strongSwan

h1. Bimodal Lattice Signature Scheme (BLISS)

BLISS is a post-quantum signature scheme based on the CRYPTO 2013 paper "Lattice Signatures and Bimodal Gaussians":https://eprint.iacr.org/2013/383 by Léo Ducas, Alain Durmus, Tancrède Lepoint, and Vadim Lyubashevsky. Starting with the strongSwan [[5.2.2]] release we offer BLISS as an IKEv2 public key authentication method and added full BLISS key and certificate generation support to the strongSwan [[IpsecPki|pki]] tool.

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h2. BLISS Private Key Generation

strongSwan currently supports the BLISS-I, BLISS-III, and BLISS-IV schemes with a cryptographic strength of 128 bits, 160 bits and 192 bits, respectively. Using the [[IpsecPki|pki]] tool a private BLISS key can be generated as follows:

<pre>

pki --gen --type bliss --size 1 --debug 2 > cakey1.der

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 240 octets

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 240 octets

l2 norm of s1||s2: 771, Nk(S): 47150 (46479 max)

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 220 octets

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 240 octets

l2 norm of s1||s2: 771, Nk(S): 43332 (46479 max)

secret key generation succeeded after 2 trials

</pre>

When generating the private key consisting of the two polynomials *s1* and *s2*, the limit for the _Nk(S)_ metric must not be exceeded. This means that often several trials are needed in order to obtain a valid BLISS private key. With the command

<pre>

pki --print --type bliss-priv --in cakey1.der

private key with:

pubkey:    BLISS 128 bits strength

keyid:     d1:a3:fb:04:8d:1b:86:4f:fa:a7:d8:45:ec:e3:e3:ec:ef:7b:85:ca

subjkey:   e3:fc:6b:59:9a:ee:81:d5:10:3a:58:9f:e2:99:f7:7f:5c:3b:1c:96

</pre>

information on the BLISS private key is displayed.

Let's now generate a BLISS-IV key with 192 bit cryptographic strength in base64-encoded PEM format

<pre>

pki --gen --type bliss --size 4 --outform pem  > cakey4.pem

secret key generation succeeded after 6 trials

</pre>

The PEM key format is printable

<pre>

cat key4.pem

-----BEGIN BLISS PRIVATE KEY-----

MIIFGgYLKwYBBAGCoCoFAgQDggOBAEOoiWS7mISnnPjzFJu6REHq1REYuLfillD2

VmmCWuB1NcL6GLTRFzwCMCw8KBLpyZhEAs6QlelSWVxPuBGMuQzQsmm9A3NjrV4U

655KXTkuiTpZP00qsfKuqh6EofkQ+89YK6qZNnxAeJ/mB9Dlkq9ELPjigNlZfUkd

Ky2fBJkwdKLb75WermM3tOYts0X06j7M3WX8DdVsGgIrzC57shAiD9nyhrUNHB15

b9IymR84GW4BJofKVW2GJVeUyLnh8YP33OUx6F5aEqweSbi5dGtbDbr0WmK6LmNw

dKQyv+hickbXGTWifAEktjpTApYjBBB6TZgiAW2P4T3dMq2ciQUbhCl1xWUlWF+2

iZbfFrcMb4dVrWOYbQRfvURmCkvJWsXHiijK8E+pmCDCruQg7TuRlIdXCRhSZrzY

+pLcY7mKBfyCvrmYmmCMRQQXeTDnGI/9VhHJ6icK6Mqy2BwRVFj9FmSsAHmF9gkL

hcaPlsgpLqaoK41FcJHjMbJIjWKaHkFXMQ0K943cM0ivB3EqRG68AptqH1QxkIi6

haUuQL6Nsl/tXo4VwyiVpm1faRQW5Re9L6KbEhLgnT3JeAft0zOOlHwx5myBDAxa

s8LP9H/EyzpO4uyd1eHlqZvGEmlt9lhOikLwEohWDoZIpWFKrtfzciQMOugLq4m4

n+ueVo25rvq6MRwncj0FCwlt0nAhWeP8hQYTzhgFsBeheM4OaWVRhRPQmqYFrLRZ

grvkgGIQd2IDKhjqCI7gpOi/KRG5RbnyvO4zaqLNy16Lk4exZ2iin19YQpmU613j

EVLsMoRTQl2tE+aB0GJ0BpE3u0Aqnrp6ZhCJmK8CybfYAGhV5sly59Cds7QtIw8r

6pXl7Wd0q2sMFsUnqadcCwqoeOciqU+AwvQ+X2g4eilxV6D2TkLMMBUOYi5BqNdj

a7pJAnUUMyEYvDXhMUYnjGlK3RFKHFCzCalQN0s5JLRTpLnTTy70TtvMaDJAWCwG

OShSbNqr0zGNfnCsFjuppZ+5tQd7GRCgjL2uG0CDTIKEq5vmaH1d3FOldJX2uYYA

O6QOKIThuiH3C0OgAQoLGoArsmFymtBXHxPZSjtE5SR+1YVCr4UEdGlSt2efJoxm

eBaYki03CF2pSm7EDHxbEjDC9E3AeOfUW6Iq4dTGThjGNGnnBIbpv0mSdXFzWcZU

3rwQo51EA4HBAACSCPjwAOUCAOCCCOP/gCDzxzweEOByCAACeEP1wF+DxhwBiQTw

AP/0AAQFx/z+SACPgR+ASePwOAOgAAQBwDiAV0OOCADwB0eOgAeAOORwMAPzwOQS

CBuMAQOB+Px/x+AuCeRweeADzwACPyCP+RyQhgBzwOCBwSBwOB+SeQAQCOBwACAB

0eByABwQBwQAAeCBwgAAB/wACDwNyPgeQOPwAAR+OAR+AB+COAQCBx+QeB+R+B+Q

CADjxwACPiARvwOBwQDjggcY7EEgAEHjcYAcogEHgccDjgcDg8fgAnjgAgfkgEgc

DnjgAEbAn8ArnAhAAcAkAEgADkgAD8DgkDgkcDnkcE88jkjkgjgD9ccAg8cc8jjr

kcgkABED8gAD8H/n8gAAj8AgEHj8D8D89Dfg4DAgEEAAgAj8HgkgAj8HgcHodAEk

cn8DAL8AD/g//gcDkAEH/AAD/gcDnj8AkD8AcDgErkHjAAcA9AAAAcAcEEEgAAAg

AYLoAgAEHgkDgYccgbk=

-----END BLISS PRIVATE KEY-----

</pre>

At last let's generate a BLISS-III key with a cryptographic strength of 160 bits with the highest debug level enabled:

<pre>

pki --gen --type bliss --size 3 --debug 4 > cakey3.der

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 380 octets

mgf1 based on sha1 is seeded with 20 octets

mgf1 generated 380 octets

l2 norm of s1||s2: 1401, Nk(S): 125552 (128626 max)

secret key generation succeeded after 1 trial

   i   f   g     a     F     G     A

   0  -1   1 11932  6730 11344  6400

   1   1   0  2227  1206  9396  6244

   2  -1   2  4844   496   414  4411

   3  -1   0  2768  2813  1412  6619

   4   0   2  3583  2753  3520  9237

   5   0   2 10160  2434  9512  8688

   6   0   2  8157  9071 10775  8990

   7   0   0  3862  5091   211  3126

   8  -1   0  3045  2278  5799  8812

   9   1   0  9942  5685  3335   541

  10   0   0  8236  1637   526  5000

  11   0   0  8638     9  9539 10618

  12   0  -2 11526 11882  8890  8976

  13  -1   0 12180 11895  3538  5231

  14  -2   0  6332  4243 11062   243

  15   0   0  4083  4302  3400  4000

  16   0   2  4545  6031  2766  1708

  17   0   0  1495  4119  8792 11954

  18   1  -2  5664  9450  5151  6621

  19   0   0  3580  1963 11193  1552

  20  -1  -2  7090  5950 10318  8445

  21   0   0  5180  8190  7147 11145

  22   0  -2  8455 12226    27 10533

  23   0   0   810  4585  6578  3333

  24   1   0  5316  9595  2034  7088

  25   0   0 10072 11746 10425  9554

  26  -1   0  4544  5888  7751  8402

  27   0   0  9529 10638  5983  9509

  28   0   0  6832  8019  5519  1124

  29   1  -2  8900  2356  4475  4326

  30   0  -2  4438  7452  2418   406

  31  -1  -2   363  9949  6078  3369

  32   0   0  6032  9713 11653 12232

  33   0   0  1342 11748 11094  4727

  34  -1  -2   780  9506  2687  5713

  35   1   0  1114 11518  5003  1173

  36   1   0 11561  8458  9766  5255

  37   0   0   932  4680  7848 10211

  38   0   0  4748  4235  6832  9975

  39   1   2  6338  9116  1371  9287

  40   0   0  9216 11714 11657  4532

  41   1  -2  1100  6203  6951  9887

  42   1   0 11955  9307   124 11984

  43   0   0  6550  6220  9948 11200

  44   1   0 10183  7920  2231  2050

  45   0   0  5858 10736 11843  4851

  46   1   0  4402  6459  5976  5509

  47   1   2  3354  2643  9397 11716

  48   1  -2  9937  3908  1174 11478

  49   2   0 11688  9298 10680  1833

  50  -1   2  5348  4731 12240  5286

  51   0   0  4594  1469 10189  5043

  52   0   2  6324  1006  6445  3268

  53   0  -4  2137  2707  4158   569

  54  -1  -2   340  2232  4643  9852

  55  -1   0  1784  8290  9620  3129

  56   1   0  7203  5610 11341   749

  57   0   0  6651 12057 10851  5621

  58  -1   0   383  5516  9861  2272

  59   2   0 10893  8086  1452   140

  60   0   0  7921 10970  6955  9293

  61   1  -2  4243 10170  5305  9178

  62  -1   2  3565  2730  3858 11021

  63   0   0  5697  1308  7157  8076

  64   0  -2  4079  5666  9079  5400

  65   1   0  3653  2895  1244 11606

  66   0   2  9829  6670  4713  3470

  67   1  -2 11728  5737  6142  2111

  68   1  -2  7403 10194  2903  2562

  69  -2   0   770  9857   301  4108

  70   0  -2  6771  2653 10239  2130

  71   0   0  7855  4463  7362  9248

  72  -1   0 10880  6688  3127   311

  73  -1   2  4691  8128   533  8290

  74  -1  -2  4037  3558   115 10006

  75   0   0  2284   389  6473  3776

  76   0   0  5390  9091  1720  7047

  77   0   2  4988  1314 11101  4376

  78   1   0  5858  6929  7217  3009

  79   0   0  8276  9115  9758  8600

  80   0   0  1719  3490  6518  2847

  81   0   0  3145    16  2434 10905

  82   1   0 12177  5643  1293  9983

  83   0   2  8860  7027  7247  4144

  84   0  -2  8029 11886  5161  8312

  85   0   0  6660  8970  4777  9518

  86   0   2  8940  2217  8996  6495

  87   0   0  4623  2243 11869 10300

  88   0   4 11841  4074  6347  3751

  89  -1   0  2220 12271  2346  3966

  90   0   2 11997   617  8162  8020

  91   0  -2  4335    73 10232  9399

  92   1   2  8016 10780 11912 11369

  93   0  -2  4302  7923   717  7152

  94   0  -2  8014  1252  8311 11638

  95   1   0 11580   975  1679  2699

  96   1   2  6246  3336   161  6745

  97  -1  -4  5081  9817 11892  6259

  98   0   0  4544 10997 12278  4499

  99   0   2  1616  9495 12225 10213

 100  -1  -2  8533  8912  6448  9929

 101   0   2  8850  8093 11649  9665

 102   1  -4  9776  4225  8805  9906

 103  -1   0 12203  5021 12232 10353

 104   1   2  1285 10557  8597  2897

 105   1   0  5553 11162  5268 10387

 106  -1  -2  6413 10365 11905  6694

 107   0  -2  1915  8797  5109 10630

 108   1   2  5668  7809 10108   689

 109   0  -2  5724  6433  9119  9062

 110   1   0  3193  2998 10987  4238

 111   0   2  3218  6756 10221 11532

 112   0   0 11475  1061  3999  2494

 113   0   0  1751 10398  9032 10926

 114   0   0  5049  4368  3557  9980

 115   0   0  6973 10707 10291  4631

 116   0   0   826  2759  8952 11976

 117  -2   0 11077  1210  8027  7898

 118   0   0  3361  8733  5169   237

 119   0   0  9447 10875 12077 11281

 120   0   0  7154   928   564 11601

 121  -1  -2  5099  1695  5523 11879

 122   0  -2  5533  6614  4882  7444

 123   0   2  2416  2221 11163  3679

 124   0  -2   683  8407  7179 11214

 125  -1   0  1698  4946  8846  5627

 126   0   2 11993  1197  5067  2037

 127  -1   2 11131 10689  4543  8346

 128   1   0 11684 12052  5700  5576

 129   0   0 11081  7285  5758  2882

 130   0   0  2204 10550 10764 10396

 131   0   0  5413  6834   237  9705

 132   0   0  3139  9589  3580  1000

 133   1   0  2435 10845 11335  4375

 134   0   0  5835  9461  5820  8967

 135   1   2  1986  7566  6638  7219

 136   1   2 12005   279  4775   854

 137   0  -2 11470  3603  1399  4755

 138   0   4  3665 10794  4373 10453

 139   0   0  6909  8265 11931 11831

 140   1   2  9201  4238  3547  9596

 141   1  -2  7577 11197  9585  4684

 142   0   0  8947  1967  2051  7873

 143   0  -2  9195  2467  6347  7903

 144   1   2 11017  8525 11401 10043

 145  -1   0  1786  7054  2174  5272

 146   0   0  2541 11091 10944 11808

 147   0   0  1685 12142  9116 11391

 148   0  -2  9324 10699 11938  1090

 149   1  -2  6706  2541  7886  7480

 150   1   0 10550  1341  3839  5373

 151  -1  -2  4665  7629  5217  2934

 152   0   2  1311  6833  4048 11099

 153   1   0 11994  1783 10226  2549

 154   1   0  9953  5962 11300 10712

 155   0  -2  2781 11449   395 11045

 156  -1   4  6768  7744  9122  6955

 157   0  -2  1288 10720  7913  9198

 158   0   2  3735  3959  3762  4924

 159   1   2  2817  4147  6807  6198

 160   0   0  2935 11500 11190  4051

 161   0   2  1193  7795 11414  3350

 162  -1  -2   757  3411  9464  4481

 163   0  -2  3830  7004 11979   593

 164   0   0 11945    57  6438  9168

 165  -2   2  1844   173  7130  9844

 166   1   0  1055  4376   673   559

 167   1   0   665  1744 11877  9442

 168  -1   0   190  3421  9077  5294

 169  -1   0  5948  4923 10003  9323

 170   0  -2    66  3154  7238 10273

 171   1   0  3608  7307  8272 11128

 172   0  -2 11068 10669  7822 12269

 173  -1  -2  2289  5725  7793 11084

 174   0  -2  2045  9528  5770  5250

 175   0  -2  5369  1937  9741  7669

 176   0   0  5495   973    32  8740

 177   1  -2   187  6219 10487 11605

 178   0   0  6664  3891  6930  9183

 179   0  -2  8951  3731  4350 10057

 180   0  -2  2119  8064  2295    14

 181   0   0  5587  7068 12132   419

 182   1   0  5551  9660  4283  5818

 183   1  -2    58   319  9240  1724

 184   0   2  9694  6238  4742 12274

 185   2   2  1752 10949  7406  7643

 186   0   2  4551  4296  5533  7516

 187   0   0  5809  2080  4616  3169

 188   0  -2  4805  9682  4940 10345

 189   0   4  5232 10223  8937  9376

 190   0  -2  4985  6043  7853   528

 191   1   0 11937  4497  1366  6015

 192   1   0  7724  7554 12130  1918

 193   0  -2  2011  4752  4070  3130

 194   0   0  8272  1015  1803  3973

 195  -1   0  7832  7988  9436  5558

 196   0  -4  8854 10413 11890  8575

 197   0   0  2277  3600   263 11719

 198  -1   2  2986  1000  9583 11721

 199   0   2  2907  8991 11579 11775

 200   0   2  7872  2207  9525  1285

 201   1   0  7562  9107  2777  2830

 202  -2  -2 10678 10608  9041 10880

 203   0   0   656 11804  3455  2400

 204  -1  -2  4799  3910  3626  6180

 205  -1   0  1998  5423  2614  5813

 206   0   0  2327 11665  8051  2567

 207   0   0   282  6807  4478  1129

 208   0  -2  7967  3811 12284  6446

 209  -1   0  3169 11501 11972 11650

 210  -1   0  2614  4186  5549 10021

 211   1  -2 11856 11417 10104  6753

 212   1   0  3692  2680  3800 12107

 213   0   2  4639  5506 11526  6189

 214   0   0  6373  9147  2814  9738

 215   0   0  1942  1124  9011  3124

 216   0   2  5163   558 11376  4381

 217   0   0 11687  9612  8623    84

 218   0   0  8537  3843 11615    35

 219   1   0 11885  4846  3711  6409

 220   0   2  9728  8703  2262  5270

 221  -1  -2  4928   745  4084  3453

 222   0   2  2383  5711  4946 10846

 223   0  -2  2480  3190 11514  2446

 224   0   2  8786  4156 10444   381

 225   0   0  7294  3059   859  5500

 226   0  -2  2793  4752  4311 11196

 227  -1   2  9428  8892  6184  2715

 228   1   0  3240  6263  8476  7279

 229   0   0  2533   993  6898  5972

 230  -1   2  6513  1130   623  3622

 231  -1   0  2175   455  8066   855

 232   1  -2  8930 11192 11277  6039

 233  -1   0 10052  9546  1723  3691

 234   0   0 12282 10488  5953 11501

 235   1   0   966  2764  1478  7550

 236   0   4  2689  4295   136  7671

 237   0   0  2735 10452  7686  5468

 238  -1   0  7155  3804 11767  4710

 239   1   2  6875  1049  8317  1238

 240  -1  -2  5800  4804 10126  7221

 241   0   2 10256  8623  4292 11309

 242  -1   0  9012  8378  9611  5688

 243  -1   2  4014  1882  3226 12134

 244   0   2 11698  2629  1993  9817

 245   1  -2  9293  4184  3392 10739

 246   0   2    93   852  8664 11953

 247   0   2  6230  8044  8507  6969

 248  -1   2  6093  7622 10297  8445

 249  -1   0 10974  7821  3675  3517

 250  -1  -2  4760 11952  9509 11495

 251   0  -2  7410  5638  8286  2604

 252   0   0   313  2955  7834  4178

 253  -1   0  9733  3273 12249 11493

 254  -1   0   682  9048  9531  3876

 255   1  -2  2283   179  4322  9567

 256   0   0 10470  1633  2290  9062

 257   0  -2 11005  5584  7880  6991

 258   1  -2  2732  7686  7623  8563

 259   0   0  8845  9994  6380  2032

 260   0  -2  9527   785  4071  4639

 261   0  -2  7141  5116   474  9863

 262   0   0  8896  9356  8790  4233

 263   0  -2  8781  5058 11323  5758

 264  -1  -2  2106  4848  5472  3773

 265   0   0 10312  2028  1706  5806

 266  -1   0 11587 11556 10433  7614

 267  -1   0  9354  4702  4673 11174

 268   1   2  4179   310  1572  9202

 269   0   0   231  7881  4637  8778

 270   0   0 10643 12282  3262 11823

 271   0   2  4803   573 11021 12201

 272  -1   0 11942  2736  1772   881

 273   1   0 10172  5565  7021  1748

 274   0   2  8091   902 11967  2343

 275   1   0  6507  2055  1543  1125

 276   0   0  8363  4684  8421  7891

 277   0   2 11435  7507  3108  1495

 278   1   0  1121  5376  1638  8545

 279   1   2  6659  7231  2291  9356

 280   2  -2 11535  5948  8451 10276

 281   0   2  9996  5929 11267 11752

 282   0   0  9341 11999 10535  9922

 283   0   0  1156   407  2491  5743

 284   1   0 10878  9742 11436  7146

 285   1   0  4269 10191  6723  1057

 286   0   0  3150  6385 11151  8222

 287  -1   2 10602 12270  1942 11540

 288  -1   0  4149  9389  5193   155

 289   1   0  2220  1914  7033  2039

 290   1  -2  5849  9681  7990 10354

 291   0  -2   578  1167  9422  2925

 292   0   2  2784  4352  1474  8850

 293   0   2  2831  7803  7941 10471

 294   1  -2  1505  5309  1529 10706

 295  -1  -2 12152  3117  1462  5319

 296   0   0 12015 10147  2163  3011

 297   0   2 12204  3215 10166   351

 298  -1   0  3251  7021  9039  9355

 299   0   0  5488  2986  1862  5927

 300   1   0  7988   280  3983 11996

 301   0  -2 11691   944  6647  7206

 302   0  -2  5811  8894 11593  4438

 303   1   2 11242  8285  3494  3099

 304   0   0  1369  3781 11946  9679

 305   0   0  4923   855 11924  2443

 306   0   0 10077  6525  5892 12143

 307   0   0  5765   923  7601  5041

 308  -1   0 11585  4403  7020  7236

 309  -1   0  9508 11281  9550  8744

 310  -1   2  8015  7011  6196   851

 311   0   0 10282  6674  7084  1139

 312  -1   0   366  5463  5297 11037

 313   0   0  3271  3185  6778 10142

 314  -1   0  6295  3530  2128  3092

 315  -1   2  2446  9761  5698  9652

 316   0   0  6414  6084 11668  2854

 317   1   0  7954 11099  5621  8453

 318   1   0  8505  3817  6471  8585

 319   0  -2 10555   260  7709  1873

 320   0   0  4679  8577  2591  3492

 321   1   0  4517 10562  7356 10826

 322   0   0  5129  7378  6792 11094

 323   1   0 11014  1117   906  7306

 324  -1  -2  8930  3044  7558  1690

 325   0  -2 12034  5641  5602  3833

 326   1   0  4468  8161 11613  1703

 327   0   0  9452  5643  6465   759

 328  -1   0  4250  1062  8885  5366

 329   0   0  2562 11062 10606 12050

 330   0   0 11004  5092  1145  9690

 331   0   0  3971  4167  9338 10914

 332   0  -2  4640  2905  8263  8180

 333  -1   2 11466 11858  4479  8686

 334  -2  -2  2263 10527 11374  8335

 335  -1   2  8803 10486  6140 10827

 336   0   0  1608 10434   277  3299

 337   0   0  8846  4037  5405 10610

 338   0   2  2025  9028 11374   249

 339   0   0  7495  5760  9448  3603

 340   0   2    15 10858 10180    53

 341   0   0  2216   822  8232 10505

 342   0   0  4552  6213  8198  2721

 343  -1   0  8537 12065  4985  6616

 344   1   0    59  1083  5343  4975

 345   0   0  6820  2485  7426  8044

 346   0  -2    79  3592   780  2094

 347   0   2  6060  2269  1661  5628

 348  -1   0   483  7927  6962  9842

 349  -1   0 10399 11975   182  8453

 350   1   2 10965  8081  9568 12240

 351  -1   0  6177  9642 10608  1217

 352   0  -2  3647  7424  6312 11588

 353   0  -2 10821  5412  7478  9670

 354   0   2  7993  8400  9262  9133

 355   0  -2 12183  9287  5467  4145

 356   1  -2 11881 11278  2062  2271

 357   0   0 11023 11205  4098  9315

 358   0   0  2486  1161  4531 11806

 359   0   2  7820  8932  2128  6164

 360   0  -2  4830  2661  6650  6782

 361   0   0  1280  8451  7065  2723

 362   1  -4  3505  2948  7690 10249

 363   0   0  1931   604   857 11619

 364  -1   0  4519  1694  1682  7386

 365   1   0  7001  5943 10006  9007

 366   1   0  6867  7829  3179  9453

 367   0   2  6439  1013  9753   968

 368   0  -2   471  7027  6703  4401

 369   0   2 10693  6320  2472  5896

 370   1   0  6616  5825  5027  4446

 371   0  -4  2610  2936 10741 11669

 372  -1   0 10505  5607  7619 11326

 373  -1   0  8796  8925  6540   641

 374   0   0  7862  9942  2067  7361

 375   2   2  5933 11598  7281  2337

 376   0   0  4397  9644  2961   575

 377   0   0 11546  3667    60   496

 378   0   2 10359   897  6655  9940

 379   0   0  8042 11627  7627  4091

 380   0   2  7229  5196 10305  4323

 381   0   2 11076  8341  5590   590

 382   1  -2  5915   587  3514 10997

 383   0   0  4235  5733  1374  7164

 384   0  -2  6883  2313  3411   910

 385   2   0  5537  5149   391 10153

 386   0   0  4786  9993 11959  7183

 387   1   0  8660  4137  8672  1422

 388  -1   0 10388  8443  6742  3136

 389  -1   0  3028  4136  7848  1024

 390  -1  -2  3013  9457  3424  5692

 391   0   2  6434 10654   246  8185

 392  -1   0  5801  5730   384  4298

 393   0   0  3559 11131  6623  3040

 394   0   2  6911  3462  6279 10768

 395   0   0  2559 11098  1487  5746

 396   0   0  6942  1081  5465  2597

 397   0   0  6852   666  5872  6467

 398   0   0 10873  4863 11256  4225

 399   1  -2  3670   513  2689  1203

 400   1   0 11066  6794  6433  4163

 401   0   2  4927 11148  7593  4700

 402   0   2  5570  7675  6432  9507

 403   0   0  9882 11756 11480  4705

 404   1   2  9553  7076  9700  2926

 405  -1   2  9678 12074  7468 11797

 406   0   2  3955  2530 10255 10763

 407   1   0 10843  8488 12022  6421

 408   0   0  2514  2611  6629  2177

 409  -2  -2  1934  6748  5463  3878

 410   1  -2  2677  5860  4847 11948

 411   1   0  2065  8327  9459  7023

 412   0   0  6908  5681   530  4705

 413   0   0 10718  6791  9883 10546

 414  -1   0 10338 11007  3468  2087

 415   1   0  7817   625 11048  7745

 416   0   0 11023  4466 10734 10811

 417   0   0  6306  7136  5359  9233

 418   0   0  1858 10575  2337 11205

 419   0   0  1118  2777  6009  7711

 420   1   0  8755  4003  5535  8938

 421  -1   0 12259  1775  2505  8171

 422   0   0  5186 12038  9054  9707

 423  -1   0  8317  9867  2073  6580

 424   0  -2  3750  7074  7221 12191

 425  -1  -2  7076  6288  3318 10214

 426   0   0  4066  8076 12163  3442

 427   1   2  5009   366 10803  1339

 428   2   0  7392  9060  4955 11591

 429  -1  -4  9381  8187  9349  5579

 430   0   0  6499  4642  5787 12187

 431   1   2 11461 11653  3278  7917

 432   1   0  8976  7597   613  6477

 433   0  -2  9335 10397  6485 11019

 434   0  -2  7590  5554  4787  9128

 435  -1  -2  7109  7497   615  8655

 436   1   2  5984   709  9806  6063

 437   1   0  4451  1057  1327  2187

 438   0   0  6532  2071  1809  9139

 439   0   0  5657  1586 11166  5121

 440   0   0  3926  7845  1167  7773

 441   0   0  6347   293  1762 11582

 442   0   0 12239 10323  4500  6461

 443   1  -2  1977  3819  4233  7946

 444   0   0  5851  9874  3996  8822

 445  -1   2  3107  3834  5546  9707

 446   1   0  5636 11215 11094  5276

 447  -1   0 12270  4649     5 11911

 448   1  -2  6452   394  1732  3872

 449  -2   0 11019   764  1006 10907

 450   0  -4 11659  6297  4922  4827

 451   1   2   890  9098 11786  3678

 452   1   2  7670  7736  2460 10669

 453   0   2  2047  7505 11511  3057

 454   0   0 12148  5933  9508  9426

 455   0   0  5596  3895  2879  7412

 456   0   2  6504  2290  4180  9071

 457   1   0  8051   946   316 11380

 458   0  -2  2479 10389  6976  2480

 459  -1   0 10512 10125  6279  6329

 460   0   0  4709  6976  7912  6808

 461   0   2  6605  9934 10200 10093

 462  -1   0   949  7882  3698  1544

 463   1  -2 10292  3467   350  3293

 464   1   0  6448  9423  1313  2345

 465   0   2   692  6812  7583  6050

 466   1   0  3635  4184  2733  3816

 467   0   0 12067  5816 10128 11192

 468   0   0  9902  8712 11275  6813

 469   0   0 10938  7970  1902  7019

 470   1   0  9568  4228   242  5633

 471   0   0  2196  5792  6794 10300

 472   0   0  4075   157  8672  2560

 473   0  -4  2110  3629  9461  9122

 474  -2  -2  3412  4091  7245  4018

 475   0   0 11653    40  5765 10897

 476   0   0 10799   728  9056 10951

 477   0   0  2114  2282  3786   314

 478  -1   2   817 10585  8784 10553

 479  -1   0  3705 12125  8654  5792

 480   0  -4  1808  8664   196  4624

 481  -1   2  5841  1907  7238  7769

 482   0   0  8769  9263  6687   676

 483   0   0  3412  9123  9517  1111

 484  -1   0  4204    49 11892  6011

 485  -1   0 11196   448  3872  2642

 486   0   0   651  2142  3834  6611

 487   1   4  7208 10823  6626 12033

 488   0  -2  8558 10995 11169  2660

 489   0   0  7955  2079  1785  7697

 490   1   0  5565 11081  6935  1449

 491   0   2 11661  2880 10737   887

 492  -1  -2  2546  3372  1543  2424

 493   1   0  1667 10715  7245 11246

 494   0   0    93   456  1273  2563

 495   0   0  3205  2733  6176  7453

 496   1   0 12191  7834  2926 12258

 497   0   0  3788  5251   935  6085

 498   0   0 10114 12224  8954 11395

 499   0  -2  7464   568  5744  7972

 500  -1   0  1992  6344 10425  3471

 501  -1   0  5249  7024   675  3466

 502   0   2  8334  3338  1945  4805

 503   0   0  8566   837  6796  2416

 504  -1   2  1905  3844  2872  1612

 505   0   2   377  8680  5459   608

 506   0   0  1990  7692 10261  6844

 507   0   2  5170  9084 10608  4433

 508   0   0 11365  3048 11553  3451

 509   0  -2 12098  6095 11214  3125

 510   1  -2  1431  2633 10329  5488

 511  -1  -2  3846  4226  8410  4614

</pre>

Shown are the 512 small coefficients of the private keys *f* = *s1* and *g* = 2 * *s2* + 1 as well as their Number Theoretic Transforms (NTT) *F* and *G*, respectively. The BLISS public key *A* is computed as the component-wise inverse of *F* * *G* and the reverse NTT gives *a* = 1/(*f* * *g*) mod q with the 14 bit modulus q = 12289. Sometime it happens that *F* * *G* is not invertible, so that the following debug message is output

<pre>

S1[91] is zero - s1 is not invertible

</pre>

and another trial run is started.

h2. BLISS Root CA Certificate Generation

A self-signed BLISS CA certificate can be generated with the following command

<pre>

pki --self --type bliss --in key4.pem --ca --dn "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA" --lifetime 3653 --debug 2 --outform pem > cacert4.pem

  file content is not binary ASN.1

  -----BEGIN BLISS PRIVATE KEY-----

  -----END BLISS PRIVATE KEY-----

L0 - BLISSPrivateKey:

L1 - keyType:

  'BLISS-IV'

L1 - public:

L1 - secret1:

L1 - secret2:

L0 - subjectPublicKeyInfo:

L1 - algorithm:

L2 - algorithmIdentifier:

L3 - algorithm:

  'blissPublicKey'

L3 - parameters:

L4 - blissKeyType:

  'BLISS-IV'

L1 - subjectPublicKey:

mgf1 based on sha256 is seeded with 32 octets

y1 = -859..738 (sigma2 = 71786, mean = -6.6)

y2 = -852..644 (sigma2 = 65618, mean =  2.0)

norm2(s1*c) + norm2(s2*c) = 63602, rejected

mgf1 generated 10304 octets

mgf1 based on sha256 is seeded with 32 octets

y1 = -942..726 (sigma2 = 81503, mean = -8.6)

y2 = -876..893 (sigma2 = 69883, mean =  2.4)

norm2(s1*c) + norm2(s2*c) = 66020, accepted

scalar(z1,s1*c) + scalar(z2,s2*c) = 86651, rejected

mgf1 generated 10528 octets

mgf1 based on sha256 is seeded with 32 octets

y1 = -862..785 (sigma2 = 72628, mean = -7.1)

y2 = -782..921 (sigma2 = 74618, mean =  4.1)

norm2(s1*c) + norm2(s2*c) = 64940, accepted

scalar(z1,s1*c) + scalar(z2,s2*c) = -176380, accepted

z1 = -873..780, z2d = -3..4

efficiency of Huffman coder is 3.4121 bits/tuple (1747 bits)

generated BLISS signature (6706 bits encoded in 839 bytes)

signature generation needed 3 rounds

mgf1 generated 10656 octets

</pre>

With a debug level of 2 you get quite a lot of debug information. Starting from the top, the automatic conversion from PEM to DER format is shown, followed by the ASN.1 encoding of the BLISS private key from which the BLISS public key is extracted. Then in order to generate the BLISS certificate signature, two vectors *y1* and *y2* with 512 random numbers tightly following a Gaussian probability distribution using rejection sampling are generated. This process usually requires several rounds and a lot of random bits are used. The BLISS signature finally consists of the random vectors *z1* and *z2* as well as the sparse challenge vector *c*.

A BLISS certificate can be displayed at any time with

<pre>

pki --print --debug 2 --in cacert4.pem

L0 - x509:

L1 - tbsCertificate:

L2 - DEFAULT v1:

L3 - version:

  X.509v3

L2 - serialNumber:

L2 - signature:

L3 - algorithmIdentifier:

L4 - algorithm:

  'BLISS-with-SHA512'

L2 - issuer:

  'C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA'

L2 - validity:

L3 - notBefore:

L4 - utcTime:

  'Dec 13 12:01:57 UTC 2014'

L3 - notAfter:

L4 - utcTime:

  'Dec 13 12:01:57 UTC 2024'

L2 - subject:

  'C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA'

L2 - subjectPublicKeyInfo:

-- > --

  L0 - subjectPublicKeyInfo:

  L1 - algorithm:

  L2 - algorithmIdentifier:

  L3 - algorithm:

    'blissPublicKey'

  L3 - parameters:

  L0 - subjectPublicKeyInfo:

  L1 - algorithm:

  L2 - algorithmIdentifier:

  L3 - algorithm:

    'blissPublicKey'

  L3 - parameters:

  L4 - blissKeyType:

    'BLISS-IV'

  L1 - subjectPublicKey:

-- < --

L2 - optional extensions:

L3 - extensions:

L4 - extension:

L5 - extnID:

  'basicConstraints'

L5 - critical:

  TRUE

L5 - extnValue:

L6 - basicConstraints:

L7 - CA:

  TRUE

L4 - extension:

L5 - extnID:

  'keyUsage'

L5 - critical:

  TRUE

L5 - extnValue:

L4 - extension:

L5 - extnID:

  'subjectKeyIdentifier'

L5 - critical:

  FALSE

L5 - extnValue:

L6 - keyIdentifier:

L1 - signatureAlgorithm:

L2 - algorithmIdentifier:

L3 - algorithm:

  'BLISS-with-SHA512'

L1 - signatureValue:

z1 = -873..780, z2d = -3..4

cert:      X509

subject:  "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA"

issuer:   "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA"

validity:  not before Dec 13 13:01:57 2014, ok

           not after  Dec 13 13:01:57 2024, ok (expires in 3652 days)

serial:    12:a0:ca:85:51:b9:f3:27

flags:     CA CRLSign self-signed

subjkeyId: 37:f4:9e:f8:b7:50:ed:d4:29:16:72:58:b4:b1:f1:f5:46:c9:54:71

pubkey:    BLISS 192 bits strength

keyid:     55:ee:7a:31:44:e5:a0:cf:b6:c9:a7:17:98:c9:60:a7:eb:d0:4e:4f

subjkey:   37:f4:9e:f8:b7:50:ed:d4:29:16:72:58:b4:b1:f1:f5:46:c9:54:71

</pre>

If you are not interested in any detailed information then just creat a self-signed BLISS CA certificate with

<pre>

pki --self --type bliss --in key1.der --ca --dn "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA" --lifetime 3653 > cacert1.der

</pre>

and view it with

<pre>

pki --print --in cacert1.der

cert:      X509

subject:  "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA"

issuer:   "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA"

validity:  not before Dec 13 12:58:21 2014, ok

           not after  Dec 13 12:58:21 2024, ok (expires in 3652 days)

serial:    5d:06:0d:4b:69:64:84:62

flags:     CA CRLSign self-signed

subjkeyId: f3:60:55:bc:0b:49:c4:8a:a6:38:cc:ad:72:67:e5:91:7c:b8:a4:f5

pubkey:    BLISS 128 bits strength

keyid:     df:78:00:c4:b4:13:e7:fd:4f:05:dd:39:1a:2e:2b:c5:65:39:10:f4

subjkey:   f3:60:55:bc:0b:49:c4:8a:a6:38:cc:ad:72:67:e5:91:7c:b8:a4:f5

</pre>

h2. BLISS End Entity Certificates

We are now going to generate a BLISS-I key pair for user Carol:

<pre>

pki --gen --type bliss --size 1 > carolKey.der

secret key generation succeeded after 2 trials

</pre>

Next we create a self-signed PKCS#10 certificate request

<pre>

 pki --req --type bliss --in carolKey.der --dn "C=CH, O=strongSwan Project, CN=carol@strongswan.org" --san carol@strongswan.org > carolReq.der

</pre>

which is used as the input for the CA to create a signed end entity certificate:

<pre>

 pki --issue --type pkcs10 --in carolReq.der --cacert cacert4.pem --cakey cakey4.pem --crl http://crl.strongswan.org/bliss.crl --flag clientAuth > carolCert.der

</pre>

and which has the following content

<pre>

pki --print --in carolCert.der

cert:      X509

subject:  "C=CH, O=strongSwan Project, CN=carol@strongswan.org"

issuer:   "C=CH, O=strongSwan Project, CN=strongSwan BLISS Root CA"

validity:  not before Dec 13 13:20:34 2014, ok

           not after  Dec 12 13:20:34 2017, ok (expires in 1094 days)

serial:    38:a9:13:10:c2:ed:ed:c3

altNames:  carol@strongswan.org

flags:     clientAuth

CRL URIs:  http://crl.strongswan.org/bliss.crl

authkeyId: 37:f4:9e:f8:b7:50:ed:d4:29:16:72:58:b4:b1:f1:f5:46:c9:54:71

subjkeyId: 8b:a3:c5:11:00:bb:84:55:dd:b8:4b:20:04:d9:58:77:57:ba:d8:3c

pubkey:    BLISS 128 bits strength

keyid:     5b:cf:17:14:a8:d8:aa:bc:40:f3:21:95:a9:67:7d:20:af:66:4e:c2

subjkey:   8b:a3:c5:11:00:bb:84:55:dd:b8:4b:20:04:d9:58:77:57:ba:d8:3c

</pre>

h2. IKEv2 Public Key Authentication using BLISS Signatures

The "ikev2/rw-ntru-bliss":http://www.strongswan.org/testresults5rc/ikev2/rw-ntru-bliss/ strongSwan remote-access VPN scenario shows the practical use of IKEv2 public key authentication using BLISS signatures. The key exchange method used is NTRU encryption so that the IPsec connection setup is not vulnerable to quantum computer based key attacks.