homomorphic signatures and applications to network codingsita/forum/article/2020/... · 2019. 9....

Homomorphic Signatures and Applications to Network Coding

Nuttapong Attrapadung （Nuts, ナッツ）産総研（AIST）

IEICEソサイエティ大会 2019.9.13

[email protected]

Linear Network Coding

• Network nodes modify packets they receive before transmitting them

• Nodes transmit linear combinations of received vectors (linear NC).

• Yields increased throughput and better resilience to packet loss. 2

v1, v2<latexit sha1_base64="JfvEXefsfhGJgWTmAagsC3SzNa8=">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</latexit><latexit sha1_base64="JfvEXefsfhGJgWTmAagsC3SzNa8=">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</latexit><latexit sha1_base64="JfvEXefsfhGJgWTmAagsC3SzNa8=">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</latexit><latexit sha1_base64="JfvEXefsfhGJgWTmAagsC3SzNa8=">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</latexit>

b = a1v1 + a2v2<latexit sha1_base64="092u06NcVDL6E5kDZfdnYKhIqL8=">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</latexit><latexit sha1_base64="092u06NcVDL6E5kDZfdnYKhIqL8=">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</latexit><latexit sha1_base64="092u06NcVDL6E5kDZfdnYKhIqL8=">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</latexit><latexit sha1_base64="092u06NcVDL6E5kDZfdnYKhIqL8=">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</latexit>

c = a3v1 + a4v2<latexit sha1_base64="iZYsnmEoCpJiyZntiOGmDMq3tGw=">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</latexit><latexit sha1_base64="iZYsnmEoCpJiyZntiOGmDMq3tGw=">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</latexit><latexit sha1_base64="iZYsnmEoCpJiyZntiOGmDMq3tGw=">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</latexit><latexit sha1_base64="iZYsnmEoCpJiyZntiOGmDMq3tGw=">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</latexit>

Source

Receiver 1

Receiver 2

d = a5b + a6c<latexit sha1_base64="jaImkATxs866S3rpefJwIL2r5T4=">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</latexit><latexit sha1_base64="jaImkATxs866S3rpefJwIL2r5T4=">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</latexit><latexit sha1_base64="jaImkATxs866S3rpefJwIL2r5T4=">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</latexit><latexit sha1_base64="jaImkATxs866S3rpefJwIL2r5T4=">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</latexit>

b, d<latexit sha1_base64="hTOcNPMgRvDqsFI2JYkTa/EcRRI=">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</latexit><latexit sha1_base64="hTOcNPMgRvDqsFI2JYkTa/EcRRI=">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</latexit><latexit sha1_base64="hTOcNPMgRvDqsFI2JYkTa/EcRRI=">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</latexit><latexit sha1_base64="hTOcNPMgRvDqsFI2JYkTa/EcRRI=">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</latexit>

c, d<latexit sha1_base64="ZYCT55v3C+7taU47aj+P4ZM04Tg=">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</latexit><latexit sha1_base64="ZYCT55v3C+7taU47aj+P4ZM04Tg=">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</latexit><latexit sha1_base64="ZYCT55v3C+7taU47aj+P4ZM04Tg=">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</latexit><latexit sha1_base64="ZYCT55v3C+7taU47aj+P4ZM04Tg=">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</latexit>

Linear Network Coding: A Model

3

• At a receiver, upon receiving independent , the file is recovered as

• Source views a file as vectors

• Source encodes them with auxiliary vectors

• At each node, upon inputs vectors it outputs ~t1, . . . ,~t` 2 Fn+mp<latexit sha1_base64="MQyIGM6UdH+5Mfe/AkTOWM/oRKo=">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</latexit><latexit sha1_base64="MQyIGM6UdH+5Mfe/AkTOWM/oRKo=">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</latexit><latexit sha1_base64="MQyIGM6UdH+5Mfe/AkTOWM/oRKo=">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</latexit><latexit sha1_base64="MQyIGM6UdH+5Mfe/AkTOWM/oRKo=">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</latexit>

a1~t1 + · · ·+ a`~t`<latexit sha1_base64="k3pNn8hzo9uAUvI7UcUk+xEj6y0=">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</latexit><latexit sha1_base64="k3pNn8hzo9uAUvI7UcUk+xEj6y0=">AAADAHichVE9axRRFD0Zv2L8yKqNYDO4RITAckeiZq2CNpb5cJNANgwzb182Q+aLmbcL67KNpZ2VhZWCoFiIlWCl2PgHLFJZi2UEGwvPzE4Qi41vePPOu++e885910/DIDci+1PWseMnTp6aPj1z5uy587O1CxfX86SXKd1SSZhkm76X6zCIdcsEJtSbaaa9yA/1hr93rzjf6OssD5L4gRmkejvyunGwEyjPMOTW5j3Xsdt9rYZmRDRvt1UnMTmB57Z1GB4eFdit1aUhNxdEHJugHATN5qLTvGU7VaSOaiwntW9oo4MECj1E0IhhiEN4yPltwYEgZWwbQ8YyoqA81xhhhtweszQzPEb3+O9yt1VFY+4LzbxkK94ScmZk2piTr/JaDuSLvJHv8nui1rDUKLwMuPpjrk7d2ceX1379lxVxNdj9yzrSs8EOFkuvAb2nZaSoQo35/YdPD9burM4Nr8kL+UH/z2VfPrOCuP9TvVzRq8+O8JNTKypfcMDZGSvyvuI9bNaVVfmaeZNVVPkGGf11WZXBiA0/7Ko9GazfaDjScFYW6kt3q9ZP4wqu4jr7extLuI9ltKj+BO/xEZ+sR9Yr6631bpxqTVWcS/hnWB/+AJ0tsnE=</latexit><latexit sha1_base64="k3pNn8hzo9uAUvI7UcUk+xEj6y0=">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</latexit><latexit sha1_base64="k3pNn8hzo9uAUvI7UcUk+xEj6y0=">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</latexit>

(~x1||~u1) , . . . , (~xm||~um) 2 Fn+mp<latexit sha1_base64="xq2yJgHZBxA+pnvxsSDYS8jzp14=">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</latexit><latexit sha1_base64="xq2yJgHZBxA+pnvxsSDYS8jzp14=">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</latexit><latexit sha1_base64="xq2yJgHZBxA+pnvxsSDYS8jzp14=">AAADKXichVG9TxRBFH+sKIgfnNqY2Ey8YCCSy1sCylkRTAwlXwckLG5254Zjws7uZnfu4rncP+A/QEGFiYXhP7C1sbUwERtNrNASExoK384tqMXhbGbnvd/8fm/ehx8HMtWIh33Wpf7LVwYGrw5du37j5nDp1u2VNGomXNR4FETJmu+lIpChqGmpA7EWJ8JTfiBW/e2n+f1qSySpjMJl3Y7FhvIaodyU3NMEuaXlUaclePai49psZ4cZp0nOmMPYuBPUI52OO+ycpP4mqTHmyJA5ytNbvp8967jx8yx8qDpuqYwVnJpEtBkZZpFRrU7b1UfMLpAyFGs+Kn0FB+oQAYcmKBAQgiY7AA9S+tbBBoSYsA3ICEvIkuZeQAeGSNskliCGR+g2/RvkrRdoSH4eMzVqTq8EtBNSMhjBT/gWj/EDHuARnvaMlZkYeS5tOv2uVsTu8Ku7Syf/VSk6NWz9UV2Ys4ZNmDa5Sso9NkheBe/qWy93j5eeLI5kD/A1/qT89/EQ31MFYesXf7MgFvcuyCelWMp0sE273o1I7+X9YFRXUvAF8XpH4aYHCeXXoKo05AM/myrrbaxMVGys2AuT5ZnZYvSDcA/uwyjN9zHMwBzMQ42iv4NvcAQ/rAPro/XZ+tKlWn2F5g78s6zvvwGEd8NC</latexit><latexit sha1_base64="xq2yJgHZBxA+pnvxsSDYS8jzp14=">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</latexit>

.

.

Threat: Pollution Attacks against Authenticity• Network coding is susceptible to pollution attacks.

• Injecting a single invalid packet may prevent the correct decoding.

4

Attacking one ⇒ affecting many

Combatting Pollution Attacks: Trivial Methods?• Trivially using digital signature schemes does not work.

• Trivial method 1: Source signs each of its released packets.

- Why it won’t work: A receiver likely obtains different packets (which are not signed!)

• Trivial method 2: Source signs the whole file at once.

- Why it won’t work: No efficient way to recover the correct file, since it cannot distinguish correct packets with bad packets.

5

• Our goal: dedicated authentication mechanism for network coding

• Which enables authenticating “vector spaces”

Approaches for Combatting Pollution Attacks• Information-theoretic approaches

• Enabling recovery from malicious fault by introducing redundancy.

• Pros: Not rely on computational assumptions.

• Cons:

• Limited classes of adversaries such as

- the number of nodes that the adversary can corrupt,

- the number of packets that can be modified, and/or

- the number of links on which the adversary can eavesdrop

• Communication overhead is also large.

6

Approaches for Combatting Pollution Attacks• Cryptographic approaches

• Providing a way for honest nodes to verify authenticity of individual packets.

• Pros: Potentially secure against adversaries that may

- eavesdrop the entire network

- control an arbitrary fraction of malicious nodes

Note: honest nodes can always detect malicious activities, and a receiver can recover the correct file if it receives sufficient vectors.

• Cons: Rely on some computational assumptions (i.e., limit the adversary’s computational power)

7

This talk is about the crypto approach.

8

accepts if

• Source signs a linear sub-space .

• A signature of accepts any vector .

Network Coding Signatures (NCS) [BFKW09]

(Homomorphic) Network Coding Signatures

[BFKW09] D. Boneh, D. Freeman, J. Katz, B. Waters. Signing a Linear Subspace: Signature Schemes for Network Coding. In PKC’09,

9

accepts if

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~v2<latexit sha1_base64="i3AQVBASaTQOAvVzQTj4wF6CuO4=">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</latexit><latexit sha1_base64="i3AQVBASaTQOAvVzQTj4wF6CuO4=">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</latexit><latexit sha1_base64="i3AQVBASaTQOAvVzQTj4wF6CuO4=">AAAC3HichVG7TtxAFD0YEgh5sJAmEo2VFRHV6hrxWiqUNJQ8sjzEY2UPw2Lhl+xZS5vVdqlAqZCgAIpEokB8Rpr8AAUVdZSSSDQUXM86ilIsGWs8d87cc+bcuU7kuYkiuu4yunuePO3te9b//MXLVwOFwaHlJKzHQlZE6IXxqmMn0nMDWVGu8uRqFEvbdzy54ux9yM5XUhknbhh8VI1Ibvp2LXB3XGErhtY2Uimaaas6Vi0UqUQT40SWyYEeHJTL01Z50rRypIh8zIeFG2xgGyEE6vAhEUBx7MFGwt86LBAixjbRZCzmyNXnEi30M7fOWZIzbEb3+F/j3XqOBrzPNBPNFnyLxzNmpokRuqILuqUfdEk/6b6jVlNrZF4avDptroyqA/tvlu7+y/J5Vdj9y3rUs8IOprVXl71HGsmqEG1++un4dmlmcaT5jr7RL/b/la7pO1cQpL/F+YJcPHnET8Javn7BBs/ttiLfl72HyXXFeb7kvM4qQr9BzP5qXJVCixv+p6tm52B5rGRRyVoYL86+z1vfh2G8xSj3dwqzmMM8Klr9EKc4M7aMz8aB8aWdanTlnNf4ZxhHD06apTg=</latexit><latexit sha1_base64="i3AQVBASaTQOAvVzQTj4wF6CuO4=">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</latexit>

a1~v1 + a2~v2<latexit sha1_base64="KyUCRoYKQDFQyOuw5qNCgzvDl8w=">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</latexit><latexit sha1_base64="KyUCRoYKQDFQyOuw5qNCgzvDl8w=">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</latexit><latexit sha1_base64="KyUCRoYKQDFQyOuw5qNCgzvDl8w=">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</latexit><latexit sha1_base64="KyUCRoYKQDFQyOuw5qNCgzvDl8w=">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</latexit>

• Source signs a linear sub-space .

• A signature of accepts any vector .

Network Coding Signatures (NCS) [BFKW09]

Homomorphic Network Coding Signatures [BFKW09]

• Given m vector-signature pairs any node can publicly compute a signature of

(~v1,�1), . . . , (~vm,�m)<latexit sha1_base64="iWqtWIbj1PDtG23XUrZ5HpRoHpQ=">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</latexit><latexit sha1_base64="iWqtWIbj1PDtG23XUrZ5HpRoHpQ=">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</latexit><latexit sha1_base64="iWqtWIbj1PDtG23XUrZ5HpRoHpQ=">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</latexit><latexit sha1_base64="iWqtWIbj1PDtG23XUrZ5HpRoHpQ=">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</latexit>

a1~v1 + · · ·+ am~vm<latexit sha1_base64="6YVw6KQY1K8xDnfkPMmBUp0GYxM=">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</latexit><latexit sha1_base64="6YVw6KQY1K8xDnfkPMmBUp0GYxM=">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</latexit><latexit sha1_base64="6YVw6KQY1K8xDnfkPMmBUp0GYxM=">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</latexit><latexit sha1_base64="6YVw6KQY1K8xDnfkPMmBUp0GYxM=">AAAC+XichVG7TtxAFD2YJDzyYAkNUhqLDShSpNV1BISlQklDySMLSCyy7NlhY+GX7FlLi7U/wA9QUBEJiZAGiSq0NPwABRV1lJJIaVLk2msUUSwZazx3ztxz5ty5dug6sSK67tP6Hz1+MjA4NPz02fMXI6XRl2tx0IqErInADaIN24ql6/iyphzlyo0wkpZnu3Ld3vmYna8nMoqdwP+k2qHc8qym72w7wlIMmaXXlmnUEynSpGMa+lu9LhqBijmwTO8O98xSmSo0M01k6Bzkg4Nqdc6ozupGgZRRjKWgdIM6Gggg0IIHCR+KYxcWYv42YYAQMraFlLGIIyc/l+hgmLktzpKcYTG6w/8m7zYL1Od9phnnbMG3uDwjZuqYpCv6Srd0Sd/oB/3pqZXmGpmXNq92lytDc2RvfPX3f1kerwqf/7Ee9Kywjbncq8PewxzJqhBdfrK7f7s6vzKZTtEX+sn+D+maLrgCP/kljpblysEDfmLW8vIXbPNsdBX5vuw9dK4rKvIl5/VWEfkbROyvyVUpdLjhd13Vewdr7yoGVYzl6fLCh6L1g3iFCbzh/r7HAhaxhBqr7+EU33GupdqhdqyddFO1voIzhntDO/sLPzSvtw==</latexit>

for any a1, . . . , am 2 Fp<latexit sha1_base64="Saq92QF5W7w32ZUj7/7WeEvr4Fk=">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</latexit><latexit sha1_base64="Saq92QF5W7w32ZUj7/7WeEvr4Fk=">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</latexit><latexit sha1_base64="Saq92QF5W7w32ZUj7/7WeEvr4Fk=">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</latexit><latexit sha1_base64="Saq92QF5W7w32ZUj7/7WeEvr4Fk=">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</latexit>

(Homomorphic) Network Coding Signatures


Using Homomorphic Network Coding Signatures

10

Source signs input vectors

~w3<latexit sha1_base64="Ygwzfe5NR/eAB/l3u/uMOJ6z9D8=">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</latexit><latexit sha1_base64="Ygwzfe5NR/eAB/l3u/uMOJ6z9D8=">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</latexit><latexit sha1_base64="Ygwzfe5NR/eAB/l3u/uMOJ6z9D8=">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</latexit><latexit sha1_base64="Ygwzfe5NR/eAB/l3u/uMOJ6z9D8=">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</latexit>

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~w3 = a1 ~w1 + a2 ~w2<latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">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</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">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</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">AAAC/nichVE7SxxRFD5OXj6SuNFGSHNxMSQEljMbE91AQJImpY+sCirDzPW6GZwXM3c3rMtC6jSWFqkSSCCEkEqw0sLGP5DCylosFdJY+M3smJBizR3u3HPOPd93v3OOE3luopkPe4xr12/cvNXb1z9w+87dwcK9ofkkrMdSVWXohfGiYyfKcwNV1a721GIUK9t3PLXgrL9K7xcaKk7cMHijm5Fa8e1a4K650tYIWYVHyw0lW+/a1hPxQtiWKS59UzyGX/7jl61CkUv8dJzZFDCyBaNSmTQrz4SZR4qUr+mwcETLtEohSaqTT4oC0rA9sinBt0QmMUWIrVALsRiWm90ralM/sHVkKWTYiK7jX4O3lEcD+ClnkqElXvGwYyAFjfEv/sanfMDf+ZjPu3K1Mo5USxOn08GqyBr8MDL3+78oH6emt39RV2rWtEaTmVYX2qMsklYhO/jGxtbp3PPZsdYD/swn0P+JD3kfFQSNM/llRs1+vEJPAi4/62ATe7XDiPfSfgjUFef5CnndWWTWgxj6aqhKUxsDv5yq6G7Ml0sml8yZ8eLUy3z0vXSfRukh5jtBU/SapqkK9k3apl3aM94bX40fxs9OqtGTY4bpn2XsXACrv7Dp</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">AAAC/nichVE7SxxRFD5OXj6SuNFGSHNxMSQEljMbE91AQJImpY+sCirDzPW6GZwXM3c3rMtC6jSWFqkSSCCEkEqw0sLGP5DCylosFdJY+M3smJBizR3u3HPOPd93v3OOE3luopkPe4xr12/cvNXb1z9w+87dwcK9ofkkrMdSVWXohfGiYyfKcwNV1a721GIUK9t3PLXgrL9K7xcaKk7cMHijm5Fa8e1a4K650tYIWYVHyw0lW+/a1hPxQtiWKS59UzyGX/7jl61CkUv8dJzZFDCyBaNSmTQrz4SZR4qUr+mwcETLtEohSaqTT4oC0rA9sinBt0QmMUWIrVALsRiWm90ralM/sHVkKWTYiK7jX4O3lEcD+ClnkqElXvGwYyAFjfEv/sanfMDf+ZjPu3K1Mo5USxOn08GqyBr8MDL3+78oH6emt39RV2rWtEaTmVYX2qMsklYhO/jGxtbp3PPZsdYD/swn0P+JD3kfFQSNM/llRs1+vEJPAi4/62ATe7XDiPfSfgjUFef5CnndWWTWgxj6aqhKUxsDv5yq6G7Ml0sml8yZ8eLUy3z0vXSfRukh5jtBU/SapqkK9k3apl3aM94bX40fxs9OqtGTY4bpn2XsXACrv7Dp</latexit>

~w4 = a3 ~w1 + a4 ~w2<latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit>

~w5 = a5 ~w3 + a6 ~w4<latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">AAAC/nichVG/axRBFP6y/orxR05tBJvFI6IIx1u9xJwgBG0s88NLAkk4dieTy5L9xe7cyXkcWNtYWlgpKIiIlWClhY3/gEUqa7GMYGPht3sbxeLiDDPz3jfve/O9eV4S+JkR2R2zDh0+cvTY+PGJEydPnZ6snDm7nMWdVOmmioM4XfXcTAd+pJvGN4FeTVLthl6gV7ydO/n9SlenmR9H90wv0Ruh2478LV+5hlCrcmW9q1X//qA1bd+yXe77/nX7Kv2ZP369ValKTabrIo5Noxg0Go1ZpzFjOyVSRTnm48pXrGMTMRQ6CKERwdAO4CLjXIMDQUJsA31iKS2/uNcYYILcDqM0I1yiO9zb9NZKNKKf58wKtuIrAVdKpo0p+SKvZE8+y2v5Jr9G5uoXOXItPZ7ekKuT1uSj80s//8sKeRps/2UdqNlgC7OFVp/akwLJq1BDfvfBk72lm4tT/UvyXL5T/zPZlU+sIOr+UC8W9OLTA/RkzBUWP9jj2hxm5Hv5f9isKy3jNeNGZ1HFH6TU12ZVBgM2fL+r9mhj+VrNkZqzUK/O3S5bP44LuIjL7O8NzOEu5tFk9sd4hw/4aD20XlpvrLfDUGus5JzDP8N6/xvOz7D3</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit>

~w4<latexit sha1_base64="AlrHFIvFGTs5S55UB1CNzt2WyLY=">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</latexit><latexit sha1_base64="AlrHFIvFGTs5S55UB1CNzt2WyLY=">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</latexit><latexit sha1_base64="AlrHFIvFGTs5S55UB1CNzt2WyLY=">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</latexit><latexit sha1_base64="AlrHFIvFGTs5S55UB1CNzt2WyLY=">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</latexit>

~w5<latexit sha1_base64="mJpk9tq4RdxY7WhhUJX1YXpSsqw=">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</latexit><latexit sha1_base64="mJpk9tq4RdxY7WhhUJX1YXpSsqw=">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</latexit><latexit sha1_base64="mJpk9tq4RdxY7WhhUJX1YXpSsqw=">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</latexit><latexit sha1_base64="mJpk9tq4RdxY7WhhUJX1YXpSsqw=">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</latexit>





Each node verifies authenticity of incoming vectors and homomorphically compute a signature of its linear combination.

Each node verifies authenticity of incoming vectors and recover the file.

Using Homomorphic Network Coding Signatures

11

Source signs input vectors


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~w3 = a1 ~w1 + a2 ~w2<latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">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</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">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</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">AAAC/nichVE7SxxRFD5OXj6SuNFGSHNxMSQEljMbE91AQJImpY+sCirDzPW6GZwXM3c3rMtC6jSWFqkSSCCEkEqw0sLGP5DCylosFdJY+M3smJBizR3u3HPOPd93v3OOE3luopkPe4xr12/cvNXb1z9w+87dwcK9ofkkrMdSVWXohfGiYyfKcwNV1a721GIUK9t3PLXgrL9K7xcaKk7cMHijm5Fa8e1a4K650tYIWYVHyw0lW+/a1hPxQtiWKS59UzyGX/7jl61CkUv8dJzZFDCyBaNSmTQrz4SZR4qUr+mwcETLtEohSaqTT4oC0rA9sinBt0QmMUWIrVALsRiWm90ralM/sHVkKWTYiK7jX4O3lEcD+ClnkqElXvGwYyAFjfEv/sanfMDf+ZjPu3K1Mo5USxOn08GqyBr8MDL3+78oH6emt39RV2rWtEaTmVYX2qMsklYhO/jGxtbp3PPZsdYD/swn0P+JD3kfFQSNM/llRs1+vEJPAi4/62ATe7XDiPfSfgjUFef5CnndWWTWgxj6aqhKUxsDv5yq6G7Ml0sml8yZ8eLUy3z0vXSfRukh5jtBU/SapqkK9k3apl3aM94bX40fxs9OqtGTY4bpn2XsXACrv7Dp</latexit><latexit sha1_base64="/UZs5tBBONjdRCrf/ibASQw2pa0=">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</latexit>

~w4 = a3 ~w1 + a4 ~w2<latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">AAAC/nichVG/TxRBFH4sKif+4ITGxGbiBQMxubyFQ+5ISIg2lvzwgATIZncYjg37K7tzR47LJdQ0lhZWmmhijLEyocLCxn/AgsraWGJiY+G3e4vG4nA2s/Pem/d98733nMhzE818OmAMXrp8Zahwdfja9Rs3R4q3RleTsBlLVZehF8brjp0ozw1UXbvaU+tRrGzf8dSas/covV9rqThxw+CJbkdqy7cbgbvjSlsjZBUnN1tKdva7VkXMC9uaFue+Ke7Dr/zxp6xiics8U2E2BYxswajVqmbtgTDzSInytRgWv9ImbVNIkprkk6KANGyPbErwbZBJTBFiW9RBLIblZveKujQMbBNZChk2onv4N+Bt5NEAfsqZZGiJVzzsGEhB4/yF3/AZf+a3/I1/9eXqZBypljZOp4dVkTVydHvl539RPk5Nu39RF2rWtEPVTKsL7VEWSauQPXzr4NnZytzyeOcev+Tv0P+CT/kTKghaP+SrJbX8/AI9Cbj8rINt7O0eI95L+yFQV5znK+T1Z5FZD2Loa6AqTV0M/Hyqor+xOlU2uWwuVUoLD/PRF+gO3aUJzHeWFugxLVId7E/pA53QR+PQeG28M973Uo2BHDNG/yzj+De4V7Du</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit><latexit sha1_base64="F/1J6YKrhbR+8sPvbcEfRiynGnI=">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</latexit>

~w5 = a5 ~w3 + a6 ~w4<latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit><latexit sha1_base64="XaCzSl2z2t9KFkXYsVxhBJNxcFI=">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</latexit>







Each node verifies authenticity of incoming vectors and homomorphically compute a signature of its linear combination.

Each node verifies authenticity of incoming vectors and recover the file.

Homomorphic scheme allows to sign each individual vector without knowing the whole sub-space.

Obtaining Homomorphic NCS: Why Difficult?• Normal digital signature schemes

12

= Alice owes Bob 100yen

• Homomorphic signature schemes: Seem to contradict security of digital signatures

Security: Unforgeability

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?

Difficult not only to construct such schemes but also to define security .


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Syntax Model of Network Coding Signature

14

Keygen(n) generates a secret signing key with a public key where is the dimension of vectors to be signed.

sk pk

n

outputs a signature on vector with an “identifier” .~v�

returns a signature on vector if each is a valid signature of with the same .

Sign(sk, id,~v)

�0 ~v 0 := a1~v1 + · · ·+ a`~v`�i

Combine(pk, id,�1, . . . ,�`,

a1, . . . , a`) ~vi id

Verify(pk, id, ~y ,�) returns True if is a valid signature of .

id

� ~y

• Homomorphic network coding signature scheme consists of 4 algorithms:

Syntax Model of Network Coding Signature

15

Keygen(n) generates a secret signing key with a public key where is the dimension of vectors to be signed.

sk pk

n

outputs a signature on vector with an “identifier” .~v�

returns a signature on vector if each is a valid signature of with the same .

Sign(sk, id,~v)

�0 ~v 0 := a1~v1 + · · ·+ a`~v`�i

Combine(pk, id,�1, . . . ,�`,

a1, . . . , a`) ~vi id

Verify(pk, id, ~y ,�) returns True if is a valid signature of .

id

� ~y

• Homomorphic network coding signature scheme consists of 4 algorithms:

• General NCS scheme is then obtained from homomorphic NCS as:

Sign(sk, id, V ) signing with the same where ~v1, . . . ,~v` V = span(~v1, . . . ,~v`)id

Security Definition for Network Coding Signature • Security goal: Unforgeability.

• From a signature on a subspace , anyone should not be able to forge a signature on vectors outside .

• Adversarial model: Chosen message attack

• Allow an adversary to obtain many signatures on subspace of its choice.

• We separate each subspace signing with unique identifier .

• The formal security definition for Unforgeability against Chosen message attack for NCS is captured on the next page.

16

V

V

id

Security Definition for Network Coding Signature • An NCS scheme is secure if any polynomial time adversary cannot win the following game:

17

Adversary Challenger

n

pk Keygen(n)! (sk, pk)

Sign(sk, idj , Vj)! �j

Vj choose unused idj�j{many

times

id⇤, ~y ⇤,�⇤

wins if for all j OR if for some j and id⇤ 6= idj id⇤ = idj ~y ⇤ 62 VjVerify(pk, id⇤, ~y ⇤,�⇤) = 1where


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Constructing Homomorphic NCS: Intuition1• Obtaining Homomorphic property: Use linear homomorphism in the exponent of

multiplicative groups

19

gv1 · gv2 = gv1+v2

• Obtaining Security (intuitively): Use the hardness of discrete log for groups.

Sign : (sk, v) 7! (gv )sk

- This is only intuition. Proving difficulty is much more involved.

easy

hard gv , (gv )sk?7! sk

To verify a signature with a public key , we check

Constructing Homomorphic NCS: Intuition 2

20

• Obtaining Verifiability: Use the pairing function of bilinear groups, where we have a useful “bilinearity” .

pk = e(g, gsk)

(e(g, gsk))v = e(g, (gv )sk)

e(ga, gb) = e(g, g)ab

e : G⇥G! G0

�?= (gv )sk

pkv?= e(g,�)

which will hold if is a valid signature since �

• Such bilinear groups exist: “pairing-friendly” elliptic curve groups (see e.g., [Cos])

Idea similar to [BLS04]

[BLS04] Dan Boneh; Ben Lynn & Hovav Shacham. "Short Signatures from the Weil Pairing". Asiacrypt 2001 and Journal of Cryptology 2004.[Cos] Craig Costello. “Pairing for Beginners". http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf

http://www.craigcostello.com.au/pairings/PairingsForBeginners.pdf

Boneh-Freeman-Katz-Waters Scheme (simplified)

21

Verify(pk, id, ~y ,�)

Keygen(n) sk = ↵ pk = (H, g, g↵)

Sign(sk, id,~v)

�0 = �a11 · · ·�annCombine(pk, id,�1, . . . ,�`, a1, . . . , a`)

� =�H(id, 1)v1 · · ·H(id, n)vn

�↵

e(g, �)?= e(g↵, H(id, 1)y1 · · ·H(id, n)yn)


[BGKW09]The first (homomorphic) NCS scheme


22

Correctness for Verify

Recall bilinearity e(ga, gb) = e(g, g)ab

e(g,�H(id, 1)y1 · · ·H(id, n)yn

�↵) = e(g↵, H(id, 1)y1 · · ·H(id, n)yn)



Sign(sk, id,~v)


� =�H(id, 1)v1 · · ·H(id, n)vn

�↵

e(g, �)?= e(g↵, H(id, 1)y1 · · ·H(id, n)yn)


[BGKW09]


23



Sign(sk, id,~v)


� =�H(id, 1)v1 · · ·H(id, n)vn

�↵

e(g, �)?= e(g↵, H(id, 1)y1 · · ·H(id, n)yn)

Correctness for Combine

which is a signature of

�0 =

Y

i=1

�H(id, 1)vi,1 · · ·H(id, n)vi,n

�↵ai

=�H(id, 1)

P`i=1 ai vi,1 · · ·H(id, n)

P`i=1 ai vi,n

�↵

a1~v1 + · · · a`~v`


[BGKW09]


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Sign(sk, id,~v)


� =�H(id, 1)v1 · · ·H(id, n)vn

�↵

e(g, �)?= e(g↵, H(id, 1)y1 · · ·H(id, n)yn)

Correctness for Combine

which is a signature of

�0 =

Y

i=1

�H(id, 1)vi,1 · · ·H(id, n)vi,n

�↵ai

=�H(id, 1)

P`i=1 ai vi,1 · · ·H(id, n)

P`i=1 ai vi,n

�↵

a1~v1 + · · · a`~v`


[BGKW09]

Goal: homomorphic iff same idIdea: hash id to the base

Intuition


25



Sign(sk, id,~v)


� =�H(id, 1)v1 · · ·H(id, n)vn

�↵

e(g, �)?= e(g↵, H(id, 1)y1 · · ·H(id, n)yn)


Security Theorem: It is secure under the co-Diffie-Hellman assumption in the random oracle model

Con: Random oracle (RO) is not instantiable in general [CGH98].

[CGH98] Ran Canetti, Oded Goldreich and Shai Halevi, The Random Oracle Methodology Revisited, STOC 1998

How to construct without RO??

[BGKW09]

H is modeled as random oracle.😣

Attrapadung-Libert Scheme (simplified)

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Keygen(n)

Sign(sk, id,~v)

Combine(pk, id,�1, . . . ,�`, a1, . . . , a`)

� = (�, �, �) =�gr , g↵F (id)r , (gv11 · · · gvnn )r

�

�0 := �1 = · · · = �n

�0 := �1 = · · · = �n

�0 = (�0, �0, �0) where

�0 := �a11 · · · �ann

where r = f (s, id)

sk = (↵, s)

e(g, g)↵ · e(�, F (id)) ?= e(g, �)

e(�, gv11 · · · gvnn )?= e(g, �)

and

pk = (F, g1, . . . , gn, e(g, g)↵)

[AL11] N. Attrapadung, B. Libert. Homomorphic Network Coding Signatures in the Standard Model. In PKC’11

[AL11]The first scheme without random oracle (=standard model)

Attrapadung-Libert Scheme (simplified)

27

Keygen(n)

Sign(sk, id,~v) � = (�, �, �) =�gr , g↵F (id)r , (gv11 · · · gvnn )r

�

where r = f (s, id)

sk = (↵, s) pk = (F, g1, . . . , gn, e(g, g)↵)

[AL11] N. Attrapadung, B. Libert. Homomorphic Network Coding Signatures in the Standard Model. In PKC’11 [W09] B. Waters. Dual System Encryption: Realizing Fully Secure IBE and HIBE under Simple Assumptions. In Crypto’09

[AL11]The first scheme without random oracle (=standard model)

Goal: homomorphic iff same idIdea: hash id to the exponent

• We show that f being pseudo-random function suffices.

• F is also a standard affine function.

• No random oracle.

• The full construction and the security proof are more involved.

• Using dual system encryption techniques [W09].

😀

Many works for Homomorphic NCS

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• Homomorphic Signature introduced by Johnson et al. (CT-RSA02)

• Homomorphic Network Coding Signature

• Boneh-Freeman-Katz-Waters (PKC09): Pairing-based

• Network Coding Sig (not homomorphic): std model

• Homomorphic NCS: RO model

• Gennaro-Katz-Krawczyk-Rabin: RSA-based (PKC10)

• Others:

• Multi-sources NCS: Agrawal-Boneh-Boyen-Freeman (PKC10)

• Homomorphic MACs: Agrawal-Boneh (ACNS09)

• Attrapadung-Libert (PKC11): First Homomorphic NCS in std model (pairing-based)

• Also, Catalano-Fiore-Warinschi (Eurocrypt11, PKC12): RSA-based

• Freeman (PKC12): framework generalized from AL11

Homomorphic Signatures with Applications Beyond Network Coding

• Further direction 1: More expressive functionality than linear homomorphicity.

• Further direction 2: Adding Security in the sense of privacy.

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Direction 1: More Expressive Functionality

30

v1

v2

v`

f (v1, v2, . . . , v`)

…

• Homomorphic NCS: F is only linear.

• We want more expressive classes of F.

f 2 F

“Computing on Authenticated Data” [BF11, ABC+12, ALP12, …]- Analogous to “computing on encrypted data” (ala. Fully homomorphic encryption [G09])

[BF11] D. Boneh, D. Freeman. Homomorphic Signatures for Polynomial Functions. In Eurocrypt’11 [ABC+12] J.-H. Ahn, D. Boneh, J. Camenisch, S. Hohenberger, a. shelat, B. Waters. Computing on Authenticated Data. In TCC 2012 [ALP12] N. Attrapadung, B. Libert, T. Peters. Computing on Authenticated Data: New Privacy Definitions and Constructions. In Asiacrypt’12 [G09] C. Gentry. Fully Homomorphic Encryption. In STOC 2009

Works on Improving Expressiveness

• Boneh-Freeman (Eurocrypt11): Low-deg polynomial functions

• Improved by Catalano-Fiore-Warinski (Crypto14)

• Ahn et al. (TCC12): Substring predicates

• Improved by Attrapadung-Libert-Peters (Asiacrypt12)

• Gorbunov-Vaikuntanathan-Wichs (STOC15): any circuits

• State-of-the-art! (as far as I know)

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[BF11] D. Boneh, D. Freeman. Homomorphic Signatures for Polynomial Functions. In Eurocrypt’11 [ABC+12] J.-H. Ahn, D. Boneh, J. Camenisch, S. Hohenberger, a. shelat, B. Waters. Computing on Authenticated Data. In TCC 2012 [ALP12] N. Attrapadung, B. Libert, T. Peters. Computing on Authenticated Data: New Privacy Definitions and Constructions. In Asiacrypt’12 [CFW14] D. Catalano, D. Fiore, B. Warinschi. Homomorphic signatures with efficient verification for polynomial functions. In Crypto’14 [GVW15] Sergey Gorbunov, Vinod Vaikuntanathan, and Daniel Wichs. Leveled fully homomorphic signatures from standard lattices. In STOC’15

• Privacy of the inputs when seeing an output signature of computed data. [ABC+12,ALP12,…]

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(1,0,1,0)

(0,1,0,1)

(2,-1,1,0)

(-1,2,0,1)

(1,1,1,1)

(1,1,1,1)

+

+} Adversary does not

know which is the case

?

Direction 2: Adding Privacy (Context-Hiding)

[ABC+12] J.-H. Ahn, D. Boneh, J. Camenisch, S. Hohenberger, a. shelat, B. Waters. Computing on Authenticated Data. In TCC 2012 [ALP12] N. Attrapadung, B. Libert, T. Peters. Computing on Authenticated Data: New Privacy Definitions and Constructions. In Asiacrypt’12

Useful for privacy-preserving data-mining

• Many levels of Context-hiding: Weak / Strong / Adaptive/ Complete

Comparisons for Linear Homomorphic Schemes

33

Scheme Contribution Unforgeability Privacy

Boneh-Freeman-Katz-Waters PKC09First

(Pairing-based)RO Strong

Gennaro-Katz-Krawczyk-Rabin PKC10 First RSA-based RO Weak?

Attrapadung-Libert PKC11First Standard (Pairing-based)

Standard Strong

Catalano-Fiore-Warinschi EC11, PKC12 First Standard RSA-based Standard Weak?

Boneh-Freeman PKC11, EC11 First lattice-based RO Weak

Freeman PKC12 Framework Standard Weak

Attrapadung-Libert-Peters AC12 Stronger privacy Standard Strong&Adaptive

Attrapadung-Libert-Peters PKC13 Strongest privacy Standard Complete…

[ALP13] N. Attrapadung, B. Libert, T. Peters. Efficient completely context-hiding quotable and linearly homomorphic signatures. In PKC’13

Concluding Remarks• Today I explained a cryptographic approach for obtaining authentication mechanisms in

network coding: Homomorphic network coding signatures.

• (Small) Open problem: Homomorphic NCS with constant public key size in the standard model

• This would yield a scheme that can deal with unbounded length vectors.

• Note: [BGKW09] has constant-size pk but is not a standard-model scheme, while [AL11] is a standard-model scheme but has non-constant-size pk.

• Remark: I am not sure if homomorphic signatures with more expressive homomorphicity and/or privacy is useful for network coding or not. Perhaps worth exploring?

• Disclaimer: I did not follow more recent schemes (if any). Sorry if I missed important schemes here.

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Thank you !!

homomorphic signatures and applications to network codingsita/forum/article/2020/... · 2019. 9....

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