 8×1,plain_secure,ok,,1.910990937,10688800,41,0.047471204,0,0.0,57,0.461214932,40,0.918620787,0,0.0,2,0.000341526,9
d10_128c,16×16,secure_secure,ok,,10.556072734,47918968,167,0.105315549,0,0.0,273,2.500224738,180,6.2264616,0,0.0,0,0.0,4
d10_128c,16×16,secure_plain,ok,,4.223417948,19169040,61,0.043932063,0,0.0,92,0.695028962,60,2.060820962,0,0.0,0,0.0,3
d10_128c,16×16,plain_secure,ok,,7.150380804,33104520,121,0.071180113,0,0.0,197,1.49732676,120,4.105489931,0,0.0,0,0.0,4
d10_128c,16×16 * 16×1,secure_secure,error,SetLevel(): The current multiplicative depth [10] is insufficient; increase it.,0.0,0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,-1
d10_128c,16×16 * 16×1,secure_plain,ok,,2.488749176,12073368,39,0.049192957,0,0.0,55,0.301773462,38,1.261234313,0,0.0,2,0.003898589,6
d10_128c,16×16 * 16×1,plain_secure,error,SetLevel(): The current multiplicative depth [10] is insufficient; increase it.,0.0,0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,-1
d10_128c,32×32 * 32×1,secure_secure,error,SetLevel(): The current multiplicative depth [10] is insufficient; increase it.,0.0,0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,-1
d10_128c,32×32 * 32×1,secure_plain,ok,,4.775395483,34603920,73,0.071106272,0,0.0,105,0.529418345,72,2.441475233,0,0.0,2,0.004060455,7
d10_128c,32×32 * 32×1,plain_secure,error,SetLevel(): The current multiplicative depth [10] is insufficient; increase it.,0.0,0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,0,0.0,-1

Benchmarks run with OMP_NUM_THREADS=1 on ubuntu-latest.